Huyen Bui

Mohamed Boutouil

Daniel Levacher

Nassim Sebaibi

Evaluation of the influence of accelerated carbonation on the microstructure and mechanical characteristics of coconut fibre-reinforced cementitious matrix

Keywords: coconut fibre, mortar, carbonation, phenolphthalein, TGA measurement. 1. Introduction

In this study, two methods were used to measure the carbonation depth of coconut fibre-reinforced mortars that were prepared from Portland cement (PC) and calcium sulfoaluminate cement (CSA cement), including traditional phenolphthalein reagent and thermogravimetric analysis (TGA). Accelerated carbonation was conducted at 4% CO2 concentration, temperature of 20°C and 65% relative humidity (RH) within 12 weeks. Complete carbonation was detected in the CSA cement-based mortar while PC-based mortar was carbonated partially according to both methods used. The difference in carbonation depth, however, is observed as soon as the reagent and TGA measurement were applied. The significant effectiveness of the carbonation process on the microstructure changes and mechanical characteristics of the cementitious matrix is highlighted through analysis of the change in the quantity of calcium carbonate (CC) and portlandite (CH) on the different mortar profiles and comparison with natural cured samples.

All these studies have suggested that the accelerated carbonation is a useful means to the sequestration of CO2 and an effective procedure to diminish the alkaline environment prematurely in the cementitious composite. This is due to the process of carbonation reaction in cementitious matrix that carries out as follows: CO2 + H2O H2CO3 (1) Ca(OH)2 + H2CO3 CaCO3 + 2H2O

