Mohammad H Afzali
email: mohammad.afzali@univ-tlse2.fr

Philippe Birmes
email: birmes.p@chu-toulouse.fr

Stéphane Vautier
email: vautier@univ-tlse2.fr

Symptoms Moderating the Association between Recent Suicide Attempt and Traumatic Morbidity: Fan-shaped Effects

Keywords: Fan-shaped effect, Traumatic morbidity, Suicide attempt, Suicidal Ideation, Depression

The present study examines variables affecting the incidence of recent suicide attempt in a large sample of participants who experienced various levels of traumatic morbidity. The sample was drawn from a French nationally representative, general population survey (N= 39,617). In the same line with previous research on the data provided by the survey, five levels of traumatic morbidity were used on the basis of answers to the post-traumatic stress disorder items from the French version of Mini International Neuropsychiatric Interview. Twenty four items, representing Axis I psychopathological symptoms and suicidality indexes, were tested to identify intervening variables moderating the suicide attempt incidence in any given traumatic morbidity level. Seven items yielded a systematic effect: four suicidality items, two major depression items and one panic disorder item. The main findings consist of (i) the suppressing effect of the absence of these symptoms, and (ii) the gradual increasing effect of their presence. These results underline the importance of comorbid symptoms in the association between trauma and suicidality.

Introduction

Suicidal behavior is among the main public health issues. According to the World Health Organization (WHO) each year approximately one million people die by suicide worldwide and 10 to 20 times more people will attempt suicide worldwide [START_REF] Nock | Cross-national analysis of the associations among mental disorders and suicidal behavior: findings from the WHO World Mental Health Surveys[END_REF]. Suicide is now among the three leading causes of death among people aged from15 to 34 years and the 14thoverall cause of death worldwide accounting for 1.5 percent of all deaths [START_REF] Gvion | Suicide and suicidal behavior[END_REF][START_REF] Nock | Suicide and suicidal behavior[END_REF]. Studies based on large community samples highlight that suicidal behavior is associated with a number of sociodemographic characteristics, such as younger age, being unemployed, and living alone, as well as a number of psychiatric diagnoses such as mood disorders, psychotic disorders, substance use disorders and anxiety disorders [START_REF] Borges | A risk index for 12-month suicide attempts in the National Comorbidity Survey Replication (NCS-R)[END_REF][START_REF] Kessler | Prevalence of and risk factors for lifetime suicide attempts in the National Comorbidity Survey[END_REF][START_REF] Nock | Cross-national analysis of the associations among mental disorders and suicidal behavior: findings from the WHO World Mental Health Surveys[END_REF]Nock, Hwang, Sampson, & Kessler, 2010).

The experience of traumatic events and psychopathological symptoms related to it are also found to be associated with suicidal behavior, while the research in the domain is no older than two decades [START_REF] Adams | Prolonged trauma and subsequent suicidal behavior: child abuse and combat trauma reviewed[END_REF]. For instance in a sample of 9,358 participants evaluated in the context of National Anxiety Disorders survey, the presence of suicidal ideation was found to be associated with the number of PTSD symptoms. According to Marshall's study (2001) the frequency of reported past month suicidal ideation is four times higher in a sample of trauma victims with four PTSD symptoms compared to a sample with no reported traumatic symptoms. In the same line [START_REF] Vaiva | Psychotraumatismes et risque suicidaire en France[END_REF] conducted a study on the French general population sample (SMPG) and showed a gradual increase in the frequency of the current month suicide attempt along with increase of the levels of traumatic morbidity. Vaiva et al.'s study used five levels of traumatic morbidity: the levels are respectively defined by (i) not having experienced a traumatic event (henceforth the first level is going to be named No trauma), having experienced (ii) a traumatic event (criterion A) without any further PTSD symptoms (henceforth this level is going to be named Trauma a ), (iii) a traumatic event (criterion A) and an intrusion symptom (criterion B) without any further PTSD symptoms (Trauma b ), (iv) a traumatic event (criterion A), an intrusion symptom (criterion B), and an avoidance symptom (criterion C) without any further PTSD symptoms (Trauma c ), and (v) a traumatic event (criterion A), an intrusion symptom (criterion B), an avoidance symptom (criterion C) along with a criterion D symptom of negative alterations in cognitions and mood (Trauma d ). Among participants without the experience of traumatic events (No trauma) only 0.36% reported a current month suicide attempt. In the sample of participants who have experienced trauma without any further PTSD symptom (Trauma a ), 0.57% reported a current month suicide attempt. The proportion is 0.72% in the case of participants characterized by Trauma b level of traumatic morbidity, 2.42% for participants characterized by Trauma c level of traumatic morbidity, and 4.6 % for participants characterized by Trauma d level of traumatic morbidity. Figure 1 displays this increasing association.

The association between experience of trauma and suicidal behavior has been the subject of several critical notes in some recent articles. In a recent review on the subject [START_REF] Krysinska | Post-traumatic stress disorder and suicide risk: a systematic review[END_REF] highlighted that studies published in the domain usually do not take into account the role of potential intervening variables in the relationship between trauma and suicidality.

That is to say, the association between experience of traumatic events and suicidality may change conditionally to presence versus absence of other symptoms. [START_REF] Jankovic | Trauma and suicidality in war affected communities[END_REF] and [START_REF] Oquendo | Association of comorbid posttraumatic stress disorder and major depression with greater risk for suicidal behavior[END_REF][START_REF] Oquendo | Posttraumatic stress disorder comorbid with major depression: factors mediating the association with suicidal behavior[END_REF] addressed the issue of variables intervening potentially on suicidality outcomes such as suicidal ideation or suicide attempt, using logistic regression. All these studies treated traumatic morbidity as a dichotomous independent variable (diagnosed with PTSD vs. not diagnosed with PTSD) whereas role of depression was specifically investigated by [START_REF] Jankovic | Trauma and suicidality in war affected communities[END_REF] and [START_REF] Oquendo | Association of comorbid posttraumatic stress disorder and major depression with greater risk for suicidal behavior[END_REF], while the specific role of borderline personality disorder was examined by [START_REF] Oquendo | Posttraumatic stress disorder comorbid with major depression: factors mediating the association with suicidal behavior[END_REF]. The logistic regression approach used in these studies allows one to test statistically the necessity of certain independent variables in the model. However, these studies assumed no interaction between the independent variables. As mentioned above, [START_REF] Vaiva | Psychotraumatismes et risque suicidaire en France[END_REF] SMPG study highlights that the incidence of the current month suicide attempt is associated with the level of traumatic morbidity (see Figure 1), where traumatic morbidity is treated as a 5-level ordinal variable. The present study aims at supplementing Vaiva et al.'s results by finding out intervening variables which affect the association between traumatic morbidity and the frequency of recent suicidal behavior in every level of traumatic morbidity.

In the first place, we are going to define intervening variable, enhancing intervening variable and fan-shaped intervening variable. Let us consider an outcome (e.g. suicidal behavior, dependent variable), which occurs with a given frequency in a sample defined by a common characteristic in its members (e.g., experience of a traumatic event, independent variable). If the outcome frequency changes when the sample is split conditionally to different modalities of another variable (e.g. depression), it is called an intervening variable. That is to say, the frequency of the outcome (suicidal behavior) is not the same with respect to different modalities of the intervening variable (experience of a traumatic event along with depression versus experience of a traumatic in the absence of depression).

If the outcome frequency is not only different but also systematically increased, the intervening variable is called enhancing. The magnitude of the increasing effect can be evaluated by an effect size measure such as Cohen's h index (1986, p. 180). If the effect size increases along with the levels of traumatic morbidity, we speak of a fan-shaped, intervening variable. Figure 2 displays an example of a fan-shaped enhancing variable where the effect size is 0.20 for the No trauma level, 0.20 for the first traumatic morbidity level, 0.22 for the second traumatic morbidity level, 0.27 for the third traumatic morbidity level, and 0.35 for the fourth traumatic morbidity level.

The main objective of the present study is to find out if there are some fan-shaped intervening variables that affect the association between the frequency of recent suicidal behavior and the level of traumatic morbidity on the large sample of French general population and mental health (SMPG) survey. Afterwards, if there are some intervening variables with fanshape characteristics, we are going to examine the frequency of suicidal behavior for each level of traumatic morbidity in the conjoint presence and in the conjoint absence of the corresponding symptoms.

Materials and methods

Materials

The French national SMPG survey used two main interviews. (i) A socio-demographic interview regarding gender, age, marital status, education, professional activity, income level, religious beliefs and practice, immigration and country of origin. (ii) The 166-item version of the French Mini International Neuropsychiatric Interview (MINI) aimed to assess the prevalence of psychiatric disorders as well as suicidality. The present study focuses on the information provided by the MINI. The MINI was designed as a brief structured interview for 12 Axis I psychiatric disorders (161 items) with the priority given to current diagnosis; moreover, five items evaluate suicidality. All items are answered in a 'yes' or 'no' format. Each psychiatric disorder is evaluated with some 'screening items' to explore the mandatory criteria. Negative answers to these items necessarily rule out the diagnosis (so complete data is only available for mandatory 'screening items'). [START_REF] Lecrubier | The Mini International Neuropsychiatric Interview (MINI). A short diagnostic structured interview: reliability and validity according to the CIDI[END_REF] intervening variables current study used the information provided by the Axis I mandatory screening items of MINI, along with five suicidality items, these items are detailed in Table 1.

Participants

The SMPG survey consisted in interviewing a nationally representative sample of 39'617 respondents in 47 sites in France. In each site, approximately 900 participants were evaluated by nursing students using interviews mentioned above. The survey data is composed of 54% women and 46% men, aged 18 and older (M = 45, SD = 18.3). According to Vaiva et al.'s (2007) increasing levels of traumatic morbidity, the size of the five sub-samples, i.e., No trauma, Trauma a , …, Trauma d , is 27'896 (70%), 8211 (21%), 1668 (4.2%), 1406 (3.5%), and 436

(1.1%), respectively.

Analyses

The reported current month suicide attempt (positive answer to item Suicidality 4 , see Table 1) was chosen as the outcome criterion. Information conveyed by 20 screening items of MINI along with the other four suicidality items (a total of 24 items) was used to identify potential fan-shaped intervening variables. In order to do that, the frequency of the current month suicide attempt was computed for each level of traumatic morbidity conditionally to the positive/negative answer to each one of 24 potential intervening variables. The difference between the resulting frequencies was expressed by using Cohen's h index of effect size (1986).

Effect sizes below 0.20, between 0.20 and 0.80 and over 0.80 are considered as small, moderate and large, respectively. In this study, fan-shaped variables with minimum h index larger than 0.20 are of particular interest.

Results

Frequency of Suicide Attempt in Each Level of Traumatic Morbidity

A total of 213 respondents reported a current month suicide attempt (0.68% of the total sample, which corresponds to 8.2% of annual suicide attempt rate). Respondents' age ranged between 18 and 68 years (M = 37, SD = 16.2). Fifty four percent of the attempters were female.

Among participants having reported a current month suicide attempt, 113 participants were in the No trauma sample, 47 were in the Trauma a sample, 20 participants were in the Trauma b sample, 34 participants were in the Trauma c sample, and 20 participants were in the Trauma d sample.

Fan-shaped Intervening Variables

Table 2 shows that among the 24 examined variables, seven variables exhibited characteristics of an intervening variable with fan-shaped pattern, with at least a moderate effect size for every level of traumatic morbidity. The corresponding items are listed on the top of Table 2. These items are four suicidality items, namely death wish (Suicidality 1 ), self harm intention (Suicidality 2 ), suicidal ideation (Suicidality 3 ), and lifetime suicide attempt (Suicidality 5 ), two major depression items, i.e. depressed mood (Depression 1 ) and loss of interest (Depression 2 ), along with one item concerning the experience of panic attack in past month (Panic Attack). Figure 2 displays the percentage of current month suicide attempt for five levels of traumatic morbidity, in the presence of the loss of interest symptom (Depression 2 ), in absence of the depression 2 symptom, and for total sample. In the same way the fan-shaped effect of the six other intervening variables is displayed in Figure 3 and Figure 4.

Suicide Attempt Incidence in the Conjoint Presence/Absence of Intervening Symptoms

In case of conjoint presence of seven aforementioned intervening symptoms the percentage of suicidal behavior goes from 42%, in the No trauma sample, to 89%, in the Trauma d sample. While in the conjoint absence of all of seven intervening symptoms the percentage of suicidal behavior remains less than 0.01% for all of different levels of traumatic morbidity, which demonstrates that conjoint absence of these symptoms suppress the association between levels of traumatic morbidity and recent suicide attempt.

Discussion

The current study employed a simple and straightforward descriptive approach to the SMPG data, without using statistical modeling, to detect the intervening variables that affect the association between traumatic morbidity and the frequency of suicide attempt in a fan-shaped way. Seven fan-shaped variables emerged from the data analysis; namely death wish, self harm intention, suicidal ideation, lifetime suicide attempt, experience of panic attack, depressed mood, and loss of interest. Conjoint absence or presence of these symptoms substantially changes the frequency of suicide attempt. In the absence of them all, the percentage of suicide attempt is less than 0.01% for every traumatic morbidity level. In the presence of all seven symptoms the percentage of suicidal behaviour goes from 42% to 89% depending on various levels of traumatic morbidity.