(2) C3S2H3 + 3H2CO3 3CaCO3 + 2SiO2 + 6H2O

(3) The degree of carbonation is the main concern of some previous studies with inconsistent observations depending on the binder type and test condition. After carbonation storage, the surface of sample could be divided into three different zones depending on CO2 absorbed amount in theory and CO2 uptake in sample, including the non-carbonated zone, the partially carbonated zone and the fully carbonated zone [START_REF] Chang | The experimental investigation of concrete carbonation depth[END_REF]. Complete carbonation was observed (i) for CEM III-based concrete under natural carbonation after 3.7 years [START_REF] Andrade | Evaluation of the degree of carbonation of concretes in three environments[END_REF], (ii) for portlandite at temperature of 20°C and 91% RH and 1% CO2 concentration [START_REF] Winnefeld | Effect of relative humidity on the carbonation rate of portlandite , calcium silicate hydrates and ettringite[END_REF]. Whereas partial carbonation was detected (i) for a polypropylene fibre-cement composite with 15% of CO2 concentration, at 60°C and with 90% relative humidity [START_REF] Henrique | Influence of the initial moisture content on the carbonation degree and performance of fiber-cement composites[END_REF], (ii) for an autoclaved aerated concrete under conditions of 20°C, 90% relative humidity and 3% of CO2 concentration after 20 days [START_REF] Matsushita | Carbonation degree of autoclaved aerated concrete[END_REF] or (iii) for OPC-based concrete with the conditions of 23°C, 70% relative humidity and 20% of CO2 concentration within 16 weeks [START_REF] Junior | The effects of the early carbonation curing on the mechanical and porosity properties of high initial strength Portland cement pastes[END_REF]. Several methods for the measurement of carbonation depth are applied, from simple to cutting-edge technology methods. However, the traditional and most popular parameter used to measure carbonation depth is the use of a suitable indicator [START_REF] Qiu | A state-of-the-art review on the carbonation process in cementitious materials: Fundamentals and characterization techniques[END_REF]. Some reagent solutions were highly recommended like phenolphthalein [START_REF] Meena | A review on carbonation study in concrete[END_REF][START_REF] Tang | Influence of axial loading and carbonation age on the carbonation resistance of recycled aggregate concrete[END_REF] or thymolphthalein [START_REF] Lee | A Study for Carbonation Degree on Concrete using a Phenolphthalein Indicator and Fourier-Transform Infrared[END_REF][START_REF] De Weerdt | Effect of carbonation on the pore solution of mortar[END_REF]. These indicators are sprayed on the tested surface of samples. The change of the pH value in samples leads to the change in colour of the indicator. But more recent conclusions [START_REF] Ashraf | Carbonation of cement-based materials : Challenges and opportunities[END_REF][START_REF] Johannesson | Microstructural changes caused by carbonation of cement mortar[END_REF] believed that even when the reduction of the pH value due to the carbonation process could be made visible clearly by the colour change of phenolphthalein pH reagent, that area can still be considered as only underwent partial carbonation. Therefore, in order to limit the drawbacks of the reagent solution, it is necessary to use simultaneously other methods for improving the accuracy of carbonation depth measurement. Some methods were suggested in previous studies such as thermogravimetric analysis [START_REF] Wei | Correlation between hydration of cement and durability of natural fiber-reinforced cement composites[END_REF][START_REF] Qin | Influence of mineral admixtures on carbonation curing of cement paste[END_REF][START_REF] De Weerdt | Effect of carbonation on the pore solution of mortar[END_REF][START_REF] Pizzol | Effect of accelerated carbonation on the microstructure and physical properties of hybrid fiber-cement composites[END_REF], images analyses [START_REF] Tomography | Monitoring the Evolution of Accelerated Carbonation of Hardened Cement Pastes by X-Ray Computed Tomography[END_REF][START_REF] Mohamed Maslehuddin | Effect of temperature and salt contamination on carbonation of cements[END_REF], Xray diffraction [START_REF] Chang | The experimental investigation of concrete carbonation depth[END_REF][START_REF] Winnefeld | Effect of relative humidity on the carbonation rate of portlandite , calcium silicate hydrates and ettringite[END_REF], infrared spectroscopy [START_REF] Winnefeld | Effect of relative humidity on the carbonation rate of portlandite , calcium silicate hydrates and ettringite[END_REF] or using fibre optic chemical sensors [START_REF] O. Mcpolin | Preliminary development and evaluation of fibre optic chemical sensor[END_REF]. Among them, digital image analysis was believed to be the most quickly and inexpensive technique for determination maximum carbonation depth [START_REF] Herrera | A comparison of methods for determining carbonation depth in fly ash-blended cement mortars[END_REF]. Among accelerated carbonation testing conditions, the CO2 concentration is considered as a main factor for the laboratory carbonation test [START_REF] Qiu | A state-of-the-art review on the carbonation process in cementitious materials: Fundamentals and characterization techniques[END_REF][START_REF] Ashraf | Carbonation of cement-based materials: Challenges and opportunities[END_REF]. It should be noted that CO2 concentration is the largest difference between natural and accelerated carbonation conditions, inducing the laboratory process used to replace natural carbonation. With the CO2 concentration of under 5%, Neves et al. [START_REF] Neves | Field assessment of the relationship between natural and accelerated concrete carbonation resistance[END_REF] have found that a linear relationship between carbonation depth and CO2 concentration was observed. Otherwise, when the CO2 concentration was beyond 5%, this relation differed significantly since the carbonation depth -CO2 concentration curve could be described by a square root function [START_REF] Hyvert | Dependency of C-S-H carbonation rate on CO2 pressure to explain transition from accelerated tests to natural carbonation[END_REF]. Moreover, Auroy et al. [START_REF] Auroy | Comparison between natural and accelerated carbonation (3% CO2): Impact on mineralogy, microstructure, water retention and cracking[END_REF] reported that 3% CO2 concentration was the most suitable content for accelerated carbonation in the laboratory to investigate and compare the consequences of the natural environment and accelerated carbonation on the properties of cementitious matrix. Statistically, most of the research was carried out with relative humidity range of 50 -75 % to get a higher carbonation rate [START_REF] Winnefeld | Effect of relative humidity on the carbonation rate of portlandite , calcium silicate hydrates and ettringite[END_REF][START_REF] De Ceukelaire | Accelerated carbonation of a blast-furnace cement concrete[END_REF][START_REF] Elsalamawy | The role of relative humidity and cement type on carbonation resistance of concrete[END_REF]. Ceukelaire et al. [START_REF] De Ceukelaire | Accelerated carbonation of a blast-furnace cement concrete[END_REF] determined the role of relative humidity condition in the carbonation process of concrete by using a series of different relative humidity levels from 40 to 90 % with an increment of 10%. They indicated that the increase in relative humidity induces an increase in the carbonation resistance. In detail, with relative humidity of 50%, a maximal rate of carbonation was obtained, expressed by maximal carbonation depth of 8.5 mm after 21 days of exposure in 10 % CO2 while only 1.4 mm was recorded for the same time exposure in natural carbonation. Otherwise, these values with the relative humidity level of 90% dropped to only 4.9 mm and 0.5 mm for accelerated and natural carbonations, respectively. The presence and abundance of free water caused by high relative humidity could diminish the rate of CO2 diffusion and leads to a slow carbonation rate [START_REF] Zeng | Progress in the research of carbonation resistance of RAC[END_REF]. As well as, Elsalamawy et al. [START_REF] Elsalamawy | The role of relative humidity and cement type on carbonation resistance of concrete[END_REF] reported that relative humidity and carbonation depth relationship corresponds to a polynomial function, showing a carbonation depth peak obtained with an approximately relative humidity of 65% for all four samples tested, regardless of the types of cement and water/cement ratio used. This seems to be corroborated by the result in ref. [START_REF] Bernal | Accelerated carbonation testing of alkaliactivated slag/metakaolin blended concretes: effect of exposure conditions[END_REF] which found that the highest carbonation rate reaches at relative humidity level of 65%, proving the ability of increase the rate of carbonation process in moderately saturated moisture conditions. Meanwhile, in the report of Weerdt et al. [START_REF] De Weerdt | Effect of carbonation on the pore solution of mortar[END_REF], the higher carbonation rate was observed in mortar based on Portland-fly ash cement (30 wt.% fly ash) compared to PC-based mortar at similar controlled conditions. The observed increase in carbonation rate could be interpreted as being a consequence of the fly ash reaction resulting in the reduction of portlandite and formation of mono-carbonate, which made lower pH value in Portland-fly ash cement-based mortar [START_REF] De Weerdt | Hydration mechanisms of ternary Portland cements containing limestone powder and fly ash[END_REF][START_REF] Plusquellec | Determination of the pH and the free alkali metal content in the pore solution of concrete: Review and experimental comparison[END_REF]. The dependence of water/binder ratio on the carbonation degree was also investigated in the literature [START_REF] Lee | A Study for Carbonation Degree on Concrete using a Phenolphthalein Indicator and Fourier-Transform Infrared[END_REF][START_REF] Ioannou | Performance characteristics of concrete based on a ternary calcium sulfoaluminate -anhydrite -fly ash cement[END_REF]. The increase in water/binder ratio enhances carbonation depth for all the types of binder used. The higher water/binder ratio provokes the formation of larger pores that contributes to the carbon ion penetration deeply into sample. This is a convenient environment to promote the carbonation process. Concerning the relationship between the mechanical strength and accelerated carbonation of composites, the carbonation process could make the compressive strength of cementitious matrix higher in comparison with the samples cured under natural conditions whatever the types of added minerals, especially at the early age [START_REF] Qin | Influence of mineral admixtures on carbonation curing of cement paste[END_REF]. This is mainly owing to the formation of CaCO3 from the transformation of calcium hydroxide Ca(OH)2 that fills pores to mitigate the porosity in the cementitious matrix. These results provided additional support for previous findings in the literature [START_REF] Chi | Effects of carbonation on mechanical properties and durability of concrete using accelerated testing method[END_REF][START_REF] Chabannes | Studying the hardening and mechanical performances of rice husk and hemp-based building materials cured under natural and accelerated carbonation[END_REF][START_REF] Zhu | Carbonation-cementation of recycled hardened cement paste powder[END_REF][START_REF] Rostami | Carbonation curing versus steam curing for precast concrete production[END_REF][START_REF] Wang | Accelerated carbonation of hardened cement pastes: Influence of porosity[END_REF].In contrast, some other studies [START_REF] Junior | The effects of the early carbonation curing on the mechanical and porosity properties of high initial strength Portland cement pastes[END_REF][START_REF] Johannesson | Microstructural changes caused by carbonation of cement mortar[END_REF][START_REF] Elsalamawy | The role of relative humidity and cement type on carbonation resistance of concrete[END_REF][START_REF] Bernal | Accelerated carbonation testing of alkaliactivated slag/metakaolin blended concretes: effect of exposure conditions[END_REF][START_REF] Ioannou | Performance characteristics of concrete based on a ternary calcium sulfoaluminate -anhydrite -fly ash cement[END_REF][START_REF] De Weerdt | Cement and Concrete Research Effect of carbonation on the pore solution of mortar[END_REF]] believed that the compressive strength of the cementitious composite was decreased at the higher carbonation degree levels. They believed that the reduction of the pH value during CO2 curing process has negative effects on the reinforcement and leads to decrease in the mechanical performance of composite. The carbonation process causes a structural change in the CSH phase of cementitious composite, which could induce both the increase in strength and carbonation cracking. This complicated change leads to the prediction effects of carbonation on the mechanical performance of cementitious materials become very difficult [60]. Regarding thermogravimetric analysis, several studies [START_REF] Pizzol | Mineralogical and microstructural changes promoted by accelerated carbonation and ageing cycles of hybrid fiber -cement composites[END_REF][START_REF] Pizzol | Effect of accelerated carbonation on the microstructure and physical properties of hybrid fiber-cement composites[END_REF] also confirmed that when samples are subjected to the carbonation process, the conversion of Ca(OH)2 into CaCO3 could result in mitigation of the pore size in the cementitious system. Therefore, at higher carbonation degree levels, samples show a better resistance to high temperature due to the change in microstructure of the carbonated sample. For instance, the amount of Ca(OH)2 diminishes as the quantity of CaCO3 enlarges along the carbonation process and decreases the porosity of composite. Additionally, the disintegration of Ca(OH)2 occurs at the temperature range from 450°C to 550°C while the weight loss at the temperature above 550°C only concerns CaCO3. The reaction of Ca(OH)2 to acquire CaCO3 could lead to the reduction in alkalinity of the cementitious composite, create a less aggressive environment to protect natural fibres in better condition [START_REF] Toledo Filho | Development of vegetable fibre-mortar composites of improved durability[END_REF]. Therefore, thermal, and chemical stabilities are considered as two main advantages of carbonated products. In present study, the influences of incorporating natural fibre and cement type on the carbonation development of mortars were determined by means of carbonation depth measurement, mechanical performance, and pyrolysis behaviour. With the usage of two different types of cement and four replacement amounts of coconut fibres, the effects of the accelerated carbonation on the microstructure change and mechanical characteristics of mortars were investigated experimentally. The outcomes of this present study could contribute to having a better understanding in carbonation resistance of fibrereinforced mortar and the influence of accelerated carbonation on behaviour performance of the cement-based materials.

2. Materials and experimental methods 2.1. Materials Coconut fibres from Vietnam were used in present study as raw fibres. Firstly, pith, coir dust, and undesirable materials were manually removed to select raw fibres. These fibres were then cut into 10 -20 mm length, as shown in Fig. 1. The specifications of fibres were identified in previous study [START_REF] Bui | Determination and Review of Physical and Mechanical Properties of Raw and Treated Coconut Fibers for Their Recycling in Construction Materials[END_REF] and given in Table . 2. 123.6 Failure strain (%) 26.9 Water content (%) 7.60 Mortars were manufactured based on two types of cement, CEM I 52.5 N type I Portland Cement (PC) and Calcium Sulfoaluminate cement (CSA Cement). CSA cement, containing approximately 55% of calcium sulfoaluminate, is considered as a green binder due to its environmentally friendly characteristics, i.e., 37% lower of CO2 emission emitted than PC [START_REF] Juenger | Advances in alternative cementitious binders[END_REF]. Indeed, CSA cement is produced at a lower calcined temperature and less energy requirement for grinding procedure than PC due to the easier grinding of CSA clinker [START_REF] Glasser | High-performance cement matrices based on calcium sulfoaluminate-belite compositions[END_REF]. In addition, CSA cement is appraised as a special cement with lower alkali content, i.e., pH range of 10.5-11 and 12 -13 for CSA cement and PC, respectively. This pH value range for CSA cement could result in reducing the degradation of natural fibres in the mortar composite. The main physical properties and oxide compositions of the two binder types and sand are presented in Table 3. CHRYSO Fluid Optima 352 EMx was added with a ratio of 1% by mass of cement as a superplasticizer for a significant water reduction and improvement of the workability of the fresh mixture in accordance with the suggestion of Yan et al. [START_REF] Li | Sisal fibre and its composites: a review of recent developments[END_REF].

Table 3

Mortar mix design

In present study, the mortar mix design was produced according to EN 196-1 standard. The incorporation of fibres consists of substituting a volume of sand by a corresponding volume of fibre with a constant total mass of fibres and sand of 1350g. The fibres incorporation ratios were 0% (without fibres), 1%, 2% and 3%, percent expressed in volume of mortar. A mixing procedure was carefully applied to achieve the best homogeneous dispersion of fibres in the mixture. With the usage of two different cement types and four replacement levels of fibres, eight mixtures were performed. The detailed compositions with the mixture references (cement symbols are followed by the ratios of fibres) are presented in Table 4.

Tab. Samples under accelerated carbonation After the period curing of 14 days, mortar samples were placed in the carbonation chamber, as showed in Fig. 2. Based on XP P 18-458 -French standard, present study proposes to use the carbonation chamber installed at 4% CO2 concentration, the temperature of 20°C and 65% relative humidity (RH). The time in the storage chamber is 12 weeks. Fig. 2. Carbonation storage chamber.

2.3.2.

pH indicator 1g phenolphthalein powder was dissolved in the mixture of 30 ml deionized water and 70ml ethyl alcohol. This solution was then used for the carbonation depth measurement in accordance with the guidance of RILEM Committee CPC-18 [START_REF]Measurement of hardened concrete carbonation depth[END_REF] and EN 13295. The clean freshly broken surface was sprayed with this solution. Due to the relative effects of the measuring time on the measured depth of carbonation after the reagent application, measurement one day after spraying was considered in present study when the limit between carbonated and non-carbonated zones was clearly demonstrated. The reagent could give a sufficiently clear colour change in sample to differentiate the neutralized areas. Phenolphthalein turns red-purple in the non-carbonated zone. Otherwise, it remains colourless in the carbonated zone (pH value in the range from 8.3 to 9.5 [START_REF] Lee | A Study for Carbonation Degree on Concrete using a Phenolphthalein Indicator and Fourier-Transform Infrared[END_REF]).