These results are consistent with findings of previous studies, concerning the intervening role of depression in the trauma-suicidal behavior association [START_REF] Jankovic | Trauma and suicidality in war affected communities[END_REF][START_REF] Oquendo | Association of comorbid posttraumatic stress disorder and major depression with greater risk for suicidal behavior[END_REF][START_REF] Oquendo | Posttraumatic stress disorder comorbid with major depression: factors mediating the association with suicidal behavior[END_REF]. However, literature addresses the issue of variables intervening in the traumatic morbidity-suicidal behavior association on the basis of logistic regression. Some methodological differences between these studies and the current study have to be underlined. Firstly, in the literature traumatic morbidity is treated as a binary variable (presence vs. absence of PTSD) whereas in the case of SMPG data [START_REF] Vaiva | Psychotraumatismes et risque suicidaire en France[END_REF] The logistic regression approach has the advantage of estimating the relation between trauma and suicidality after taking into account the effect of intervening variables; moreover it is capable of providing the possibility to reproduce frequencies of the outcome given different combinations of the independent variables. However, the validity of such a model can be questioned because the interaction terms are systemically set at zero. This is probably due to the large number of interaction terms that one would have to insert into the model. For instance, in their analyses, [START_REF] Jankovic | Trauma and suicidality in war affected communities[END_REF] set 502 interaction terms at zero; [START_REF] Oquendo | Association of comorbid posttraumatic stress disorder and major depression with greater risk for suicidal behavior[END_REF][START_REF] Oquendo | Posttraumatic stress disorder comorbid with major depression: factors mediating the association with suicidal behavior[END_REF] set 26 interaction terms at zero. The descriptive approach used here encounters some practical obstacles in order to find out the frequency of the outcome with respect to different combinations of intervening variables. For instance, in our study 640 samples have to be taken into account (the seven binary intervening variables and the five point-scaled traumatic morbidity variable require the calculation of 2 7 × 5 = 640 combinations). The data reveals that 209 of these different combinations are not instantiated in the overall sample (amongst the 39,617 cases of the SMPG data, no relevant case was observed). Nevertheless, a statistical model could allow one to estimate the frequencies of interest. A logistic regression model (with 12 parameters, i.e., one intercept, four beta coefficients for the dummy variables used to describe traumatic morbidity and seven beta coefficients for the intervening variables) yields the 12 estimates reported in the Table 3.

All of the four traumatic morbidity levels have non-significant association with suicidal behaviour. However, these variables cannot be removed from the model because the nested model which sets the corresponding parameters at zero (see Table 4) has a significantly worse fit: Δ-2LL (4) = 15.28 , p = 0.004. As mentioned above, considering the fact that in such a model all of the 628 interaction terms (640 combinations minus 12 parameters) have to be set at zero the estimates are doubtful. Table 5 shows the discrepancies between the estimated percentage and the observed percentage of recent suicide attempt in the presence and the absence of each intervening symptom. In addition, the three bottom lines compare the estimated percentage with the observed percentage in the conjoint presence and in the conjoint absence of the intervening symptoms, and the number of suicide attempts. The model over-estimates the number of recent suicide attempts: 260 predicted vs. 213 observed. Figure 5 illustrates that the model reproduces the fan-shaped effect with a systematic discrepancy between the observed and the predicted suicide frequencies.

Finally, coming back to the association between traumatic morbidity and suicidal behavior, the SMPG data show that even in the case of highest traumatic morbidity level in the presence of each intervening symptom, the frequency of suicidal behaviour remains less than 50%. Thus it suggests that future research could investigate the variables, or reasons for living, which may protect people with traumatic morbidity from suicide. In the same vein the absence of intervening variables can be considered as a protective factor that decreases the frequency of the suicide attempt: as mentioned at the beginning of the Discussion, in conjoint absence the seven intervening variables, the percentage of suicide attempt is less than 0.01% for every traumatic morbidity level.

  and Sheehan et al. (1998) reported a high inter-rater agreement between the diagnosis proposed by MINI and the clinical diagnosis based on the Forth version of Diagnostic and Statistical Manual of Mental Disorders (DSM-IV) or the Tenth version of International Statistical Classification of Diseases (ICD-10). In order to identify

  traumatic morbidity is a five-level ordinal scale. This is much like the case of Marshal et al.'s (2001) study. Neither Marshal et al. nor Vaiva et al. looked for fan-shaped intervening variables.

Figure 1 .

 1 Figure 1. The percentages of current month suicide attempt in five sub-samples with different number of traumatic symptoms.

Figure 2 .

 2 Figure 2. The percentages of current month suicide attempt for five levels of traumatic morbidity, in presence of a depression symptom, in absence of a depression symptom, and for total sample, an example of fan-shaped intervening variable.

Figure 3 .

 3 Figure 3. The percentages of current month suicide attempt for five levels of traumatic morbidity in presence of three intervening symptoms (Suicidality 1 at the top, Suicidality 2 in the middle, and Suicidality 3 on the bottom) in their absence, and for total sample.

Figure 4 .

 4 Figure 4. The percentages of current month suicide attempt for five levels of traumatic morbidity in presence of three intervening symptoms (Suicidality 4 at the top, Depression 1 in the middle, and panic attack on the bottom) in their absence, and for total sample.

Figure 5 .

 5 Figure 5. The percentages of current month suicide attempt for five levels of traumatic morbidity in presence of a depression symptom, and in absence of a depression symptom, frequencies predicted by model versus frequencies in the actual data.

Table 2 h

 2 index of difference and current month suicide attempt percentage given positive or negative answer to potential intervening variables (in parenthesis).

	Item		h (%suicide attempt item=no, %suicide attempt in item=yes)			h index
		No trauma	Traumaa	Traumab	Traumac	Traumad	Min	Max	Average
	Suicidality2	0.65 (0.2, 11.37)	0.69 (0.1, 13.3)	0. 78 (0.3, 20.7)	0.83 (1.6, 24.3)	1.06 (1.2, 36.3)	0.6	1.1	0.83
	Suicidality3	0.45 (0.08, 6.99)	0.54(0.1, 8.8)	0.67 (0.2, 14.1)	0.86 (0.7, 24.5)	1.25 (0, 3.5)	0.4	1.3	0.80
	Suicidality5	0.38 (0.06, 3.17)	0.40 (0, 4)	0.44 (0.1, 6.1)	0.60 (0.5, 13.1)	0.81 (0.3 ,20.2)	0.4	0.8	0.55
	Suicidality1	0.30 (0.08, 4.69)	0.37 (0.1, 6)	0.45 (0.2, 7.5)	0.50 (1.3, 12.6)	0.94 (0.3, 25.7)	0.3	1.0	0.54
	Depression1	0.21 (0.14, 1.56)	0.23 (0.1, 2.1)	0.26 (0.3, 3.3)	0.37 (0.9, 7.6)	0.47 (1.8, 13.2)	0.2	0.5	0.31
	Panic Attack	0.20 (0.15, 1.7)	0.20 (0.1, 2)	0.21 (0.4, 2.3)	0.29 (1.3, 6.6)	0.52 (1.5, 13.7)	0.2	0.5	0.27
	Depression2	0.2 0(0.06, 1.46)	0.20 (0.1, 1.5)	0.22 (0.2, 2.4)	0.27(1, 5.6)	0.35 (2, 10)	0.2	0.4	0.24
	Psychotic Disorder 2	0.08 (0.35, 1.2)	0.1 (0.4, 1.3)	0.23 (0.1, 1.9)	0.1 (2, 3.7)	0.41 (1.9, 11.4)	0.1	0.4	0.20
	Psychotic Disorder 7	0.11 (0.44, 1.85)	0.14 (0.5, 2)	0.07 (0.4, 1)	0.17 (2.1, 5.2)	0.36 (3.4, 12.7)	0.1	0.4	0.18
	Maniac Episode 2	0.06 (0.35, 1.02)	0.08 (0.4, 1.1)	0.15 (0.2, 1.3)	0.18 (1.6, 4.6)	0.34 (2.5, 10.3)	0.1	0.3	0.17
	Psychotic Disorder 6	0.16 (0.35, 2.5)	0.21 (0.4, 2.7)	0.14 (0.3, 1.5)	0.12 (2.1, 4.1)	0.24 (3.7, 9.5)	0.1	0.2	0.17
	Depression3	0.11 (0.27, 1.48)	0.14 (0.3, 1.6)	0.04 (0.3, 0.6)	0.21 (1.2, 4.4)	0.3 (2.2, 8.7)	0.0	0.3	0.16
	Drug Abuse	0.08 (0.35, 1.3)	0.11 (0.4, 1.4)	0.11 (0.3, 1.2)	0.15 (2, 4.7)	0.34 (3.0, 11.5)	0.1	0.3	0.16
	Psychotic Disorder 1	0.05 (0.44, 0.93)	0.06 (0.5, 1)	0.14 (0.2, 1.3)	0.09 (2.1, 3.6)	0.39 (2.4, 11.6)	0.1	0.4	0.16
	Psychotic Disorder 3	0.05 (0.44, 1.02)	0.06 (0.5, 1.1)	0.13 (0.3, 1.3)	0.25 (1.7, 6.3)	0.21 (3.8, 8.8)	0.1	0.3	0.15
	Maniac Episode 1	0.12 (0.27, 1.67)	0.16 (0.3, 1.8)	0.12 (0.2, 1.1)	0.17 (1.6, 4.5)	0.17 (3.5, 7.3)	0.1	0.2	0.14
	Psychotic Disorder 4	0.08 (0.44, 1.3)	0.1 (0.5, 1.4)	0.03 (0.4, 0.6)	0.03 (2.4, 2.9)	0.47 (3.1, 1)	0.0	0.5	0.14
	Dysthymia	0.11 (0.09, 1.2)	0.15 (0.1, 1.3)	0.17 (0, 0.7)	0.09 (1.1, 2.1)	0.12 (2, 4.1)	0.1	0.2	0.13
	Agoraphobia	0.05 (0.44, 0.93)	0.07 (0.5, 1)	0.1 (0.2, 0.9)	0.21 (1.2, 4.5)	0.13(3.6, 6.3)	0.1	0.2	0.11
	Psychotic Disorder 5	0.08 (0.44, 1.39)	0.1 (0.5, 1.5)	0.1 (0.4, 1.2)	0.1 (2.3, 4)	0.12 (4.4, 7.1)	0.1	0.1	0.10
	Alcohol Abuse	0.02 (0.44, 0.65)	0.02 (0.5, 0.7)	0.2 (0, 1)	0.08 (2, 3.2)	0.09 (3.8, 5.6)	0.0	0.2	0.08
	Generalized Anxiety	0.08 (0.18, 0.93)	0.11 (0.2, 1)	0.05 (0.1, 0.4)	0.03 (0.8, 1.1)	0.09 (1.5, 2.9)	0.0	0.1	0.07
	Social Phobia	0.03 (0.44, 0.74)	0.04 (0.5, 0.8)	0.07 (0.3, 0.7)	0.11 (1.7, 3.5)	0.12 (3.6, 6.2)	0.0	0.1	0.07

Table 5

 5 Estimated vs. observed (in parenthesis) percentages of current month suicide attempt given presence vs. absence of intervening symptoms, frequencies predicted by model versus frequencies in the actual data

		No trauma	Traumaa	Traumab	Traumac	Traumad
	Item	Item=no	Item=yes	Item=no	Item=yes	Item=no	Item=yes	Item=no	Item=yes	Item=no	Item=yes
	Suicidality2	0.1 (0.2)	8.7 (11.4)	0.1 (0.1)	7.8 (13.3)	0.1 (0.3)	13.6 (20.7)	0.3 (1.6)	16.1 (24.3)	0.4 (1.3)	23.2 (36.4)
	Suicidality3	0.3 (0.1)	22.1 (7)	0.4 (0.1)	25 (8.8)	0.7 (0.2)	36.9 (14.1)	1.3 (0.7)	36.7 (24.5)	2 (0)	39.8 (34.5)
	Suicidality5	0.1 (0.1)	15.5 (3.2)	0.2 (0)	20 (4)	0.4 (0.1)	30.7 (6.1)	0.6 (0.5)	31.3 (13.1)	0.5 (0.4)	37.4 (20.2)
	Suicidality1	0.2 (0.1)	8 (4.7)	0.3 (0.1)	10 (6.1)	0.5 (0.2)	14.9 (7.5)	1 (1.3)	18.5 (12.6)	0.9 (0.3)	28 (25.7)
	Depression1	0.2 (0.1)	2.9 (1.6)	0.3 (0.1)	4.1 (2.1)	0.6 (0.3)	5.8 (3.3)	1.2 (0.9)	9.7 (7.6)	2.3 (1.8)	15.1 (13.2)
	Panic Attack	0.4 (0.1)	4.3 (1.7)	0.6 (0.1)	5.3 (2)	1.2 (0.4)	7.7 (2.3)	2.2 (1.3)	11.8 (6.6)	3.3 (1.5)	19.2 (13.8)
	Depression2	0.2 (0.1)	4.4 (1.5)	0.4 (0.1)	5.8 (1.5)	0.8 (0.2)	8.8 (2.4)	1.1 (1)	13.9 (5.6)	2.9 (2)	17.5 (10)
	Conjoint Absence	0.03 (0)	0.04 (0.02)		0.05 (0)	0.05 (0)		0.05 (0)
	Conjoint Presence	69.1 (41.3)	71.3 (43.2)	76.2 (60)	77.3 (56.4)	77.1 (88.9)
	Total suicide attempt	132 (105)	58 (47)		17 (7)		30 (34)		23 (20)

Have you been consistently depressed or down, most of the day, nearly every day, for the past two weeks? Depression 2

In the past two weeks, have you been much less interested in most things or much less able to enjoy the things you used to enjoy most of the time? Depression 3

Did you feel tired or without energy almost every day? Dysthimia

Have you felt sad, low or depressed most of the time for the last two years? Maniac Episode 1

Have you ever had a period of time when you were feeling 'up' or 'high' or 'hyper' or so full of energy or full of yourself that you got into trouble? Maniac Episode 2

Have you ever been persistently irritable, for several days, so that you had arguments or verbal or physical fights, or shouted at people outside your family? Agoraphobia Do you feel anxious or uneasy in places or situations where you might have a panic attack or the panic-like symptoms we just spoke about, or where help might not be available or escape might be difficult: like being in a crowd, standing in a line (queue)? Panic Attack

Have you, on more than one occasion, had spells or attacks when you suddenly felt anxious, frightened, uncomfortable or uneasy? Social Phobia

In the past month, were you fearful or embarrassed being watched, being the focus of attention, or fearful of being humiliated? This includes things like speaking in public, eating in public or with others, writing while someone watches, or being in social situations. Generalized Anxiety Have you worried excessively or been anxious about several things over the past 6 months? Alcohol Abuse

In the past 12 months, have you had 3 or more alcoholic drinks within a 3 hour period on 3 or more occasions? Drug Abuse

In the past 12 months, did you take any drugs more than once, to get high, to feel better, or to change your mood? Psychotic Disorder 1

Have you ever had thoughts that your entourage considered strange or unusual, and they did not share with you? Psychotic Disorder 2

Have you ever believed that people were spying on you, or that someone was plotting against you, or trying to hurt you? Psychotic Disorder 3

Have you ever believed that someone was reading your mind or could hear your thoughts, or that you could actually read someone's mind or hear what another person was thinking? Psychotic Disorder 4

Have you ever believed that someone or some force outside of yourself put thoughts in your mind that were not your own, or made you act in a way that was not your usual self? Have you ever felt that you were possessed? Psychotic Disorder 5

Have you ever believed that you were being sent special messages through the TV, radio, or newspaper? Psychotic Disorder 6

Have you ever had visions when you were awake or have you ever seen things other people could not see? Psychotic Disorder 7 Suicidality 1

Have you ever heard things other people could not hear, such as voices? Did you think that would be better off dead or wish you were dead? Suicidality 2 Suicidality 3 Suicidality 4 Suicidality 5 PTSD 1 PTSD 2 PTSD 3 PTSD 4 Did you wish to harm or hurt or injure yourself? (Current month) Did you think about suicide? (Current month) Did you attempt suicide? (Current month) Had you ever make a suicide attempt in your life? Have you ever experienced or witnessed or had to deal with an extremely traumatic event? During the past month, have you re-experienced, thought, or dreamt of the event in a distressing way? Have you avoided thinking about or talking about the event?