TGA-DSC measurement

During the sample preparation process for TGA-DSC (Thermogravimetric analysis -Differential scanning calorimetry), the grinding along the profile was carried out from the external (outside) surface to the central part of sample with an increment of 2 mm. Considering the 40 mm sample height, ten profile ground sections were obtained for each sample. Thermal gravimetric analysis of unreinforced and fibre-reinforced mortars was performed on a thermal analyser (STA 449 F5, NETZSCH) in the temperature range of 20°C -900°C in nitrogen atmosphere with the speed of 50ml/min. The rate of heating was kept constant at 20°C/min. Sample mass between 50 mg and 60 mg was put in a platinum crucible for this measurement. Due to the thermal decomposition of portlandite (CH) at the temperature ranging between 450°C and 550°C and that of calcium carbonate (CC) at the temperature ranging of 550°C -850°C, the amount of two minerals is expressed in following equations in accordance with refs. [START_REF] Wei | Correlation between hydration of cement and durability of natural fiber-reinforced cement composites[END_REF][START_REF] De Weerdt | Effect of carbonation on the pore solution of mortar[END_REF][START_REF] Deschner | Cement and Concrete Research Hydration of Portland cement with high replacement by siliceous fl y ash[END_REF]:

(1) (2)
When the carbonated process occurs, CH converts gradually into CC, depending on the rate of CO2 diffusivity. This means that the mass of CH decreases while the amount of CC increases when the carbonation process of mortar samples is happening. The carbonation degree could be determined based on the quantity of CC and CH. At the depth on which the appearance of both CH and CC is obtained, this zone is considered as carbonation zone. Otherwise, the amount of CC is negligible at the greater depth of sample from which the carbonation process almost does not occur. Hence, TGA analyses constitute a suitable and accurate method to evaluate the change in carbonation degree along the profile of sample sections. In this study, phenolphthalein indicator and TGA-DSC measurement were performed on mortar without fibres and with 2% fibres.

2.3.4.

Mechanical strength testing Mechanical strengths of mortars were conducted based on NF EN 196-1 standard. The bending test was performed on hardened 40 × 40 × 160 mm 3 prisms by means of an electromechanical press. Three samples were tested after the exposure period at a constant rate deformation of 0.05 kN/s with a capacity of 50 kN load cell. The compression test was performed on halves of the prismatic broken samples after the bending test, i.e., six samples. The same press machine was used. However, the load cell of 250kN was installed. As recommended by the NF EN 196-1 standard, the loading rate applied was 2.5 kN/s. The half-prisms tested were placed laterally in the centre of the bearing plate of the machine.

Experimental results and discussions 3.1. Depth of carbonation Reagent method

Fig. 3 and Fig. 4 show the typical images of the observed surface of samples after being covered by phenolphthalein reagent. As can be seen in these images, PC-based mortar makes the colour change in a part of sample, while CSA cementbased mortar does not present any change in colour in its whole surface. This means that PC-based mortar was only partly carbonated, while complete carbonation was observed in the mortar based on CSA cement. PC-based mortars showed relatively higher carbonation resistance but have a lower ratio of Ca/Si in comparison with CSA cement-based mortars. This behaviour could be justified in part by the difference in the amount of CaO and SiO2 between PC and CSA cement. In fact, CaO and SiO2 in cement convert into calcium silicate hydrate gel (CSH) and portlandite (Ca(OH)2) due to the hydration acceleration. More CH content results in higher alkalinity in the pore solution of mortar based on PC, which results in change of its colour to red-purple in the inner part of surface after the exposure to phenolphthalein reagent. On the other hand, due to whole CH content converting into CC, remaining colourless in carbonated mortars in CSA samples was obtained as shown in Fig. 3. This result also evidences that the carbonation resistance in CSA cement-based mortar was insignificantly governed by the dense microstructure formed by ettringite [START_REF] Ioannou | Performance characteristics of concrete based on a ternary calcium sulfoaluminate -anhydrite -fly ash cement[END_REF]. Some previous studies [START_REF] Winnefeld | Effect of relative humidity on the carbonation rate of portlandite , calcium silicate hydrates and ettringite[END_REF][START_REF] Black | Surface carbonation of synthetic C-S-H samples : A comparison between fresh and aged C-S-H using X-ray photoelectron spectroscopy[END_REF][START_REF] Li | Effects of Ca/Si ratio, aluminum and magnesium on the carbonation behavior of calcium silicate hydrate[END_REF]] also highlighted the ratio of Ca/Si influences on the carbonation behaviour. They reported that with a higher ratio of Ca/Si, a faster carbonation degree was obtained in consequence of the rapid decalcification of calcium silicate hydrate gel (CSH). Additionally, CaCO3 formation results from the decomposition of ettringite [START_REF] Fernández-Carrasco | Carbonation of ternary building cementing materials[END_REF][START_REF] Zhang | Carbonation resistance of sulphoaluminate cement-based high performance concrete[END_REF], which is the main phase of CSA cement, and induces to boost the depth of carbonation in CSA sample. Other studies [START_REF] Ashraf | Carbonation of cement-based materials: Challenges and opportunities[END_REF][START_REF] Ioannou | Performance characteristics of concrete based on a ternary calcium sulfoaluminate -anhydrite -fly ash cement[END_REF] also indicated that carbonation depth in concrete with CSA cement as the main binder is considerably higher than that in concrete manufactured with other binders such as Portland or blast furnace cement. This value, for example, in ref. [START_REF] Ioannou | Performance characteristics of concrete based on a ternary calcium sulfoaluminate -anhydrite -fly ash cement[END_REF], was up to 15 mm after a seven-week period of carbonation exposure compared with 6 mm and 5 mm of PC-based concrete and blast furnace cement-based concrete, respectively. Concerning the CO2 penetration, the fibres could improve the carbonation rate of mortar due to the high air content (the fibres act an air-entraining admixture), which induces CO2 penetration becomes easier. It can be observed from the images in Fig. 4, the carbonation degree of the control mortar is higher than that of mortar incorporating fibres, showed by a weaker colouration on the surface of reinforced mortar. When mortar with fibres addition is employed, the carbonation degree of mortar could be increased. To confirm the pH value reduction due to accelerated carbonation, an additional pH measurement was done with samples under natural curing. The technique used to obtain pH value of mortars bases on the recommendation of ASTM D4972 standard. A mixture of 10 g air dried mortar powders and 10 g distilled water was stirred within 12 hours at ambient temperature using a hotplate stirrer (UC125D, STUART) to facilitate lime powder in mortar to dissolve totally in water. This Carbonated zone Non-carbonated zone Carbonated zone solution obtained was then filtered with the filter paper of 25 μm pore size. The pH measurement of these mixtures was carried out by means of pH meter. The results of the pH value for mortars based on two types of cement are given in Fig. 5. It is observed that the pH of mortar based on CSA cement is significantly smaller than that of PC-based mortar due to the difference in pH of CSA cement and PC. In addition, the pH value of mortar seems to be not governed by fibre content. Fig. 5. Effect of type of cement on pH values for PC and CSA cement-based mortars. TGA-DSC measurement Fig. 6 illustrates the measured CC and CH contents using TGA-DSC measurement, depending on the depth from the surface to centre of sample. For PC2 sample, where a part of area was exposed by carbonation, a clear decrease in CC content was detected while the amount of CH measured was increased gradually. This means that the exposure to carbonation has significant effects on the degree of clinker reaction. This induces the two opposite trends in CC and CH contents from the outer surface to the middle part of the carbonated samples observed. More deeply, the CO2 penetration was more complicated and almost restricted in the outer sections due to the dense microstructure impeding. On the contrary, the concentration distribution of CH and CC in CSA2 sample, which was carbonated completely, was almost constant over the sample whole depth. Additionally, the amount of CC in the carbonated area of PC2 sample was higher than that of CSA2 sample, but lower in the non-carbonated area. A stronger carbonation degree for PC2 sample compared to CSA2 sample in outer sections was obtained. This observation could be explained by CaO content in the two types of binders. The higher content of CaO in PC could induce the higher content of CC in the carbonation area. Also, because of the low pH value and low porosity of CSA cement-based mortar [START_REF] Juenger | Advances in alternative cementitious binders[END_REF], the CO2 diffusivity becomes more difficult even in the outer sections. Nevertheless, the difference happened upon the non-carbonated area and the lower content of CC in PC2 sample was observed. The cause of this phenomenon is that clinker reaction under carbonation curing during 12 weeks increases the amount of CC created, which makes the higher content of CC in CSA2 sample than in PC2 sample within non-carbonation sections. Otherwise, due to the low CH content [START_REF] Kleib | The use of calcium sulfoaluminate cement to mitigate the alkali silica reaction in mortars[END_REF], the consumption of CH in CSA samples were almost completed after 12 weeks of accelerated carbonation. It may be viewed in Fig. 6, the different depth of carbonation according to two methods, i.e., one using phenolphthalein reagent and the other, the thermogravimetric analysis for sample powder profiles, was observed. The ratio between depth on which both CC and CH were detected, and the maximal depth of colourless zone was approximately three. A reasonable explanation for this difference may be the heterogeneity of carbonation front depth in the entire sample induced by the differences in the internal structure of composite and diffusivity of CO2 and the powder sampling approach [START_REF] De Weerdt | Effect of carbonation on the pore solution of mortar[END_REF]. Another possible cause is the carbonation degree. While phenolphthalein indicator remained colourless only in the fully carbonated zone, where the carbonation degree was above 50%, the appearance of both CH and CC was detected by TGA technique even in partly carbonation zone, i.e. where the carbonation degree is less than 50% [START_REF] Chang | The experimental investigation of concrete carbonation depth[END_REF]. The restricted use of the traditional method is to focus only on the change in the pore solution alkalinity of the carbonated sample, and apparently, it lacks the correlation with other criteria [START_REF] Bernal | Accelerated carbonation testing of alkaliactivated slag/metakaolin blended concretes: effect of exposure conditions[END_REF]. Therefore, the depth of colourless zone is mostly underestimated [START_REF] Qiu | A state-of-the-art review on the carbonation process in cementitious materials: Fundamentals and characterization techniques[END_REF][START_REF] Herrera | A comparison of methods for determining carbonation depth in fly ash-blended cement mortars[END_REF] and smaller considerably than the carbonation depth identified by other measurements. However, phenolphthalein method could offer a simple way to investigate the carbonation-induced difference concerning alkalinity that occurred at the surface of sample applied. In contrast, TGA method shows a sufficient level of measurement accuracy in relation to CH and CC concentrations distribution of CH and CC in each profile section. Concerning the PC mortar samples with and without coconut fibres, the 12-week carbonation reaction process improved the compressive strength by 8.7% and 32.7%, respectively. For CSA cement-based mortars, which were higher carbonation depth, the compressive strength of control and fibre-reinforced samples decreased slightly by 3% and 1.3%, respectively. The carbonation process of CSA cement-based mortars occurred on complete area tested, and then led to the degradation of ettringite in a short time. This could result in a slight decrease of compressive strength in CSA samples after 12 weeks of storage in carbonation chamber as shown in Fig. 7. Chi et al. [START_REF] Chi | Effects of carbonation on mechanical properties and durability of concrete using accelerated testing method[END_REF] also indicated a logical relationship between carbonation depth and mechanical strength since they concluded that the compressive strength decreased at higher carbonation depth. This could be contributed by the pore structure of composite, which gains remarkable changes after accelerated carbonation, controlling both the carbonation degree and compressive strength totally. It is clearly seen that this relation depends significantly on the type of binder used in composite [START_REF] Leemann | Carbonation resistance of recycled aggregate concrete[END_REF][START_REF] Elsalamawy | The role of relative humidity and cement type on carbonation resistance of concrete[END_REF][START_REF] Fattuhi | Carbonation of concrete as affected by mix constituents and initial water curing period[END_REF]. Thus, compressive resistance should not be used as a main determinant of carbonation resistance since samples contain different binder types. The effects of carbonation on flexural strength are different for both CSA cement and PC-based mortars. A relative drop in flexural resistance was observed in mortar containing CSA cement after carbonation storage. Accelerated carbonationinduced flexural strength of CSA cement-based mortar was only approximately 6 -6.5 MPa, either incorporating fibres or Curing in air Carbonation curing without fibres. Flexural strength of CSA mortar samples incorporating fibres decreased significantly by nearly a half. This may be due to the cumulated effects of incorporating fibres and tensile strength of cementitious matrix (loss of ettringite), which affects remarkably the bending behaviour. Otherwise, flexural strength of PC-based mortar showed a slight upward trend, regardless of fibre content.