Have you had trouble recalling some important part of what happened?


Pascal Michel
email: michel@math.univ-paris-diderot.fr

Simulation of the Collatz

Keywords: Collatz function, 3x + 1 function, Turing machines Mathematics Subject Classification (2010): Primary 03D10, Secondary 68Q05, 11B83

come

Introduction

Turing machines can be classified according to their numbers of states and symbols. It is known (see [START_REF] Woods | The complexity of small universal Turing machines: a survey[END_REF] for a survey) that there are universal Turing machines in the following sets (number of states × number of symbols): 2 × 18, 3 × 9, 4 × 6, 5 × 5, 6 × 4, 9 × 3, 18 × 2.

On the other hand, all the Turing machines in the following sets are decidable:

1 × n, 2 × 3, 3 × 2, n × 1.
In order to refine the classification of Turing machines between universal and decidable classes, properties in connection with the 3x + 1 function have been considered. The 3x+1 function is also called the Collatz function, and Collatz-like functions are functions on integers with a definition of the following form: there exist integers d ≥ 2, a i , b i , 0 ≤ i ≤ d -1, such that, for all integers x,

f (x) = a i x + b i d if x ≡ i (mod d).
With these definitions, we can state the following properties of Turing machines, that have been used to refine the classification according to the numbers of states and symbols (see [START_REF] Michel | Generalized 3x+1 functions and the theory of computation[END_REF] for a survey).

• Turing machines that simulate the iteration of the 3x + 1 function and never halt. It is known that there are such machines in the sets

2 × 8, 3 × 5, 4 × 4, 5 × 3, 10 × 2.
We improve these results by giving a 3 × 4 Turing machine.

• Turing machines that simulate the iteration of the 3x + 1 function and halt when the loop 2 → 1 → 2 is reached. It is known that there is such a machine in the set 6 × 3.

In this article, we give four new Turing machines, in the classes 3 × 10, 4 × 6, 5 × 4 and 13 × 2.

• Turing machine that simulate the iteration of a Collatz-like function. It is known that there are such machines in the sets 2 × 4, 3 × 3, 5 × 2.

Preliminaries: Turing machines

The Turing machines we use have

• one tape, infinite on both sides, made of cells containing symbols,

• one reading and writing head, [START_REF] Margenstern | Frontier between decidability and undecidability: a survey[END_REF][START_REF] Margenstern | Frontier between decidability and undecidability: a survey[END_REF], Ba = Baiocchi [START_REF] Baiocchi | 3N+1, UTM e Tag-systems[END_REF], M i 1 = Michel [START_REF] Michel | Busy beaver competition and Collatz-like problems[END_REF], M i 2 = Michel (this paper). In roman boldface, halting machines.

• a set Q = {A, B, . . .}
• a set Σ = {b, 0, 1, . . .} of symbols, where b is the blank symbol (or Σ = {0, 1}, when 0 is the blank symbol),

• a next move function

δ : Q × Σ → Σ × {L, R} × (Q ∪ {H}).
If δ(p, a) = (b, D, q), then the Turing machine, reading symbol a in state p, replaces a by b, moves in the direction D ∈ {L, R} (L for Left, R for Right), and comes into state q. On an input x k . . . x 0 ∈ Σ k+1 , the initial configuration is ω b(Ax k ) . . . x 0 b ω . This means that the word x k . . . x 0 is written on the tape between two infinite strings of blank symbols, and the machine is reading symbol x k in state A.

The known Turing machines

Let us give some more precisions about the Turing machines that simulate the 3x+1 function.

The following results are displayed in Table 1.

Michel [START_REF] Michel | Busy beaver competition and Collatz-like problems[END_REF] gave a 6×4 Turing machine that halts when number 1 is reached. This machine works on numbers written in binary. Division by 2 of even integers is easy and multiplication by 3 is done by the usual multiplication algorithm.

Margenstern [START_REF] Margenstern | Frontier between decidability and undecidability: a survey[END_REF][START_REF] Margenstern | Frontier between decidability and undecidability: a survey[END_REF] gave never halting 5 × 3 and 11 × 2 Turing machines in binary, and never halting 2 × 10, 3 × 6, 4 × 4 Turing machines in unary, that is working on numbers n written as strings of n 1s.

Baiocchi [START_REF] Baiocchi | 3N+1, UTM e Tag-systems[END_REF] gave five never halting Turing machines in unary, including 2 × 8, 3 × 5 and 10 × 2 machines that improved Margenstern's results.

In this article, we give a never halting 3 × 4 Turing machine that works on numbers written in base 3. Multiplication by 3 is easy and division by 2 is done by the usual division algorithm. Note that Baiocchi and Margenstern [START_REF] Baiocchi | Cellular automata about the 3x + 1 problem[END_REF] already used numbers written in base 3 to define cellular automata that simulate the 3x + 1 function.

By adding two states to this 3 × 4 Turing machine, we derive a 5 × 4 Turing machine that halts when number 1 is reached.

We also give three other Turing machines that halt when number 1 is reached:

• A 3 × 10 Turing machine obtained by adding one state to the 2 × 10 Turing machine of Margenstern [START_REF] Margenstern | Frontier between decidability and undecidability: a survey[END_REF][START_REF] Margenstern | Frontier between decidability and undecidability: a survey[END_REF].

• A 4 × 6 Turing machine obtained by adding one state to the 3 × 6 Turing machine of Margenstern [START_REF] Margenstern | Frontier between decidability and undecidability: a survey[END_REF][START_REF] Margenstern | Frontier between decidability and undecidability: a survey[END_REF].

• A 13 × 2 Turing machine obtained by adding two states to the 11 × 2 Turing machine of Margenstern [START_REF] Margenstern | Frontier between decidability and undecidability: a survey[END_REF][START_REF] Margenstern | Frontier between decidability and undecidability: a survey[END_REF].

A never halting 3 × Turing machine

This Turing machine M 1 is defined as follows:

M 1 b 0 1 2 A bLC 0RA 0RB 1RA B 2LC 1RB 2RA 2RB C bRA 0LC 1LC 2LC
The idea is simple. A positive integer is written on the tape, in base 3, in the usual order. Initially, in state A, the head reads the most significant digit, at the left end of the number. The initial configuration on input x = k i=0 x i 3 i is ω b(Ax k ) . . . x 0 b ω . Then the machine performs the division by 2, using the usual division algorithm. Partial quotients are written on the tape. Partial remainders are stored in the states: 0 in state A, 1 in state B. When the head passes the right end of the number, reading a b, then

• if the remainder is 0, nothing is done: 2n → n,

• if the remainder is 1, a 2 is concatenated to the number: 2n + 1 → n → 3n + 2.

Then the head comes back, in state C, to the left end of the number and is ready to perform a new division by 2.

We have the following theorem.

Theorem 4.1 The 3x + 1 conjecture is true iff, for all positive integer x = x k . . . x 0 written in base 3, there exists an integer n ≥ 0 such that, on input x k . . . x 0 , the Turing machine M 1 eventually reaches the configuration ω b0 n (A1)b ω .

5 Turing machines that halts on the final loop

A 5 × 4 Turing machine

This Turing machine M 6 is defined as follows.

M 6 b 0 1 2 A bLC 0RA 0RB 1RA B 2LE 1RB 2RA 2RB C bRD 0LC 1LC 2LC D bRA bRB 1RA E bRH 0LC 1LC 2LC
Turing machine M 6 is obtained from Turing machine M 1 by adding a state D that wipes out the useless 0s, and a state E that detects the partial configuration b(Bb).

We have the following theorem.

Theorem 5.3 The 3x + 1 conjecture is true iff Turing machine M 6 halts on all input x = x k . . . x 0 representing a positive integer written in base 3.

A 13 × 2 Turing machine

Margenstern [5, Fig. 8] gave the following never halting 11 × 2 Turing machine M 7 (in this table, H is not a halting state).

M 7 0 1 A 1RI 0RB B 0RA 0RG C 0RA 1RD D 0RC 1RE E 1RI 1RF F 1RC 0RG G 1RC 1RH H 0RE 1RG I 1LJ J 0RB 1LK K 0LJ 1LJ
This machine works on numbers written in binary, with the least significant bit at the left end of the number, and digits 0 and 1 coded by 10 and 11, so that the initial configuration on number n = x k . . . x 0 = k i=0 x i 2 i is ω 0(A1)x 0 1x 1 . . . 1x k 0 ω . Division by 2 of even integers is easy, and multiplication by 3 is done by the usual algorithm.

By adding two new states L and M , we can detect the partial configuration (A1)10, and we obtain the following 13 × 2 Turing machine M 8 , where Z is the halting state.

M 8 0 1 A 1RI 0RL B 0RA 0RG C 0RA 1RD D 0RC 1RE E 1RI 1RF F 1RC 0RG G 1RC 1RH H 0RE 1RG I 1LJ J 0RB 1LK K 0LJ 1LJ L 0RA 0RM M 0LZ 1RH
We have the following theorem.

Theorem 5.4 The 3x + 1 conjecture is true iff, for all positive number n = x k . . . x 0 = k i=0 x i 2 i , Turing machine M 8 halts on the initial configuration ω 0(A1)x 0 1x 1 . . . 1x k 0 ω .

Conclusion

We have given a new 3 × 4 never halting Turing machine that simulates the iteration of the 3x + 1 function. It seems that it will be hard to improve the known results on never halting machines.

On the other hand, for Turing machines that halt on the conjectured final loop of the 3x + 1 function, more researches are still to be done.

5. 1 A 3 ×

 13 10 Turing machine Margenstern [5, Fig. 11] gave the folowing never halting 2 × 10 Turing machine M 2 . kRB xRA xRA rLA rLA yRA B zLB uRB xRB yRB vLB uRA tLB 1LA xRB bRB Turing machine M 2 works on numbers written in unary, so that the initial configuration on number n ≥ 1 is ω b(A1)1 n-1 b ω . By adding a new state C, we can detect the partial configuration (A1)b, and we obtain the following 3 × 10 Turing machine M 3 . kRB xRA xRA rLA rLA yRA B zLB uRB xRB yRB vLB uRA tLB 1LA xRB bRB C bLH uRB yRB We have the following theorem Theorem 5.1 The 3x + 1 conjecture is true iff, for all positive integers n, Turing machine M 3 halts on the initial configuration ω b(A1)1 n-1 b ω . 5.2 A 4 × 6 Turing machine Margenstern [5, Fig. 10] gave the folowing never halting 3 × 6 Turing machine M 4 (Note that transition (1, z) → (xR2) in this figure should be (1, z) → (rR2)).

Theorem 5 . 2

 52 1LA rRB B 1LB aRC 1LB 1LA xRB bRA C zLC xRC 1LC aRA rRC zLC Turing machine M 4 works on numbers written in unary, with initial configuration ω b(A1)1 n-1 b ω . By adding a new state D, we can detect the partial configuration (A1)b, and we obtain the following 4 × 6 Turing machine M 5 . 1LA 1LA rRB B 1LB aRC 1LB 1LA xRB bRA C zLC xRC 1LC aRA rRC zLC D bLH aRC xRB We have the following theorem. The 3x + 1 conjecture is true iff, for all positive integers n, Turing machine M 5 halts on the initial configuration ω b(A1)1 n-1 b ω .

Table 1 :

 1 Turing machines simulating the 3x + 1 function: M a = Margenstern

	of states, plus a halting state H (or Z),


Une Approche

Lagrangienne Pour

L E Syst

Ème De

Ë L Danchin

A LAGRANGIAN APPROACH FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS

Keywords: . 1991 Mathematics Subject Classification. 35Q35, 76N10 Compressible fluids, uniqueness, critical regularity, Lagrangian coordinates. Fluides compressibles, unicité, régularité critique, coordonnées lagrangiennes

, ce qui permet d'établir entre autres que des vitesses initiales ayant une régularité Besov (pas trop) négative génèrent une solution unique. La densité initiale est à régularité critique et doit juste être strictement positive et tendre vers une constante à l'infini. Les coefficients de viscosité preuvent dépendre de la densité. L'usage de coordonnées lagrangiennes est la clef de toutes ces améliorations car il permet de résoudre le système par itérations de Picard. Comme corollaire immédiat, on obtient que les conditions pour l'unicité sont les mêmes que pour l'existence, ainsi que la continuité de l'opérateur solution (pour le système écrit en coordonnées lagrangiennes).

Introduction

We address the well-posedness issue for the barotropic compressible Navier-Stokes equations with variable density in the whole space R n : (0.1)

     ∂ t ρ + div (ρu) = 0, ∂ t (ρu) + div (ρu ⊗ u) -2div (µ(ρ)D(u)) -∇(λ(ρ)div u) + ∇(P (ρ)) = 0, ρ| |t=0 = ρ 0 , u| |t=0 = u 0 .
Above ρ = ρ(t, x) ∈ R + stands for the density, u = u(t, x) ∈ R n , for the velocity field. The space variable x belongs to the whole R n . The notation D(u) designates the deformation tensor which is defined by D(u) := 1 2 (Du + ∇u) with (Du) ij := ∂ j u i and (∇u) ij := ∂ i u j .

The pressure function P and the viscosity coefficients λ and µ are given suitably smooth functions of the density. With no loss of generality, one may assume that P is defined over R and vanishes at 0. As we focus on viscous fluids, we suppose that (0.2) α := min inf ρ>0 (λ(ρ) + 2µ(ρ)), inf ρ>0 µ(ρ) > 0, which ensures the second order operator in the velocity equation of (0.1) to be uniformly elliptic.

We supplement System (0.1) with the condition at infinity that u tends to 0 and ρ, to some positive constant (that may be taken equal to 1 after suitable normalization). The exact meaning of those boundary conditions will be given by the functional framework in which we shall consider the system.

In the present paper, we aim at solving (0.1) in critical functional spaces, that is in spaces which have the same invariance with respect to time and space dilation as the system itself (see e.g. [START_REF] Danchin | Local theory in critical spaces for compressible viscous and heat-conductive gases[END_REF] for more explanations about this nowadays classical approach). In this framework, it has been stated [START_REF] Danchin | Local theory in critical spaces for compressible viscous and heat-conductive gases[END_REF][START_REF] Danchin | On the uniqueness in critical spaces for compressible Navier-Stokes equations[END_REF] in the constant coefficients case that, for data (ρ 0 , u 0 ) such that

a 0 := (ρ 0 -1) ∈ Ḃn/p p,1 (R n ), u 0 ∈ Ḃn/p-1 p,1 (R n )
and that, for a small enough constant c, (0.3) a 0 Ḃn/p p,1 (R n ) ≤ c, we have for any p ∈ [1, 2n):

• existence of a local solution (ρ, u) such that a := (ρ -1) ∈ C b ([0, T ]; Ḃn/p p,1 ), u ∈ C b ([0, T ];

Ḃn/p-1 p,1

) and ∂ t u, ∇ 2 u ∈ L 1 (0, T ; Ḃn/p-1 p,1

); • uniqueness in the above space if in addition p ≤ n. If p ≤ n then the viscosity coefficients may depend (smoothly) on ρ and the smallness condition (0.3) may be replaced by the following positivity condition (see [START_REF] Chen | Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities[END_REF][START_REF] Danchin | Well-posedness in critical spaces for barotropic viscous fluids with truly nonconstant density[END_REF]): (0.4) inf x∈R n ρ 0 (x) > 0. Those results have been somewhat extended in [START_REF] Haspot | Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces[END_REF] where it has been noticed that a 0 may be taken in a larger Besov space, with another Lebesgue exponent.

The above results are based on maximal regularity estimates in Besov spaces for the evolutionary Lamé system, and on the Schauder-Tychonoff fixed point theorem. In effect, owing to the hyperbolicity of the density equation, there is a loss of one derivative in the stability estimates thus precluding the use of the contraction mapping (or Banach fixed point) theorem. As a consequence, with this method it is found that the conditions for uniqueness are stronger than those for existence.

Following our recent paper [START_REF] Danchin | A Lagrangian approach for the incompressible Navier-Stokes equations with variable density[END_REF] dedicated to the incompressible density-dependent Navier-Stokes equation, and older works concerning the compressible Navier-Stokes equations (see [START_REF] Mucha | The Cauchy problem for the compressible Navier-Stokes equations in the L p -framework[END_REF][START_REF] Valli | An existence theorem for compressible viscous fluids[END_REF][START_REF] Valli | Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case[END_REF]), we here aim at solving System (0.1) in the Lagrangian coordinates. The main motivation is that the mass is constant along the flow hence, to some extent, only the (parabolic type) equation for the velocity has to be considered. After performing this change of coordinates, we shall see that solving (0.1) may be done by means of the Banach fixed point theorem. Hence, the condition for uniqueness is the same as that for the existence, and the flow map is Lipschitz continuous. In addition, in the case of fully nonhomogeneous fluids with variable viscosity coefficients, the analysis turns out to be simpler than in [START_REF] Chen | Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities[END_REF][START_REF] Danchin | Well-posedness in critical spaces for barotropic viscous fluids with truly nonconstant density[END_REF] even for density-dependent viscosity coefficients and in the case where the density is not close to a constant. Indeed, our proof relies essentially on a priori estimates for a parabolic system (a suitable linearization of the momentum equation in Lagrangian coordinates) with rough constant depending only on the initial density hence time-independent. In contrast, in [START_REF] Chen | Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities[END_REF][START_REF] Danchin | Well-posedness in critical spaces for barotropic viscous fluids with truly nonconstant density[END_REF] tracking the time-dependency of the coefficients was quite technical.

We now come to the plan of the paper. In the next section, we introduce the compressible Navier-Stokes equations in Lagrangian coordinates and present our main results. Section 2 is devoted to the proof of our main existence and uniqueness result in the simpler case where the density is close to a constant and the coefficients, density independent. In Section 3, we treat the general fully nonhomogeneous case with nonconstant coefficients. A great deal of the analysis is contained in the study of the linearized momentum equation for (0.1) (see Subsection 3.1) which turns out to be a Lamé type system with variable rough coefficients. This will enable us to define a self-map Φ on a suitably small ball of some Banach space E p (T ) and to apply the contraction mapping theorem so as to solve the compressible Navier-Stokes equations in Lagrangian coordinates. In the Appendix we prove several technical results concerning the Lagrangian coordinates and Besov spaces.

Notation: Throughout, the notation C stands for a generic constant (the meaning of which depends on the context), and we sometimes write A B instead of A ≤ CB. For X a Banach space, p ∈ [1, +∞] and T > 0, the notation L p (0, T ; X) or L p T (X) designates the set of measurable functions f : [0, T ] → X with t → f (t) X in L p (0, T ), endowed with the norm f L p T (X) := f X L p (0,T ) . We agree that C([0, T ]; X) denotes the set of continuous functions from [0, T ] to X.

Main results

Before deriving the Lagrangian equations corresponding to (0.1), let us introduce more notation. We agree that for a

C 1 function F : R n → R n × R m then div F : R n → R m with (div F ) j := i ∂ i F ij for 1 ≤ j ≤ m,
and that for A = (A ij ) 1≤i,j≤n and B = (B ij ) 1≤i,j≤n two n × n matrices, we denote

A : B = TrAB = i,j A ij B ji .
The notation adj(A) designates the adjugate matrix that is the transposed cofactor matrix. Of course if A is invertible then we have adj(A) = (det A) A -1 . Finally, given some matrix A, we define the "twisted" deformation tensor and divergence operator (acting on vector fields z ) by the formulae

D A (z) := 1 2 Dz • A + T A • ∇z and div A z := T A : ∇z = Dz : A.
Let X be the flow associated to the vector-field u, that is the solution to

(1.1) X(t, y) = y + t 0 u(τ, X(τ, y)) dτ.
Denoting ρ(t, y) := ρ(t, X(t, y)) and ū(t, y) = u(t, X(t, y))

with (ρ, u) a solution of (0.1), and using the chain rule and Lemma 1 from the Appendix, we gather that (ρ, ū) satisfies

(1.2)    ∂ t (J ρ) = 0 ρ 0 ∂ t ū -div adj(DX) 2µ(ρ)D A (ū) + λ(ρ) div A ū Id + P (ρ)Id = 0
with J := det DX and A := (D y X) -1 . Note that one may forget any reference to the initial Eulerian vector-field u by defining directly the "flow" X of ū by the formula

(1.3) X(t, y) = y + t 0 ū(τ, y) dτ.
We want to solve the above system in critical homogeneous Besov spaces. Let us recall that, for 1 ≤ p ≤ ∞ and s ≤ n/p, a tempered distribution u over R n belongs to the homogeneous Besov space Ḃs

p,1 (R n ) if u = j∈Z ∆j u in S ′ (R n ) and (1.4) u Ḃs p,1 (R n ) := j∈Z 2 js ∆j u L p (R n ) < ∞.
Here ( ∆j ) j∈Z denotes a homogeneous dyadic resolution of unity in Fourier variables -the socalled Littlewood-Paley decomposition (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 2 for more details on the Littlewood-Paley decomposition and Besov spaces).

Loosely speaking, a function belongs to Ḃs p,1 (R n ) if it has s derivatives in L p (R n ). In the present paper, we shall mainly use the following classical properties:

• the Besov space Ḃn/p p,1 (R n ) is a Banach algebra embedded in the set of continuous functions going to 0 at infinity, whenever 1 ≤ p < ∞; • the usual product maps Ḃn/p-1

p,1 (R n ) × Ḃn/p p,1 (R n ) in Ḃn/p-1 p,1 (R n ) whenever 1 ≤ p < 2n; • Let F : I → R be a smooth function (with I an open interval of R containing 0)
vanishing at 0. Then for any s > 0, 1 ≤ p ≤ ∞ and interval J compactly supported in I there exists a constant C such that

(1.5) F (a) Ḃs p,1 (R n ) ≤ C a Ḃs p,1 (R n )
for any a ∈ Ḃs p,1 (R n ) with values in J. In addition, if a 1 and a 2 are two such functions and s = n/p then we have

(1.6) F (a 2 ) -F (a 1 ) Ḃn/p p,1 (R n ) ≤ C a 2 -a 1 Ḃn/p p,1 (R n ) .
From now on, we shall omit R n in the notation for Besov spaces. We shall obtain the existence and uniqueness of a local-in-time solution (ρ, ū) for (1.2), with ā := ρ -1 in C([0, T ]; Ḃn/p p,1 ) and ū in the space

E p (T ) := v ∈ C([0, T ]; Ḃn/p-1 p,1 ), ∂ t v, ∇ 2 v ∈ L 1 (0, T ; Ḃn/p-1 p,1
)

• That space will be endowed with the norm

v Ep(T ) := v L ∞ T ( Ḃn/p-1 p,1 ) + ∂ t v, ∇ 2 v L 1 T ( Ḃn/p-1 p,1
) . Let us now state our main result. Theorem 1. Let 1 < p < 2n and n ≥ 2. Let u 0 be a vector-field in Ḃn/p-1 p,1

. Assume that the initial density ρ 0 satisfies a 0 := (ρ 0 -1) ∈ Ḃn/p p,1 and (1.7) inf

x ρ 0 (x) > 0.

Then If ρ 0 is close enough to some positive constant then the statement holds true for all p ∈ [1, ∞) and n ≥ 1.

In Eulerian coordinates, this result recasts in: Theorem 2. Under the hypotheses of Theorem 1 with 1 < p < 2n and n ≥ 2, System (0.1) has a unique local solution (ρ, u) with u ∈ E p (T ), ρ bounded away from 0 and (ρ-1) ∈ C([0, T ]; Ḃn/p p,1 ). Let us make a few comments concerning the above assumptions.

• We expect the Lagrangian method to improve the uniqueness conditions given in e.g. [START_REF] Danchin | Local theory in critical spaces for compressible viscous and heat-conductive gases[END_REF] for the full Navier-Stokes equations. We here consider the barotropic case for simplicity. • The condition 1 ≤ p < 2n is a consequence of the product laws in Besov spaces. It implies that the regularity exponent for the velocity has to be greater than -1/2 (to be compared with -1 for the homogeneous incompressible Navier-Stokes equations). It would be interesting to see whether introducing a modified velocity as in B. Haspot's works [START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fluids[END_REF][START_REF] Haspot | Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces[END_REF] allows to consider different Lebesgue exponents for the Besov spaces pertaining to the density and the velocity so as to go beyond p = 2n for the velocity. • The regularity condition over the density is stronger than that for density-dependent incompressible fluids (see [START_REF] Danchin | A Lagrangian approach for the incompressible Navier-Stokes equations with variable density[END_REF]). In particular, in contrast with incompressible fluids, it is not clear that combining Lagrangian coordinates and critical regularity approach allows to consider discontinuous densities. • Owing to the fact that the density satisfies a transport equation, we do not expect Lipschitz continuity of the flow map in high norm for the Eulerian formulation to be true. • It is worth comparing our results with those of P. Germain in [START_REF] Germain | Weak-strong uniqueness for the isentropic compressible Navier-Stokes system[END_REF], and D. Hoff in [START_REF] Hoff | Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow[END_REF] concerning the weak-strong uniqueness problem. In both papers, the idea is to show that, in the constant viscosity case, a finite energy weak solution coincides with a strong one under some additional assumptions. The weak solution turns out to have less regularity than in Theorem 2. At the same time, the assumptions on the strong solution (ρ, u) are much stronger. In both papers, ∇u has to be in L 1 (0, T ; L ∞ ), and to satisfy additional conditions: roughly ∇ 2 u or ∂ t u have to be in L 2 (0, T ; L d ) in Germain's work, while √ tD 2 u ∈ L r (0, T ; L 4 ) with r = 4/3 if n = 2, and r = 8/5 if n = 3 in Hoff's paper. Some regularity conditions are required on the density but they are, to some extent, weaker than ours.

The simple case of almost homogeneous compressible fluids

As a warm up and for the reader convenience, we here explain how local well-posedness may be proved for the system in Lagrangian coordinates in the simple case where:

(1) The viscosity coefficients are constant, (2) The density is very close to one. Let µ ′ := λ + µ. Keeping in mind the above two conditions and using the fact that the first equation of (1.2) implies that (2.1)

J(t, •)ρ(t, •) ≡ ρ 0 , with J := | det DX| and (2.2) X(t, y) := y + t 0 ū(τ, y) dτ,
we rewrite the equation for the Lagrangian velocity as (recall that A := (DX) -1 ):

(2.3) ∂ t ū -µ∆ū -µ ′ ∇div ū = (1 -ρ 0 )∂ t ū + 2µ div adj(DX)D A (ū) -D(ū)
+ λdiv adj(DX) div A ū -div ū Id -div adj(DX)P (J -1 ρ 0 ) .

The left-hand side of the above equation is the linear Lamé system with constant coefficients, the solvability of which may be easily deduced from that of the heat equation in the whole space (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 2 or [START_REF] Danchin | Fourier analysis methods for compressible flows, topics on compressible Navier-Stokes equations, états de la recherche SMF[END_REF]). We get:

Proposition 1. Let the viscosity coefficients (µ, µ ′ ) ∈ R 2 satisfy µ > 0 and µ + µ ′ > 0. Let p ∈ [1, ∞] and s ∈ R. Let u 0 ∈ Ḃs p,1 and f ∈ L 1 (0, T ; Ḃs p,1
). Then the Lamé system

(2.4) ∂ t u -µ∆u -µ ′ ∇div u = f in (0, T ) × R n u| t=t 0 = u 0 on R n has a unique solution u in C([0, T ); Ḃs p,1 ) such that ∂ t u, ∇ 2 u ∈ L 1 (0, T ; Ḃs p,1
) and the following estimate is valid:

(2.5) u L ∞ T ( Ḃs p,1 ) + min(µ, µ + µ ′ ) ∇ 2 u L 1 T ( Ḃs p,1 ) ≤ C( f L 1 T ( Ḃs p,1 ) + u 0 Ḃs p,1
)

where C is an absolute constant with no dependence on µ, µ ′ and T.