Mechanical properties

Comparison with PC based-mortar without fibres, PC based-mortar incorporating fibres showed a higher development in both compressive and flexural strengths. This phenomenon can be caused by the higher porosity in fibre-reinforced mortar, which is a convenient environment for CO2 to penetrate deeply into sample, and the carbonation degree occurs more rapidly than in reference mortar. It is reported that the carbonation reaction seems to be governed considerably by CO2 diffusion process. Porosities in sample are filled and the sample becomes denser after accelerated carbonation is applied in mortar samples [START_REF] Pizzol | Mineralogical and microstructural changes promoted by accelerated carbonation and ageing cycles of hybrid fiber -cement composites[END_REF][START_REF] Pizzol | Effect of accelerated carbonation on the microstructure and physical properties of hybrid fiber-cement composites[END_REF]. It is clear that the presence of fibres plays a critical role in the change of mechanical characteristics of mortar samples. The establishment of carbonation could result in the decrease of the alkaline environment prematurely in cementitious matrix. This critical advantage provides more chemical stability to the cellulose fibres due to a less aggressive environment [START_REF] Pizzol | Effect of accelerated carbonation on the microstructure and physical properties of hybrid fiber-cement composites[END_REF]. In addition, after the accelerated carbonation is applied, the bonding between natural fibres and cementitious matrix is improved, resulting in a drop of the voids surrounding fibres. Due to this, mortars at higher fibre content could give better mechanical performance.

Thermogravimetric analyse

Fig. 8. TGA and DSC graphs for different profiles of PC2 sample after accelerated carbonation. The results of thermogravimetric analysis and differential scanning calorimetry (TGA-DSC) from the surface to centre part of PC2 sample used in the carbonation test are presented in Fig. 8. With these graphs, the weight loss is identified in Fig. 9. Concerning the derivative of the thermogravimetric curve, three decomposition peaks were obtained, including CSH, CH and CC. The first two peaks indicate that CSH and CH content in the inner (inside) profile ground sections are higher than in outer ones (near the surface) while the last peak concerns the decomposition of CC, showing the amount of CC in outer sections is higher than that in inner ones. Due to the negligible mass of fibre powder compared to cementitious matrix powder, the decomposition peak regarding cellulose of fibre, which occurs in the temperature range from 300 to 360°C [START_REF] Henrique | Influence of the initial moisture content on the carbonation degree and performance of fiber-cement composites[END_REF][START_REF] Bui | Determination and Review of Physical and Mechanical Properties of Raw and Treated Coconut Fibers for Their Recycling in Construction Materials[END_REF], was almost observed. When the coconut fibres are employed in reinforcement, these fibres are the lightest components in the mortar mixture, and the influence on the powder components is difficult to recognize. These quantity changes result from the reaction between HCO 3-ions, which dissociated by the reaction of CO2, and Ca 2+ ions, and released by dissolution of CH, to form CC within the carbonation curing. The decomposition peaks of CSH and CH mainly occurred in non-carbonated and part-carbonated sections in the temperature range from 100 -150°C to 400 -500°C, respectively, and the strongest in PC2-16-18mm and PC2-18-20 mm profiles. While CC peaks almost occurred in carbonated areas in the temperature range from 700 -800°C, the strongest peaks were found in PC2-0-2 mm and PC2-2-4mm profiles. In terms of weight loss, at the first of the decomposition period, both non-carbonated and carbonated areas have very similar thermal behaviour. The pyrolysis behaviours of ten profiles were almost the same. As can be seen from the data results, carbonated sections present higher resistance to temperature than other ones from ambient to 550°C. This could be explained because carbonated sections have higher CC content decomposed only after 550°C. A significant difference in the pyrolysis behaviour was observed from 550°C when the weight loss after this temperature level was characterized by carbonation. The steeper curves after 550°C for the outer profiles thus clearly indicate the attack of CO2 and the higher rate of weight loss due to accelerated carbonation [START_REF] Rostami | Microstructure of cement paste subject to early carbonation curing[END_REF]. The endothermal effects could be divided into four main steps: (I) 20 -110°C (evaporation and CSH), (II) 110 -460°C (CSH and cellulose), (III) 460 -760°C (portlandite and poor-crystallized carbonates) and (IV) 760 -900°C (well-crystallized carbonates). In detail, at the first step from ambient temperature to 110°C, decomposition of sections occurs slowly since the mass loss was about 2% for all profiles and mainly concerned free water content. The weight

I II III IV

Poor-crystalized carbonates loss changes at the temperature range between 110 and 200 °C significantly relate to the consumption of calcium silicate hydrate gel (CSH) advanced by the carbonation process with the CO2 absorption capacity of the cement matrix. However, the degradation process happens mainly between 110°C and 760°C that a mass loss of 8% was observed. The step, happening between 350°C and 400°C, relates to cellulose degradation, and nearly no change was found between the tested profiles.

Then the development of endothermal peaks from 450 to 550 °C correlates with the decomposition of CH. From 550°C, the changes were observed between the carbonated areas. The degradation process continues slowly at the last stage with the weight loss of about 2% for outer profiles and approximately 1% for inner ones. It gets a stable value of residue left at 900°C. At the end of the decomposition period, residual weight at 900°C of outermost profiles ranged of 86-90% is against slightly more than 90% of most in-depth profiles. A possible explanation for this could be that at the high temperature, the decomposition was in regarded to CC formed from the carbonation process in carbonated areas. Otherwise, the decomposition at this temperature in uncarbonated areas is mainly due to the amount of calcium-carbonated filler, which is already in the component of conventional cement to decrease the cost in cement production industry [START_REF] Pizzol | Effect of accelerated carbonation on the microstructure and physical properties of hybrid fiber-cement composites[END_REF]. As a result of thermal analysis, it is observed that carbonated areas have a poorer ability to resist high temperature than other ones due to the strong CC decomposition in outer profile ground sections.

Fig. 9: Mass loss for different profiles of PC2 sample and transition temperature ranges.

Conclusion

Coconut fibre-reinforced mortar was prepared using two different cement types, i.e., PC and CSA cement. The carbonation resistance was investigated in laboratory conditions. The comparison in terms of carbonation degree and mechanical performance between types of mortars was carried out. According to the test results, several conclusions can be summarised: -Using CSA cement with lower alkali content could lead to a notable reduction in carbonation resistance due to the lower content of CaO compared to conventional cement. While the large inner surface area of PC-based mortar presents in redpurple, all broken surface of CSA cement-based mortar shows in colourless when they are sprayed by phenolphthalein reagent after the carbonation exposure period of 12 weeks.