In the rest of this section, we drop the bars on the Lagrangian velocity field. Granted with the above proposition, we define a map Φ : v → u on E p (T ) where u stands for the solution to

(2.6) ∂ t u -µ∆u -µ ′ ∇div u = I 1 (v) + 2µdiv I 2 (v, v) + λdiv I 3 (v, v) -div I 4 (v)
with initial data u 0 and (2.3). We claim that the existence of such points is a consequence of the standard Banach fixed point theorem in a suitable closed ball of E p (T ).

I 1 (w) = -a 0 ∂ t w, I 2 (v, w) = adj(DX v )D Av (w) -D(w), I 3 (v, w) = div Av w adj(DX v ) -div w Id, I 4 (v) = adj(DX v )P (J -1 v ρ 0 ). Note that any fixed point of Φ is a solution in E p (T ) to
First step: estimates for I 1 , I 2 , I 3 and I 4 . Throughout we assume that for a small enough constant c,

(2.7) T 0 Dv Ḃn/p p,1 dt ≤ c. It is obvious that (2.8) I 1 (w) L 1 T ( Ḃn/p-1 p,1 ) ≤ a 0 M( Ḃn/p-1 p,1 ) ∂ t w L 1 T ( Ḃn/p-1 p,1
)

where the multiplier norm M( Ḃs p,1 ) for Ḃs p,1 , is defined by (2.9)

f M( Ḃs p,1 ) := sup ψf Ḃs p,1
.

The supremum is taken over those functions ψ in Ḃs p,1 with norm 1.

Next, taking advantage of the fact that Ḃn/p p,1 is an algebra if 1 ≤ p < ∞, of (A.12), (A.13) and (2.7), we readily get (2.10)

I 2 (v, w) L 1 T ( Ḃn/p p,1 ) + I 3 (v, w) L 1 T ( Ḃn/p p,1 ) ≤ C Dv L 1 T ( Ḃn/p p,1 ) Dw L 1 T ( Ḃn/p p,1
) . As regards the pressure term (that is I 4 (v)), we use the fact that under assumption (2.7), we have, by virtue of the composition inequality (1.5) and of flow estimates (see (A.9) and (A.11)), (2.11)

I 4 (v) L ∞ T ( Ḃn/p p,1 ) ≤ C 1 + Dv L 1 T ( Ḃn/p p,1 ) 1 + a 0 Ḃn/p p,1
.

Second step: Φ maps a suitable closed ball in itself. At this stage, one may assert that if v ∈ E p (T ) satisfies (2.7) then the right-hand side of (2.6) belongs to L 1 (0, T ; Ḃn/p-1 p,1

). Hence Proposition 1 implies that Φ(v) is well defined and maps E p (T ) to itself. However it is not clear that it is contractive over the whole set E p (T ). So we introduce u L the "free solution" to

∂ t u L -µ∆u L -µ ′ ∇div u L = 0, u L | t=0 = u 0 .
Proposition 1 guarantees that u L belongs to E p (T ) for all T > 0.

We claim that if T is small enough (a condition which will be expressed in terms of the free solution u L ) and if R is small enough (a condition which will depend only on the viscosity coefficients and on p, n and

P ) then v ∈ BEp(T ) (u L , R) implies that (2.7) is fulfilled and that u ∈ BEp(T ) (u L , R). Indeed u := u -u L satisfies ∂ t u -µ∆ u -µ ′ ∇div u = I 1 (v) + 2µdiv I 2 (v, v) + λdiv I 3 (v, v) -div I 4 (v), u| t=0 = 0. So Proposition 1 yields 1 u Ep(T ) I 1 (v) L 1 T ( Ḃn/p-1 p,1 ) + I 2 (v, v) L 1 T ( Ḃn/p p,1 ) + I 3 (v, v) L 1 T ( Ḃn/p p,1 ) + T I 4 (v) L ∞ T ( Ḃn/p p,1
) . Inserting inequalities (2.8), (2.10) and (2.11), we thus get:

u Ep(T ) Dv 2 L 1 T ( Ḃn/p p,1 ) + a 0 M( Ḃn/p-1 p,1 ) ∂ t v L 1 T ( Ḃn/p-1 p,1 ) + T (1 + a 0 Ḃn/p p,1
).

That is, keeping in mind that v is in BEp(T

) (u L , R), u Ep(T ) ≤ C a 0 M( Ḃn/p-1 p,1 ) (R + ∂ t u L L 1 T ( Ḃn/p-1 p,1
)

) + Du L 2 L 1 T ( Ḃn/p p,1 ) + R 2 + T (1 + a 0 Ḃn/p p,1
) .

So we see that if T satisfies (2.12) CT (1 + a 0 Ḃn/p p,1

) ≤ R/2 and Du L L 1 T ( Ḃn/p p,1 ) + ∂ t u L L 1 T ( Ḃn/p-1 p,1 ) ≤ R then we have u Ep(T ) ≤ 2C a 0 M( Ḃn/p-1 p,1
) R + 2CR 2 + R/2. Hence there exists a small constant η = η(n, p) such that if

(2.13) a 0 M( Ḃn/p-1 p,1
) ≤ η, and if R has been chosen small enough then u is in BEp(T ) (u L , R). Of course, taking R and T even smaller ensures that (2.7) is satisfied for all vector-field of BEp(T ) (u L , R).

Third step: contraction properties. We claim that under Conditions (2.13) and (2.12) (with a smaller R if needed), the map Φ is 1/2-Lipschitz over BEp(T ) (u L , R). So we are given v 1 and v 2 in BEp(T ) (u L , R) and denote

u 1 := Φ(v 1 ) and u 2 := Φ(v 2 ).
Let X 1 and X 2 be the flows associated to v 1 and v 2 . Set A i = (DX i ) -1 and J i := det DX i for i = 1, 2. The equation satisfied by δu := u 2 -u 1 reads ∂ t δu -µ∆δu -µ ′ ∇div δu = δf := δf 1 + div δf 2 + 2µdiv δf 3 + λdiv δf 4 1 For simplicity, we do not track the dependency of the coefficients with respect to µ and µ ′ .

with δf 1 := -a 0 ∂ t δu, δf 2 := adj(DX 1 )P (ρ 0 J -1 1 ) -adj(DX 2 )P (ρ 0 J -1 2 ), δf 3 := adj(DX 2 )D A 2 (u 2 ) -adj(DX 1 )D A 1 (u 1 ) -D(δu), δf 4 := adj(DX 2 ) T A 2 : ∇u 2 -adj(DX 1 ) T A 1 : ∇u 1 -div δu Id.
Once again, bounding δu in E p (T ) stems from Proposition 1, which ensures that (2.14) δu Ep(T )

δf 1 L 1 T ( Ḃn/p-1 p,1 ) + T δf 2 L ∞ T ( Ḃn/p p,1 ) + δf 3 L 1 T ( Ḃn/p p,1 ) + δf 4 L 1 T ( Ḃn/p p,1
) . In order to bound δf 1 , we just have to use the definition of the multiplier space M( Ḃn/p-1 p,1

). We get

δf 1 L 1 T ( Ḃn/p-1 p,1 ) ≤ a 0 M( Ḃn/p-1 p,1 ) ∂ t δu L 1 T ( Ḃn/p-1 p,1
) . (2.15)

Next, using the decomposition

δf 2 = (adj(DX 1 ) -adj(DX 2 ))P (ρ 0 J -1 2 ) + adj(DX 1 )(P (ρ 0 J -1 1 ) -P (ρ 0 J -1 2 )
), together with composition inequalities (1.5), (1.6) )

Dδv L 1 T ( Ḃn/p p,1 )
Finally, we have

δf 4 = (adj(DX 2 )-adj(DX 1 )) T A 2 : ∇u 2 +adj(DX 1 ) T (A 2 -A 1 ) : ∇u 2 +(adj(DX 1 ) T A 1 -Id) : ∇δu,
whence, by virtue of (A.9), (A.10), (A.18) and (A. [START_REF] Krylov | Lectures on elliptic and parabolic equations in Sobolev spaces[END_REF]),

(2.17)

δf 4 L 1 T ( Ḃn/p p,1 ) Dδv L 1 T ( Ḃn/p p,1 ) Du 2 L 1 T ( Ḃn/p p,1 ) + Dδu L 1 T ( Ḃn/p p,1 ) Dv 1 L 1 T ( Ḃn/p p,1
) . Bounding δf 3 works exactly the same. So we see that if Conditions (2.12) and (2.13) are satisfied (with smaller η and larger C if need be) then we have

δu Ep(T ) ≤ 1 2 δv Ep(T ) .
Hence, the map Φ : BEp(T ) (u L , R) → BEp(T ) (u L , R) is 1/2-Lipschitz. Therefore, Banach' fixed point theorem ensures that Φ admits a unique fixed point in BEp(T ) (u L , R). This completes the proof of existence of a solution in E p (T ) for System (1.2). A tiny variation over the proof of the contraction properties yields uniqueness and Lipschitz continuity of the flow map. We eventually get:

Theorem 3. Assume that n ≥ 1. Let p ∈ [1, ∞) and u 0 be a vector-field in Ḃn/p-1 p,1
. Assume that the initial density ρ 0 satisfies a 0 := (ρ 0 -1) ∈ Ḃn/p p,1 . There exists a constant c depending only on p and on n such that if ) and u ∈ E p (T ).

We do not give here more details on how to complete the proof of Theorem 3 and its Eulerian counterpart, Theorem 4, as it will done in the next section under much more general assumptions.

The fully nonhomogeneous case

For treating the general case where ρ 0 only satisfies (1.7), just resorting to Proposition 1 is not enough because the term I 1 (v, v) in the r.h.s. of (2.6) need not be small. One has first to establish a similar statement for a Lamé system with nonconstant coefficients. More precisely, keeping in mind that ρ = J -1 u ρ 0 (we still drop the bars for notational simplicity), we recast the velocity equation of (1.2) in:

L ρ 0 (u) = ρ -1 0 div I 1 (u, u) + I 2 (u, u) + I 3 (u, u) + I 4 (u) with (3.1) L ρ 0 (u) := ∂ t u -ρ -1 0 div 2µ(ρ 0 )D(u) + λ(ρ 0 )div u Id and I 1 (v, w) := (adj(DX v ) -Id) µ(J -1 v ρ 0 )(Dw A v + T A v ∇w) + λ(J -1 v ρ 0 )( T A v : ∇w)Id I 2 (v, w) := (µ(J -1 v ρ 0 ) -µ(ρ 0 ))(Dw A v + T A v ∇w) + (λ(J -1 v ρ 0 ) -λ(ρ 0 ))( T A v : ∇w)Id I 3 (v, w) := µ(ρ 0 ) Dw(A v -Id) + T (A v -Id)∇w + λ(ρ 0 )( T (A v -Id) : ∇w)Id I 4 (v) := -adj(DX v )P (ρ 0 J -1 v )
. Therefore, in order to solve (1.2) locally, it suffices to show that the map

(3.2) Φ : v -→ u
with u the solution to

L ρ 0 (u) = ρ -1 0 div I 2 (v, v) + I 3 (v, v) + I 4 (v, v) + I 5 (v) , u| t=0 = u 0
has a fixed point in E p (T ) for small enough T.

As a first step, we have to study the properties of the linear Lamé operator L ρ 0 . This is done in the following subsection.

3.1. Linear parabolic systems with rough coefficients. As a warm up, we consider the following scalar heat equation with variable coefficients:

(3.3) ∂ t u -adiv (b∇u) = f.
We assume that

(3.4) α := inf (t,x)∈[0,T ]×R n (ab)(t, x) > 0.
Let us first consider the smooth case.

Proposition 2. Assume that a and b are bounded functions satisfying (3.4) and such that b∇a and a∇b are in L 2 (0, T ; Ḃn/p p,1 ) for some 1 < p < ∞. There exist two constants κ = κ(p) and C = C(s, n, p) such that the solutions to (3.3) satisfy for all t ∈ [0, T ], Proof. We first rewrite the equation for u as follows:

u L ∞ t ( Ḃs p,1 ) + κα u L 1 t ( Ḃs+2 p,1 ) ≤ u 0 Ḃs p,1 + f L 1 t ( Ḃs
∂ t u -div (ab∇u) = f -b∇a • ∇u,
then localize the equation in the Fourier space, according to Littlewood-Paley decomposition:

∂ t u j -div (ab∇u j ) = f j -∆j (b∇a • ∇u) + R j
with u j := ∆j u, f j := ∆j f and R j := div ([ ∆j , ab]∇u).

Next, we multiply the above equation by u j |u j | p-2 and integrate over R n . Taking advantage of Lemma 8 in the appendix of [START_REF] Danchin | On the well-posedness of the incompressible density-dependent Euler equations in the L p framework[END_REF] (here 1 < p < ∞ comes into play) and of Hölder inequality, we get for some constant c p depending only on p:

1 p d dt u j p L p + c p α2 2j u j p L p ≤ u j p-1 L p f j L p + ∆j (b∇a • ∇u) L p + R j L p ,
which, after time integration, leads to

(3.5) u j L ∞ t (L p ) + c p α2 2j u j L 1 t (L p ) ≤ u 0,j L p + f j L 1 t (L p ) + t 0 ∆j (b∇a • ∇u) L p + R j L p dτ.
According to Lemmas 4 and 5 in Appendix, there exist a positive constant C and some sequence (c j ) j∈Z with c ℓ 1 (Z) = 1, satisfying .

(3.6) ∆j (b∇a • ∇u) L p + R j L p ≤ Cc j 2 -
So plugging this in (3.7) and applying Gronwall lemma completes the proof of the proposition.