-Incorporating natural fibres into mortar could lead to an increase in the carbonation degree due to the additional presence of air entrainment that supports strongly for CO2 penetration into fibre-reinforced mortar. -A difference in carbonation degree was obtained between two measurement methods used: phenolphthalein reagent and TGA measurement. The depth of colourless zone determined by phenolphthalein is nearly three times smaller than the carbonation depth obtained by TGA technique. Several drawbacks of the conventional method by the reagent were reported, but this simple method still suits to evaluate the change in alkalinity of pore solution in samples after the carbonation exposure. Meanwhile, TGA technique could give accurate results even if in the case powdered samples are representative only of different small areas, not of the entire sample.

-The carbonation has different effects on the mechanical performance of mortar. While carbonation-induced compressive strength of PC-based mortar increases to 32.7%, that of all mortars based on CSA cement decreases insignificantly. However, in bending, the flexural strength ratio of CSA cement-based mortar before and after carbonation storage is nearly two due to the cumulated effect of fibres added and loss of ettringite of CSA cement-base samples during the carbonation process. It should be noted that for composites containing different binders, mechanical behaviour could be not a key factor in evaluating carbonation resistance.

-Three decomposition peaks were clearly observed corresponding three decompositions, namely of CSH, CH and CC in the PC2 sample, respectively. Meanwhile, inner profile (centre part of sample), are concerned by the two first decomposition peaks and outer (near surface) ones are responsible for the last decomposition peak of CC. This means that from the external surface to centre part of PC2 sample, the increase in CSH and CH contents and decrease CC content were obtained.

-The thermal stability of the non-carbonated area is slightly higher than that of the carbonated area. For instance, at 900°C, the mass loss of the carbonated area is approximately 10 -14%, while this value of the non-carbonated area is less than 10%. This could be due to the formation of CC, which is decomposed at the higher temperature during the carbonation process in the carbonated area. Other, the decomposition at this temperature mainly depends on the calcium-carbonated filler content in the production of conventional cement.

Fig. 1 .

 1 Fig. 1. Appearance of fibres before mixing. Tab. 2. Fundamental properties of coconut fibre [62]. Parameters Value Diameter (mm) 0.250 Length (mm) 10 -20 Absolute density (g/cm 3 ) 1.41 Water absorption (%) (after immersion of 48 hours) 133 Tensile failure (MPa)123.6 Failure strain (%) 26.9 Water content (%) 7.60 Mortars were manufactured based on two types of cement, CEM I 52.5 N type I Portland Cement (PC) and Calcium Sulfoaluminate cement (CSA Cement). CSA cement, containing approximately 55% of calcium sulfoaluminate, is considered as a green binder due to its environmentally friendly characteristics, i.e., 37% lower of CO2 emission emitted than PC[START_REF] Juenger | Advances in alternative cementitious binders[END_REF]. Indeed, CSA cement is produced at a lower calcined temperature and less energy requirement for grinding

Fig. 3

 3 Fig. 3 Images of CSA (left) and CSA2 (right) samples surfaces after application of phenolphthalein solution.

Fig. 4 .

 4 Fig. 4. Images of PC (left) and PC2 (right) samples surfaces after application of phenolphthalein solution.To confirm the pH value reduction due to accelerated carbonation, an additional pH measurement was done with samples under natural curing. The technique used to obtain pH value of mortars bases on the recommendation of ASTM D4972 standard. A mixture of 10 g air dried mortar powders and 10 g distilled water was stirred within 12 hours at ambient temperature using a hotplate stirrer (UC125D, STUART) to facilitate lime powder in mortar to dissolve totally in water. This

Fig. 6 .

 6 Fig.6. CH and CC concentrations distribution over the depth of PC2 and CSA2 samples.3.2. Mechanical properties

Fig. 7 .

 7 Fig. 7. Mechanical strengths of mortar samples cured in two different conditions.Concerning the PC mortar samples with and without coconut fibres, the 12-week carbonation reaction process improved the compressive strength by 8.7% and 32.7%, respectively. For CSA cement-based mortars, which were higher carbonation depth, the compressive strength of control and fibre-reinforced samples decreased slightly by 3% and 1.3%, respectively. The carbonation process of CSA cement-based mortars occurred on complete area tested, and then led to the degradation of ettringite in a short time. This could result in a slight decrease of compressive strength in CSA samples after 12 weeks of storage in carbonation chamber as shown in Fig.7. Chi et al.[START_REF] Chi | Effects of carbonation on mechanical properties and durability of concrete using accelerated testing method[END_REF] also indicated a logical relationship between carbonation depth and mechanical strength since they concluded that the compressive strength decreased at higher carbonation depth. This could be contributed by the pore structure of composite, which gains remarkable changes after accelerated carbonation, controlling both the carbonation degree and compressive strength totally. It is clearly seen that this relation depends significantly on the type of binder used in composite[START_REF] Leemann | Carbonation resistance of recycled aggregate concrete[END_REF][START_REF] Elsalamawy | The role of relative humidity and cement type on carbonation resistance of concrete[END_REF][START_REF] Fattuhi | Carbonation of concrete as affected by mix constituents and initial water curing period[END_REF]. Thus, compressive resistance should not be used as a main determinant of carbonation resistance since samples contain different binder types. The effects of carbonation on flexural strength are different for both CSA cement and PC-based mortars. A relative drop in flexural resistance was observed in mortar containing CSA cement after carbonation storage. Accelerated carbonationinduced flexural strength of CSA cement-based mortar was only approximately 6 -6.5 MPa, either incorporating fibres or

  CC (%wt)The amount of CH (%wt) Depth from surface to center of sample (mm)

Table 1 .