In the rough case where the coefficients are only in Ḃn/p p,1 , the above proposition has to be modified as follows: Proposition 3. Let a and b be bounded positive and satisfy (3.4). Assume that b∇a and a∇b are in L ∞ (0, T ; Ḃn/p-1 p,1

) with 1 < p < ∞. There exist three constants η, κ and C such that if for some m ∈ Z we have

inf (t,x)∈[0,T ]×R n Ṡm (ab)(t, x) ≥ α/2, (3.9) (Id -Ṡm )(b∇a, a∇b) L ∞ T ( Ḃn/p-1 p,1
) ≤ ηα (3.10) then the solution to (3.3) satisfies for all t ∈ [0, T ], , which may be obtained by taking σ = 1 and ν = 1 in Lemmas 4 and 5. However, when bounding R j , in addition to (3.11), one has to assume that p ≤ n. Also, as it involves the highest regularity of u, we cannot expect to absorb this "remainder term" any longer, unless a∇b and b∇a are small in Ḃn/p-1 p,1

u L ∞ t ( Ḃs p,1 ) + ακ u L 1 t ( Ḃs+2 p,1 ) ≤ u 0 Ḃs p,1 + f L 1 t ( Ḃs p,1 ) exp C α t 0 Ṡm (b∇a, a∇b)
(which would correspond to the case that has been treated in the previous section). So we rather rewrite the heat equation as follows:

∂ t u -div ( Ṡm (ab)∇u) = f + div ((Id -Ṡm )(ab)∇u) -Ṡm (b∇a) • ∇u -(Id -Ṡm )(b∇a) • ∇u.
Now, using the infimum bound for Ṡm (ab) and arguing as for proving (3.5), we get

u j L ∞ t (L p ) + c p α2 2j u j L 1 t (L p ) ≤ u 0,j L p + f j L 1 t (L p ) + t 0 ∆j div ((Id -Ṡm )(ab)∇u) L p dτ + t 0 ∆j ( Ṡm (b∇a) • ∇u) L p + ∆j ((Id -Ṡm )(b∇a) • ∇u) L p + div ([ Ṡm (ab), ∆j ]∇u) L p dτ.
The idea is to apply the procedure of the "smooth" case for the low frequency part of the coefficients (that is the part containing Ṡm ) and the "perturbation" approach for the other part. More precisely, appealing to Lemmas 4 and 5, we get under Condition (3.11) and for some sequence (c j ) j∈Z with c ℓ 1 (Z) = 1: .

∆j div ((Id -Ṡm )(ab)∇u) L p c j 2 -
Let us plug those four inequalities in the above inequality for u j . After multiplying by 2 js and summing up over j, we get

u L ∞ t ( Ḃs p,1 ) + c p α u L 1 t ( Ḃs+2 p,1 ) ≤ u 0 Ḃs p,1 + f L 1 t ( Ḃs p,1 ) +C (Id -Ṡm )(ab) L ∞ t ( Ḃn/p p,1 ) + (Id -Ṡm )(b∇a) L ∞ t ( Ḃn/p-1 p,1 ) u L 1 t ( Ḃs+2 p,1 ) +C t 0 Ṡm (a∇b, b∇a) Ḃn/p p,1 ∇u Ḃs p,1 dτ.
It is clear that, under Condition (3.10), the second line may be absorbed by the left-hand side.

Hence the desired inequality follows from the interpolation inequality (3.8), exactly as in the smooth case.

We now look at the following Lamé system with nonconstant coefficients:

(3.13) ∂ t u -2adiv (µD(u)) -b∇(λdiv u) = f.
Note that u and f are valued in R n . We assume throughout that the following uniform ellipticity condition is satisfied:

(3.14) α := min inf (t,x)∈[0,T ]×R n (aµ)(t, x), inf (t,x)∈[0,T ]×R n (2aµ + bλ)(t, x) > 0.
Let us first study the "smooth case": Proposition 4. Assume that a, b, λ and µ are bounded functions satisfying (3.14) and such that a∇µ, b∇λ, µ∇a and λ∇b are in L 2 (0, T ; Ḃn/p p,1 ) for some 1 < p < ∞. There exists a constant C such that the solutions to (3.13) satisfy for all t ∈ [0, T ], Proof. We introduce the following functions:

u L ∞ t ( Ḃs p,1 ) + α u L 1 t ( Ḃs+2 p,1 ) ≤ C u 0 Ḃs p,1 + f L 1 t ( Ḃs
d := |D| -1 div u and Ω := |D| -1 curl u with (curl u) ij := ∂ i u j -∂ j u i .
Owing to the use of homogeneous Besov space, and because the Fourier multipliers A(D) := |D| -1 div and B(D) := |D| -1 curl are of degree 0, it is equivalent to estimate u or (d, Ω) in

L ∞ T ( Ḃs p,1 ) ∩ L 1 T ( Ḃs+2 p,1
). So the basic idea is to show that d and Ω satisfy heat equations similar to (3.3). More precisely, applying A(D) to (3.13) yields

(3.15) ∂ t d -(2aµ + bλ)∆d = A(D)(f + 2a∇µ • D(u) + b∇λ div u) + [A(D), aµ]∆u + [A(D), aµ + bλ]∇div u.
Given Condition (3.14), we see that arguing exactly as for proving (3.7) and because A(D) maps Ḃs p,1 in itself, Similarly, the vorticity part Ω of u satisfies

d L ∞ t ( Ḃs p,1 ) + κα d L 1 t ( Ḃs+2 p,1 ) ≤ d 0 Ḃs p,1 + A(D)f L 1 t ( Ḃs p,1 ) + C t 0 2a∇µ • D(u) + b∇λdiv u Ḃs
∂ t Ω -aµ∆Ω = B(D)(f + 2a∇µ • D(u) + b∇λdiv u) + [B(D), aµ]∆u + [B(D), aµ + bλ]∇div u.
So arguing exactly as for bounding d, and resorting to the interpolation inequality (3.8) and to Gronwall lemma, we easily get the desired inequality. It is just a matter of following the proof for the case of the heat equation.

Let us finally focus on the "rough case" where the coefficients of (3.13) are only in L ∞ T ( Ḃn/p p,1 ).

Proposition 5. Let a, b, λ and µ be bounded functions satisfying (3.14). Assume that a∇µ, b∇λ, µ∇a and λ∇b are in L ∞ (0, T ; Ḃn/p-1 p,1

) for some 1 < p < ∞. There exist two constants η and κ such that if for some m ∈ Z we have

min inf (t,x)∈[0,T ]×R n Ṡm (2aµ + bλ)(t, x), inf (t,x)∈[0,T ]×R n Ṡm (aµ)(t, x) ≥ α 2 , (3.16) (Id -Ṡm )(µ∇a, a∇µ, λ∇b, b∇λ) L ∞ T ( Ḃn/p-1 p,1
) ≤ ηα (3.17) then the solutions to (3.13) satisfy for all t ∈ [0, T ],

u L ∞ t ( Ḃs p,1 ) + α u L 1 t ( Ḃs+2 p,1 ) ≤ C u 0 Ḃs p,1 + f L 1 t ( Ḃs p,1 ) exp C α t 0 Ṡm (µ∇a, a∇µ, λ∇b, b∇λ) 2 Ḃn/p p,1 dτ whenever -min(n/p, n/p ′ ) < s ≤ n/p -1.
Proof. As for the heat equation, we split the coefficients of the system into a smooth (but large) low frequency part and a rough (but small) high frequency part. It turns out to be more convenient to work directly on the equations for d and Ω. More precisely, as regards d, we write (starting from (3.15) and denoting c := 2aµ + bλ) that , ∆j ((Id -Ṡm )∇c • ∇d)

∂ t d -div (c∇d) = -∇c • ∇d + A(D)(f + 2a∇µ • D(u) + b∇λ div u) +[A(D), aµ]∆u + [A(D), aµ + bλ]∇div u, whence, denoting d j := ∆j d, ∂ t d j -div ( Ṡm c∇d j ) = div ([ ∆j , Ṡm c]∇d) + ∆j div ((Id -Ṡm )c∇d) -Ṡm ∇c • ∇d -(Id -Ṡm )∇c • ∇d + ∆j A(D) f + 2 Ṡm (a∇µ) • D(u) + 2(Id -Ṡm )(a∇µ) • D(u) + Ṡm (b∇λ) div u + (Id -Ṡm )(b∇λ) div u + ∆j [A(D), Ṡm (aµ)]∆u +
c j 2 -js (Id -Ṡm )∇c Ḃn/p-1 p,1 ∇d Ḃs+1 p,1 , ∆j [A(D), Ṡm (aµ)]∆u L p c j 2 -js ∇ Ṡm (aµ) Ḃn/p p,1 ∆u Ḃs-1 p,1 , ∆j [A(D), (Id -Ṡm )(aµ)]∆u L p c j 2 -js ∇(Id -Ṡm )(aµ) Ḃn/p-1 p,1 ∆u Ḃs p,1
, and similar estimates for ∆j

A(D)( Ṡm (a∇µ) • D(u)), ∆j A(D)((Id -Ṡm )(a∇µ) • D(u)), ∆j A(D)( Ṡm (b∇λ) div u), ∆j A(D)((Id -Ṡm )(b∇λ) div u), ∆j [A(D), Ṡm (aµ + bλ)]∇div u, ∆j [A(D), (Id -Ṡm )(aµ + bλ)]∇div u.
The curl part Ω of the velocity may be treated in the same way. Therefore we get

u L ∞ t ( Ḃs p,1 ) + α u L 1 t ( Ḃs+2 p,1 ) u 0 Ḃs p,1 + f L 1 t ( Ḃs p,1 ) + t 0 Ṡm (a∇µ, µ∇a, b∇λ, λ∇b) Ḃn/p p,1 u Ḃs+1 p,1 dτ + t 0 (Id -Ṡm )(a∇µ, µ∇a, b∇λ, λ∇b) Ḃn/p-1 p,1 u Ḃs+2 p,1
dτ.

Obviously the last term may be absorbed by the left-hand side if η is small enough in (3.17) and the last-but-one term may be handled by interpolation according to (3.8). So applying Gronwall lemma yields the desired inequality.

For the sake of completeness, we still have to justify the existence of a solution to (3.13). More precisely, we want to establish the following result: Proposition 6. Let p be in (1, +∞). Let a, b, λ and µ be bounded functions satisfying (3.14). Assume in addition that there exist some constants ā, b, λ and μ such that (3.18) 2āμ + bλ > 0 and āμ > 0,

and such that a -ā, b -b, µ -μ and λ -λ are in C([0, T ]; Ḃn/p p,1 ). Finally, suppose that

(3.19) lim m→+∞ (Id -Ṡm )(a -ā, b -b, λ -λ, µ -μ) L ∞ T ( Ḃn/p
p,1 ) = 0. Then for any data u 0 ∈ Ḃs p,1 and f ∈ L 1 (0, T ; Ḃs p,1 ) with s satisfying (3.11), System (3.13) admits a unique solution u ∈ C([0, T ]; Ḃs p,1 ) ∩ L 1 (0, T ; Ḃs+2 2,1 ). Besides, the estimates of Proposition 5 are fulfilled for all large enough m ∈ Z.

Proof. The proof is based on the continuity method as explained in e.g. [START_REF] Krylov | Lectures on elliptic and parabolic equations in Sobolev spaces[END_REF] (and used in [START_REF] Danchin | On the solvability of the compressible Navier-Stokes system in bounded domains[END_REF] in a similar context as ours). For θ ∈ [0, 1], we introduce the following second order operator P θ acting on vector-fields u as follows:

P θ u := -2a θ div (µ θ D(u)) -b θ ∇(λ θ div u),
where a θ := (1 -θ)ā + θa, b θ := (1 -θ) b + θb, and so on. We claim that one may find some m ∈ Z independent of θ such that for all θ ∈ [0, 1], the conditions (3.16) and (3.17) are satisfied by a θ , b θ , µ θ and λ θ . Indeed, we notice that

a θ -ā = θ(a -ā).
Hence, for all θ ∈ [0, 1],

(Id -Ṡm )(a θ -ā) L ∞ T ( Ḃn/p p,1 ) ≤ (Id -Ṡm )(a -ā) L ∞ T ( Ḃn/p p,1
) , and similar properties hold for b θ , λ θ and µ θ . In particular, owing to the continuous embedding of Ḃn/p p,1 in the set of continuous bounded functions, and to (3.19), we deduce that there exists some m ∈ Z so that the ellipticity condition (3.16) is satisfied by operator P θ for all θ ∈ [0, 1].

Likewise, we have for instance

µ θ ∇a θ = θ(1 -θ)μ∇a + θ 2 µ∇a
and similar relations for the other coefficients. Hence one may find some large enough m so that (3.17) is satisfied for all θ ∈ [0, 1]. In addition, the above relation shows that Ṡm (µ θ ∇a θ ) Ḃn/p .

Hence all the terms appearing in the exponential term of the estimate in Proposition 5 may be bounded by a constant depending only on m and on the coefficients a, b, λ and µ. As a conclusion, one may thus find some constant C independent of θ such that any solution w of

∂ t w -P θ w = g, w| t=0 = w 0 satisfies (3.20) w L ∞ T ( Ḃs p,1 ) + α w L 1 T ( Ḃs+2 p,1 ) ≤ C w 0 Ḃs p,1 + g L 1 T ( Ḃs p,1
) . After this preliminary work, one may start with the proof of existence (uniqueness follows from the estimates of Proposition 5). Let E be the set of those θ in [0, 1] such that for every data u 0 and f (as in the statement of the theorem), System Note that according to Proposition 1, the set E contains 0 hence is nonempty. So it suffices to find a fixed ε > 0 such that for all θ 0 ∈ E, we have (3.22) [θ 0 -ε, θ 0

+ ε] ∩ [0, 1] ⊂ E.
So let us fix some θ 0 ∈ E, u 0 ∈ Ḃs p,1 , f ∈ L 1 (0, T ; Ḃs p,1 ) and v ∈ F s p (T ) and consider the solution u to the system ∂ t u -P θ 0 u = f + (P θ -P θ 0 )v with θ ∈ [0, 1] such that |θ -θ 0 | ≤ ε. Given that θ 0 is in E, the existence of u in F s p (T ) is granted if (P θ -P θ 0 )v ∈ L 1 (0, T ; Ḃs p,1 ). So let us first check this: we have

(P θ -P θ 0 )v = (θ -θ 0 ) 2a θ 0 div (μ -µ)D(v) + 2(ā -a)div (µ θ D(v)) +b θ 0 ∇ ( λ -λ)div v + ( b -b)∇ λ θ div v .
Under Condition (3.11), one may thus conclude thanks to product estimates in Besov spaces (see Lemma 4) that (P θ -P θ 0 )v ∈ L 1 (0, T ; Ḃs p,1 ). Furthermore

(P θ -P θ 0 )v Ḃs p,1 ≤ Cε (ā + a θ 0 -ā Ḃn/p p,1
) µ -μ Ḃn/p p,1

+ (μ + µ θ -μ Ḃn/p p,1
) a -ā Ḃn/p p,1

+( b + b θ 0 -b Ḃn/p p,1
) λ -λ Ḃn/p p,1

+ ( λ + λ θ -λ Ḃn/p p,1
) a -ā Ḃn/p p,1

Dv Ḃs+1 p,1
.

The coefficients may be bounded in terms of the initial coefficients a, b, λ and µ. Hence, applying (3.20) we get for some constant independent of θ 0 and of θ,

u L ∞ T ( Ḃs p,1 ) + α u L 1 T ( Ḃs+2 p,1 ) ≤ C ε v L 1 T ( Ḃs+2 p,1 ) + w 0 Ḃs p,1 + f L 1 T ( Ḃs p,1
) . Taking ε small enough, it becomes clear that the linear map Ψ θ : v → u is contractive on the Banach space F s p (T ). Hence it has a (unique) fixed point u ∈ F s p (T ). In other words, u satisfies (3.21).

Given that E is nonempty and that ε is independent of θ 0 , one may now conclude that 1 is in E. Therefore, there exists a solution u ∈ F s p (T ) to (3.13). Remark 1. Under the assumptions of the above proposition, the constructed solution u satisfies ∂ t u ∈ L 1 (0, T ; Ḃs p,1 ). Indeed, it suffices to notice that

∂ t u = f + (ā + (a -ā))div (μ + (µ -μ)D(u)) + ( b + (b -b))∇( λ + (λ -λ)div u),
and to use Lemma 4 together with the facts that ∇u is in L 1 (0, T ; Ḃs+1 p,1 ). Moreover we have where C may depend also on the norm of a -ā, b -b, λ -λ and µ -μ in L ∞ (0, T ; Ḃn/p p,1 ). 3.2. Proof of Theorem 1. As we want to consider (possibly) large velocities, we introduce, as in the almost homogeneous case the free solution to the Lamé system corresponding to ρ ≡ 1, that is the vector-field u L in E p (T ), given by Proposition 1, satisfying2 L 1 u L = 0, u| t=0 = u 0 .

∂ t u L 1 T ( Ḃs p,1 ) ≤ C u 0 Ḃs p,1 + f L 1 t ( Ḃs
We claim that the Banach fixed point theorem applies to the map Φ defined in (3.2) in some closed ball BEp(T ) (u L , R) with suitably small T and R. Denoting u := u -u L , we see that u has to satisfy

(3.23) L ρ 0 u = ρ -1 0 div I 1 (v, v) + I 2 (v, v) + I 3 (v, v) + I 4 (v) + (L 1 -L ρ 0 )u L , u| t=0 = 0. If the right-hand side is in L 1 (0, T ; Ḃn/p-1 p,1
) and if there exists some m ∈ Z so that (3.16) and (3.17) are fulfilled then Proposition 6 and Remark 1 ensure the existence of u in E p (T ). Now, the existence of m so that min inf

x Ṡm 2

µ(ρ 0 ) ρ 0 + λ(ρ 0 ) ρ 0 , inf x Ṡm µ(ρ 0 ) ρ 0 > α 2 and (Id -Ṡm ) µ(ρ 0 ) ρ 2 0 ∇ρ 0 , µ ′ (ρ 0 ) ρ 0 ∇ρ 0 , λ(ρ 0 ) ρ 2 0 ∇ρ 0 , λ ′ (ρ 0 ) ρ 0 ∇ρ 0 Ḃn/p-1 p,1 ≤ ηα.
is ensured by the fact that all the coefficients (minus some constant) belong to the space Ḃn/p p,1

which is defined in terms of a convergent series and embeds continuously in the set of bounded continuous functions. The study of the right-hand side of (3.23) will be carried out below. ) ensure that

(3.24) u Ep(T ) ≤ Ce Cρ 0 ,mT (L 1 -L ρ 0 )u L L 1 T ( Ḃn/p-1 p,1 ) + ρ -1 0 M( Ḃn/p-1 p,1 ) I 1 (v, v) L 1 T ( Ḃn/p p,1 ) + I 2 (v, v) L 1 T ( Ḃn/p p,1 ) + I 3 (v, v) L 1 T ( Ḃn/p p,1 ) + I 4 (v) L 1 T ( Ḃn/p p,1 )
for some constant C ρ 0 ,m depending only on ρ 0 and on m.

In what follows, we assume that T and R have been chosen so that (2.7) is satisfied by v. Using the decomposition ) and

(L 1 -L ρ 0 )u L = (ρ -1 0 -1)div 2µ(ρ 0 )D(u L ) + λ(ρ 0 )div u L Id +div 2(µ(ρ 0 ) -µ(1))D(u) + (λ(ρ 0 ) -λ( 1
(3.25) (L 1 -L ρ 0 )u L L 1 T ( Ḃn/p-1 p,1
) a 0 Ḃn/p

v into v + u L : u Ep(T ) ≤ Ce Cρ 0 ,mT (1 + a 0 Ḃn/p p,1 ) 2 (T + a 0 Ḃn/p p,1 Du L L 1 T ( Ḃn/p p,1 ) ) + Du L 2 L 1 T ( Ḃn/p p,1 ) + Du L L 1 T ( Ḃn/p p,1 ) + D v L 1 T ( Ḃn/p p,1 ) D v L 1 T ( Ḃn/p p,1 ) . So, because v ∈ B Ep(T ) (u L , R), u Ep(T ) ≤ Ce Cρ 0 ,mT (1 + a 0 Ḃn/p p,1 ) 2 (T + a 0 Ḃn/p p,1 Du L L 1 T ( Ḃn/p p,1 ) ) +(R + Du L L 1 T ( Ḃn/p p,1 ) ) Du L L 1 T ( Ḃn/p p,1 ) + R 2 .
Therefore, if we first choose R so that for a small enough constant η, (3.28)

(1 + a 0 Ḃn/p p,1

) 2 R ≤ η and then take T so that

(3.29) C ρ 0 ,m T ≤ log 2, T ≤ R 2 , a 0 Ḃn/p p,1 Du L L 1 T ( Ḃn/p p,1 ) ≤ R 2 , Du L L 1 T ( Ḃn/p p,1 )
≤ R, then we may conclude that Φ maps BEp(T ) (u L , R) into itself.

Second step: contraction estimates. Let us now establish that, under Condition (3.29), the map Φ is contractive. We consider two vector-fields v 1 and v 2 in BEp(T ) (u L , R), and set u 1 := Φ(v 1 ) and u 2 := Φ(v 2 ). Let δu := u 2 -u 1 and δv := v 2 -v 1 . In order to prove that Φ is contractive, it is mainly a matter of applying Proposition 5 to

L ρ 0 δu = ρ -1 0 div (I 1 (v 2 , v 2 ) -I 1 (v 1 , v 1 )) +(I 2 (v 2 , v 2 ) -I 2 (v 1 , v 1 )) + (I 3 (v 2 , v 2 ) -I 3 (v 1 , v 1
I 1 (v 2 , v 2 ) -I 1 (v 1 , v 1 ) = λ(J -1 v 2 ρ 0 ) T A v 2 : ∇v 2 adj(DX v 2 )
-adj(DX v 1 ) + adj(DX v 1 ) -Id λ(J -1 v 2 ρ 0 ) -λ(J -1 v 1 ρ 0 ) T A v 2 : ∇v 2 + adj(DX v 1 ) -Id λ(J -1 v 1 ρ 0 ) ( T A v 2 -T A v 1 ) : ∇v 1 + T A v 2 : ∇δv + terms pertaining to µ.

Taking advantage of product laws in Besov spaces, of composition estimates (1.5) and (1.6), and of the flow estimates in the appendix, we deduce that for some constant C ρ 0 depending only on ρ 0 : Given that v 1 and v 2 are in BEp(T ) (u L , R), our hypotheses over T and R (with smaller η in (3.28) if need be) thus ensure that, say, δu Ep(T ) ≤ 1 2 δv Ep(T ) .

I 1 (v 2 , v 2 ) -I 1 (v 1 , v 1
One can thus conclude that Φ admits a unique fixed point in BEp(T ) (u L , R). .

Proof. The result relies on Bony decomposition (A.21). The standard continuity results for the paraproduct and remainder operators ensure that (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], Chap. 2): There exists a constant C depending only on ν, p, σ and n such that for all k ∈ {1, • • • , n}, we have for some sequence (c j ) j∈Z with c ℓ 1 (Z) = 1: Arguing as in the proof of Lemma 6 in [START_REF] Danchin | On the well-posedness of the incompressible density-dependent Euler equations in the L p framework[END_REF], we get

T f g Ḃσ p,1 f Ḃ-ν ∞,
∂ k [a, ∆j ]w L p ≤ Cc
R 1 j L p ≤ C |j ′ -j|≤4 ∇ Ṡj ′ -1 a L ∞ ∆j ′ w L p .
Now, for ν ≥ 0, we have

∇ Ṡj ′ -1 a L ∞ ≤ C2 j ′ ν ∇a Ḃ-ν ∞,1
.

Therefore, for some sequence (c j ) j∈Z in the unit sphere of ℓ 1 (Z),

(A.24) R 1 j L p ≤ Cc j 2 -jσ ∇a Ḃ-ν ∞,1 w Ḃσ+ν p,1
.

To deal with R 2 j , we use the fact that, owing to the localization properties of the Littlewood-Paley decomposition, we have

R 2 j = j ′ ≥j-2
∂ k S j ′ +2 ∆j w ∆j ′ a .

Hence, using the Bernstein and Hölder inequalities,

R 2 j L p ≤ C j ′ ≥j-2 S j ′ +2 ∆j w L ∞ ∆j ′ ∇a L p , ≤ C2 -jσ j ′ ≥j-2 2 (j-j ′ )( n p -ν) 2 j(σ+ν-n p ) ∆j w L ∞ 2 j ′ ( n p -ν) ∆j ′ ∇a L p .

( 1 . 2 ) 1 )

 121 has a unique local solution (ρ, ū) with (ā, ū) ∈ C([0, T ]; Ḃn/p p,1 ) × E p (T ). Moreover, the flow map (a 0 , u 0 ) -→ (ā, ū) is Lipschitz continuous from B × E p (T ).In Eulerian coordinates, this result recasts in: Theorem 4. Under the above assumptions with in addition n ≥ 2 and p < 2n, System (0.1) has a unique local solution (ρ, u) with density bounded away from vacuum and a ∈ C([0, T ]; Ḃn/p-1 p,1

p, 1

 1 ) exp C α t 0 (b∇a, a∇b) 2 Ḃn/p p,1 dτ whenever -min(n/p, n/p ′ ) < s ≤ n/p.

dτ

  whenever -min(n/p, n/p ′ ) < s ≤ n/p.

1 ≤ 1 ,

 11 D), aµ]∆u + [A(D), aµ + bλ]∇div u Ḃs Note that applying Lemma 6 with σ = s -1, ν = 0 and Lemma 4 with σ = s and ν = 0 yields [A(D), aµ]∆u Ḃs p,C ∇(aµ) Ḃn/p and analogous estimates for [A(D), aµ + bλ]∇div u and b∇λdiv u.

1 , 1 ,

 11 [A(D), Ṡm (aµ + bλ)]∇div u + [A(D), (Id -Ṡm )(aµ)]∆u + [A(D), (Id -Ṡm )(aµ + bλ)]∇div u . Under Condition (3.11), Lemmas 4, 5 and 6 imply that div ([ ∆j , Ṡm c]∇d) L p c j 2 -js Ṡm ∇c Ḃn/p ∆j div ((Id -Ṡm )c∇d) L p c j 2 -js (Id -Ṡm )c Ḃn/p ∆j ( Ṡm ∇c • ∇d) L p c j 2 -js Ṡm ∇c Ḃn/p

( 3

 3 .21) ∂ t u -P θ u = f, u| t=0 = u 0 has a solution u in the set F s p (T ) := C([0, T ]; Ḃs p,1 ) ∩ L 1 (0, T ; Ḃs+2 p,1 ).

First step:

  Stability of BEp(T ) (u L , R) for small enough R and T . Proposition 5 and the definition of the multiplier space M( Ḃn/p-1 p,1

  ))div u Id , and composition inequalities (1.5) and (1.6), we see that (L 1 -L ρ 0 )u L ∈ L 1 (0, T ; Ḃn/p-1 p,1

1 )

 1 )) + (I 4 (v 2 ) -I 4 (v 1 )) • So we have, given that C ρ 0 ,m T ≤ log 2, (3.30) δu Ep(T ) ≤ C(1 + a 0 Ḃn/p p,I 1 (v 2 , v 2 ) -I 1 (v 1 , v 1 ) L 1 T ( Ḃn/p p,1 ) + I 2 (v 2 , v 2 )-I 2 (v 1 , v 1 ) L 1 T ( Ḃn/p p,1 ) + I 3 (v 2 , v 2 )-I 3 (v 1 , v 1 ) L 1 T ( Ḃn/p p,1 ) + I 4 (v 2 )-I 4 (v 1 ) L 1 T ( Ḃn/p p,1 )• In order to deal with the first term of the right-hand side, we use the decomposition

p, 1 )

 1 ≤ C ρ 0 (Dv 1 , Dv 2 ) L 1 T ( Ḃn/p p,1 ) Dδv L 1 T ( Ḃn/p p,1) . Similar estimates may be proved for the next two terms of the right-hand side of (3.30). Concerning the last one, we use the decompositionI 4 (v 2 ) -I 4 (v 1 ) = adj(DX v 1 ) -adj(DX v 2 ) P (J -1 v 2 ρ 0 ) -adj(DX v 1 ) P (J -1 v 2 ρ 0 ) -P (J -1 v 1 ρ 0 ) . Hence I 4 (v 2 ) -I 4 (v 1 )We end up with δu Ep(T ) ≤ C(1 + a 0 Ḃn/p p,1

) 2 T

 2 + (Dv 1 , Dv 2 )

Lemma 4 .

 4 Let p be in[1, +∞] and the real numbers ν and σ satisfy ν ≥ 0 and -Then the following estimate holds true for all tempered distributions f and g over R n :

j 2 - 1 w B σ+ν p, 1 for 1 j+ ∂ k T ′ ∆j w a R 2 j-∂ k ∆j T ′ w a R 3 j

 211123 jσ ∇a B n/p-ν p,all j ∈ Z. Proof. Taking advantage of the Bony decomposition (A.21), we rewrite the commutator as 3 (A.23) ∂ k ([a, ∆j ]w) = ∂ k ([T a , ∆j ]w) R .

  and (A.19), and product laws in Besov space yields

	(2.16)	δf 2 L ∞ T (	Ḃn/p p,1 )	T (1 + a 0 Ḃn/p p,1

  js b∇a Ḃn/p

										p,1	+ a∇b Ḃn/p p,1	∇u Ḃs p,1
						p,1	+ f L 1 t ( Ḃs p,1 )
										t
										+ C	0	(b∇a, a∇b) Ḃn/p p,1	u Ḃs+1 p,1	dτ.
	From the interpolation inequality
	(3.8)	u Ḃs+1 p,1	≤ u	1/2 Ḃs p,1	u	1/2 p,1 Ḃs+2	,
	we gather that
	C (b∇a, a∇b) Ḃn/p p,1	u Ḃs+1 p,1	≤	αc p 2	u Ḃs+2 p,1	+	C 2 2αc p	(b∇a, a∇b) 2 Ḃn/p p,1	u Ḃs p,1

.

Then inserting

(3.6

) in (3.5), multiplying by 2 js and summing up over j yields

(3.7) u L ∞ t ( Ḃs p,1 ) + c p α u L 1 t ( Ḃs+2 p,1 ) ≤ u 0 Ḃs

  L p + R j L p ≤ Cc j 2 -js b∇a Ḃn/p-1

	whenever
	(3.11) -min(n/p, n/p Proof. Given the new assumptions, it is natural to replace (3.6) by the inequality
	(3.12)	∆j (b∇a • ∇u) p,1	+ a∇b Ḃn/p-1 p,1	∇u Ḃs+1 p,1
				2 Ḃn/p p,1	dτ

′ ) < s ≤ n/p -1.

  1 if σ > -min(n/p, n/p ′ ).Lemma 5. Assume that σ, ν and p are such that

		T g f Ḃσ p,1 R(f, g) Ḃσ p,1	g Ḃσ+ν p,1 g Ḃσ+ν-n/p ∞,1 f Ḃn/p-ν p,1 f Ḃn/p-ν p,1 g Ḃσ+ν p,1	if ν ≥ 0, if σ + ν -n/p ≤ 0,
	So the result follows once noticed that Ḃs p,1 ֒→	Ḃs-n/p
	(A.22)	1 ≤ p ≤ +∞, 0 ≤ ν ≤	n p	and -min	n p	,	n p ′ -1 < σ ≤	n p	-ν.

∞,1

for any s ∈ R.

See (3.1) for the definition of operator L1.

Here we use the notation T ′ u v := Tuv + R(u, v).

Third step: Regularity of the density. Granted with the above velocity field u in E p (T ), we set ρ := J -1 u ρ 0 . By construction, the couple (ρ, u) satisfies (1.2). Let us now prove that a := ρ -1 is in C([0, T ]; Ḃn/p p,1 ). We have a = (J -1 u -1)a 0 + a 0 . Given (A.11) and using the fact that Du ∈ L 1 (0, T ; Ḃn/p p,1 ), it is clear that J -1 u -1 belongs to C([0, T ]; Ḃn/p p,1 ). Hence a belongs to C([0, T ]; Ḃn/p p,1 ), too. Because Ḃn/p p,1 is continuously embedded in L ∞ , Condition (1.7) is fulfilled on [0, T ] (taking T smaller if needed).

Last step: Uniqueness and continuity of the flow map. We now consider two couples (ρ 1 0 , u 1 0 ) and (ρ 2 0 , u 2 0 ) of data fulfilling the assumptions of Theorem 1 and we denote by (ρ 1 , u 1 ) and (ρ 2 , u 2 ) two solutions in E p (T ) corresponding to those data. Setting δu := u 2 -u 1 , we see that

((I 2 j -I 1 j )(u 1 , u 1 ) + (I 2 4 -I 1 4 )(u 1 ) ,

where I i 1 , I i 2 , I i 3 and I i 4 correspond to the quantities that have been defined just above (3.2), with density ρ i 0 . Note that those terms may be bounded exactly as in the second step. So the only definitely new terms are (L ρ 1 0 -L ρ 2 0 )(u 2 ) and the last line. As regards (L

Id . Hence, combining composition, flow and product estimates, we get for t ≤ T,

Du 2

) . It is not difficult to show that the other "new" terms satisfy analogous estimates. Hence, applying Proposition 5 to the system that is satisfied by δu, we discover that for t ≤ T,

) . An obvious bootstrap argument thus shows that if t, δu 0 and δρ 0 are small enough then

As regards the density, we have So we eventually get uniqueness and continuity of the flow map on a small enough time interval. Then iterating the proof yields uniqueness on the initial time interval [0, T ]. Note that it also yields Lipschitz continuity of the flow map for the velocity as for fixed data (ρ 1 0 , u 1 0 ), one may find some neighborhood and common time interval on which all the solutions constructed in the previous steps exist. and ρ 0 ∈ (1 + Ḃn/p p,1 ), the local existence for (0.1) may be proved directly (see [START_REF] Chen | Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities[END_REF][START_REF] Danchin | Local theory in critical spaces for compressible viscous and heat-conductive gases[END_REF]) but only under the assumption that p ≤ n in the case of nonconstant viscosity coefficients. Here we get the result (including uniqueness) from Theorem 1, and under the sole assumption that p < 2n. This is a mere corollary of the following proposition which states the equivalence of the systems (0.1) and (1.2) in our functional setting.

Proposition 7. Assume that the couple (ρ, u) with (ρ -1) ∈ C([0, T ]; Ḃn/p p,1 ) and u ∈ E p (T ) (with 1 ≤ p < 2n) is a solution to (0.1) such that

Let X be the flow of u defined in (1.1). Then the couple (ρ, ū) := (ρ • X, u • X) belongs to the same functional space as (ρ, u), and satisfies (1.2).

Ḃn/p p,1 ) × E p (T ) and (ρ, ū) satisfies (1.2) and, for a small enough constant c,

and in fact a locally

Ḃn/p+1 p,1

) diffeomorphism over R n and the couple (ρ, u) := (ρ • X -1 , ū • X -1 ) satisfies (0.1) and has the same regularity as (ρ, ū).

Proof. Let us first consider a solution (ρ, u) to (0.1) with the above properties. Then, the definition of X implies that DX -Id is in C([0, T ]; Ḃn/p p,1 ). In addition, Proposition 8 ensures that (ρ, ū) := (ρ • X, u • X) belongs to the same functional space as (ρ, u), and (A.9), (A.10), (A.11) below imply that A -Id, adj(DX) -Id and J -1 -1 are in C([0, T ]; Ḃn/p p,1 ). Therefore the product laws for Besov spaces enable us to use the algebraic relations (A.5), (A.6), (A.7) and (A.8) whenever p < 2n. Therefore (ρ, ū) fulfills (1.2).

Conversely, if we are given some solution (ρ, ū) in C([0, T ]; (1 + Ḃn/p p,1 )) × E p (T ) to (1.2) then one may check (see the appendix of [START_REF] Danchin | Fourier analysis methods for compressible flows, topics on compressible Navier-Stokes equations, états de la recherche SMF[END_REF]) that, under condition (2.7), the "flow" X(t, •) of ū defined by

). Hence one may construct the Eulerian vector-field u and Eulerian density by setting

As above, the algebraic relations (A.5), (A.6), (A.7) and (A.8) hold whenever p < 2n. Hence (ρ, u) is a solution to (0.1). That (ρ, u) has the desired regularity stems from Proposition 8.

Proof of Theorem 2. We consider data (ρ 0 , u 0 ) with ρ 0 bounded away from 0, (ρ 0 -1) ∈ Ḃn/p p,1 and u 0 ∈ Ḃn/p-1

Then Theorem 1 provides a local solution (ρ, ū) to System (1.2) in

) is a solution of (0.1) in the desired functional space.

In order to prove uniqueness, we consider two solutions (ρ 1 , u 1 ) and (ρ 2 , u 2 ) corresponding to the same data (ρ 0 , u 0 ), and perform the Lagrangian change of variable (pertaining to the flow of u 1 and u 2 respectively). The obtained vector-fields ū1 and ū2 are in E p (T ) and both satisfy (1.2) with the same ρ 0 and u 0 . Hence they coincide, as a consequence of the uniqueness part of Theorem 1.

Appendix A

A.1. Change of coordinates. Here we establish a result of regularity concerning changes of variables in Besov spaces. Even though this is somewhat classical (at least in nonhomogeneous Besov spaces), we did not find any reference in the literature of the estimates that we need. We here give a result in general Besov spaces Ḃs p,q , the definition of which may be found in e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF].

Proposition 8. Let X be a globally bi-Lipschitz diffeomorphism of R n and (s, p, q) with 1 ≤ p < ∞ and -n/p ′ < s < n/p (or just -n/p ′ < s ≤ n/p if q = 1 and just

Then a → a • X is a self-map over Ḃs p,q in the following cases: (1) s ∈ (0, 1), (2) s ∈ (-1, 0] and J X -1 is in the multiplier space M( Ḃ-s p ′ ,q ′ ) defined in (2.9), (3) s ≥ 1 and (DX -Id) ∈ Ḃn/p p,1 . Proof: Let us first assume that s ∈ (0, 1) and q = p. Then one may use the classical characterization of the norm of Ḃs p,p in terms of finite differences (see e.g. [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]) so as to write:

• Hence performing the change of variable x ′ = X(x) and y ′ = X(y), we get

. The condition that s < n/p ensures in addition that u belongs to some Lebesgue space L p * (R n ) with p * < ∞ (or in the set of continuous functions going to 0 at infinity if q = 1 and s = n/p). Hence u • X ∈ L p * (R n ) too and one may thus conclude that u • X ∈ Ḃs p,p (R n ). An interpolation argument then yields the desired result for any s ∈ (0, 1) and q ∈ [1, +∞].

The result for negative s may be achieved by duality: we have

. So the definition of the multiplier space and the first part of the lemma allows to conclude.

Finally, let us examine the cases of larger values of s. If 1 < s < 2 then one may write

As 0 < s -1 < 1, the first part of the proof ensures that Du • X ∈ Ḃs-1 p,q . As moreover (DX -Id) ∈ Ḃn/p p,1 , the standard product laws in Besov spaces give the result. If 2 < s < 3 then we use the algebraic relation,

Hence the result follows from product laws and the previous result applied with s -1 or s -2.

The higher values of s may be achieved by induction, and the remaining cases (s an integer) follow by interpolation. The details are left to the reader. A.2. Some properties of Lagrangian coordinates. Let us first derive a few algebraic relations involving changes of coordinates. We are given a C 1 -diffeomorphism X over R n . For H : R n → R m , we agree that H(y) = H(x) with x = X(y). With this convention, the chain rule writes

Hence we have

Lemma 1. Let K be a C 1 scalar function over R n and H, a C 1 vector-field. Let X be a C 1 diffeomorphism such that J := det(D y X) > 0. Then the following relations hold true:

where adj(D y X) stands for the adjugate of D y X.

Proof: The first item stems from the following series of computations (based on integrations by parts, changes of variable and (A.2)) which hold for any vector-field φ with coefficients in

Proving the second item is similar. Combining (A.2), (A.4) and (A.3), we deduce that if u : R n → R n and P : R n → R then ∆ x u = J -1 div y (adj(D y X)∇ x u) = J -1 div y (adj(D y X) T A∇ y ū), (A.5)

Note that we will use the above relations in the case where X is the flow of some time-dependent vector field u, defined by the relation

Hence we will also have

Let us now establish some estimates for the flow X v of some given "Lagrangian" vector field (that is X v is defined by (3.33)).

Lemma 2. Let p ∈ [1, +∞) and v be in E p (T ) satisfying (2.7). Let X v be defined by (3.33). Then we have for all t ∈ [0, T ],

Dv L 1 t ( Ḃn/p p,1 ) , (A.10)

Furthermore, if w is a vector field such that D w ∈ L 1 (0, T ; Ḃn/p p,1 ) then

Proof: Recall that (see e.g. the appendix of [START_REF] Danchin | A Lagrangian approach for the incompressible Navier-Stokes equations with variable density[END_REF]) for any n × n matrix C we have

where the entries of the matrix P 2 (C) are at least quadratic polynomials. Applying this relation to the matrix DX(t), and using the fact that (A. Given that Ḃn/p p,1 is a Banach algebra and that (2.7) holds, we readily get (A.9). In order to prove (A.10), we just use the fact that, under assumption (2.7), we have

Dv dτ, and that Ḃn/p p,1 is a Banach algebra. As regards (A.11), we write (A.17)

Hence, if Condition (2.7) holds then we have (A.11) for J v . In order to get the inequality for J -1 v , it suffices to use the fact that

For proving (A.12), we use the decomposition

Hence the desired inequality stems from (A.9) and (A.10), and from the fact that Ḃn/p p,1 is a Banach algebra. Inequality (A.13) is similar. This completes the proof of the lemma. Lemma 3. Let v1 and v2 be two vector-fields satisfying (2.7), and δv := v2 -v1 . Then we have for all p ∈ [1, +∞) and all t ∈ [0, T ] (with obvious notation): ) . Proof: In order to prove the first inequality, we use the fact that, for i = 1, 2, we have

So using the fact that Ḃn/p p,1 is a Banach algebra, it is easy to conclude to (A.18). The second inequality is a consequence of the decomposition (A.14) and of the Taylor formula which ensures that, denoting δC :

where the coefficients of P 2 are polynomials of degree n -1. As the sum is finite and Ḃn/p p,1 is a Banach algebra, we get (A. [START_REF] Krylov | Lectures on elliptic and parabolic equations in Sobolev spaces[END_REF].

Proving the third inequality relies on similar arguments. It is only a matter of using (A.17). The details are left to the reader. A.3. Commutator and product estimates. This last paragraph is devoted to the proof of commutator and product estimates that have been used for investigating the Lamé system. Those proofs rely on the following Bony decomposition (first introduced in [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF]) for the product of two functions:

The paraproduct and remainder operators T and R are defined by Proof. Taking advantage once again of Bony's decomposition, we decompose the commutator into (A.27) [A(D), q]w = [A(D), T q ]w + A(D)T ′ w q -T ′ A(D)w q. According to Lemma 2.99 in [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF], we have for ν ≥ 0, [A(D), T q ]w Ḃσ+1 Next, given that A(D) is a homogeneous multiplier of degree 0, it maps any homogeneous Besov space in itself. Therefore the last two terms of (A.27) may be just bounded according to standard continuity results for the paraproduct and remainder operators.