 1 A literature review concerning accelerated carbonation.

	Reference	Binder(s)		Testing conditions		Properties
				Concentration of	Relative	Temperature	Max.	investigated
				CO2 (%)	humidity	(°C)	exposure
					RH (%)		time
	XP P 18-458			50 ± 5	65 ± 5	20 ± 3	1 week	Carbonation
	Accelerated							depth
	carbonation test -
	French standard
	EN 12390-10 -			0.040 ± 0.001	65 ± 2	20 ± 2	1 year	Carbonation
	European							depth
	standard
	EN 13295 -			1	60 ± 10	21 ± 2	56 days	Carbonation
	European							depth
	standard
	Lee et al.[11]	PC, fly ash	5	60	20	1 year	Carbonation
								degree
	Georget et al.[12]	PC		0.040	70	Not	6 months Micro-structure
						mentioned		characterisation
	RILEM			0.03	65	20	4 years	Carbonation
	Committee CPC-							depth
	18 [13]
	PERFDUB			3 ± 0.5	65 ± 5	20 ± 2	70 days	Carbonation
	project [15]							depth
	EN 12390-12 -			3 ± 0.5	57 ± 3	20 ± 2	70 days	Carbonation
	European							depth
	standard
	Tang et al.[16]	42.5 PC		20 ± 2	70 ± 5	20 ± 5	4 weeks	Carbonation
								depth
	Qin et al.[17]	42.5 PC, fly	20	60 ± 5	20 ± 3	4 hours	Mineral
		ash,	GGBS,					composition,
		limestone					mechanical
		powder						properties,
	De Weerdt et	PC, fly ash	1	60	20	9 weeks	Carbonation
	al.[18]							depth, moisture
								content, mineral
								composition
	Ashraf et al.[19]	Calcium		15	94	35, 45 and	145	Carbonation
		silicates				60	hours	kinetics
				100	94	60	120	Microscale
							hours	mechanical
								performance
				100	94	60	145	Macroscale
							hours	mechanical
								performance
	Leemann et	PC, fly ash,	4	57	20	9 weeks	Mechanical
	al.[20]	slag						properties,
								carbonation
								coefficient
	Zhu et al.[21]	PC		100	70	20	24 hours	Mechanical
								properties,
								mineral
								composition
	Romildo et al.[22] OPC		9.8	64.3	26.7	109 days	Durability
	Tonoli et al.[23]	PC		100	75	20	7 days	Physical	and
								mechanical
								properties
	Tomography et	PC		20 ± 3	70 ± 5	20 ± 5	28 days	Carbonation
	al.[24]							degree,	crack
								distribution, pore
								structure
	Neves Junior et	High	initial	20	60	25	24 hours	Mechanical and
	al.[25]	strength	and					porosity
		sulfate-resistant					properties
		PC
	Mi et al.[26]	Cement		0.04	70	20	28 days	Distribution of
								carbonation

Note: (O)PC = (Ordinary)Portland cement.

  [START_REF] Sindhu | Degradation studies of coir fiber/polyester and glass fiber/polyester composites under different conditions[END_REF]. Mix proportion of coconut fibres cementitious composite.

	Mortar reference	PC (control)	PC1 (1% fibre)	PC2 (2% fibre)	PC3 (3% fibre)	CSA (control)	CSA1 (1% fibre)	CSA2 (2% fibre)	CSA3 (3% fibre)
	PC (g)	450	450	450	450
	CSA cement					450	450	450	450
	(g)
	Water (g)	225	225	225	225	225	225	225	225
	Sand (g)	1350	1339.20	1328.41	1317.61	1350	1339.20	1328.41	1317.61
	Superplasticizer	4.5	4.5	4.5	4.5	4.5	4.5	4.5	4.5
	(g)
	Raw fibres (g)	0	10.80	21.59	32.39	0	10.80	21.59	32.39
	2.3. Experimental methods
	2.3.1.

Acknowledgements

The first author would like to thank Vietnamese government scholarship -911 Project -for the provision of PhD study funding.


Giovanni Brigati
email: brigati@ceremade.dauphine.fr

Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

Nikita Simonov
email: nikita.simonov@sorbonne-universite.fr

Sobolev Stability

Inequality

Stability for the logarithmic Sobolev inequality

Keywords: logarithmic Sobolev inequality, stability, log-concavity, heat flow, entropy, carré du champ. 2020 MSC: Primary: 39B62; Secondary: 47J20, 49J40, 35A23, 35K85

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Introduction and main results

Let us consider the Gaussian logarithmic Sobolev inequality

∇u 2 L 2 (R d ,d γ) ≥ 1 2 R d u 2 log ⎛ ⎝ u 2 u 2 L 2 (R d ) ⎞ ⎠ d γ ∀u ∈ H 1 (R d , d γ) ( 1
)
where d γ = γ(x) d x is the normalized Gaussian probability measure with density

γ(x) = (2π) -d 2 e -1 2 x 2 ∀ x ∈ R d .
In this paper we are interested in stability results, that is, in estimating the difference of the two terms in (1) from below, by a distance to the set of optimal functions. According to [START_REF] Carlen | Superadditivity of Fisher's information and logarithmic Sobolev inequalities[END_REF][START_REF] Émery | Diffusions hypercontractives[END_REF], equality in (1) is achieved by functions in the manifold

M ∶= w a,c ∶ (a,c) ∈ R d × R
where w a,c (x) = c e -a⋅x ∀ x ∈ R d and only by these functions. The ultimate goal of stability estimates is to find a notion of distance d, an explicit constant β > 0 and an explicit exponent α > 0, which may depend on d, such that

∇u 2 L 2 (R d ,d γ) - 1 2 R d u 2 log ⎛ ⎝ u 2 u 2 L 2 (R d ) ⎞ ⎠ d γ ≥ β inf w∈M d(u, w) α (S )
for any given u ∈ H 1 (R d , d γ). In this paper we consider the slightly simpler question of finding a specific w u ∈ M such that

∇u 2 L 2 (R d ,d γ) - 1 2 R d u 2 log ⎛ ⎝ u 2 u 2 L 2 (R d ) ⎞ ⎠ d γ ≥ β d(u, w u ) α , (⋆)
which provides us with no more than an estimate for (S ): any estimate of α and β for (⋆) is also an estimate for (S ). In order to illustrate the difference between the two questions, let us consider the following elementary example. Assume that d = 1 and consider the functions u ε (x) = 1 + ε x in the limit as ε → 0. With d(u, w) = u ′w ′ L 2 (R,dγ) , which is the strongest possible notion of distance that we can expect to control in (⋆), elementary computations show that the deficit of the logarithmic Sobolev inequality, i.e., the left hand-side in (⋆), is

∇u ε 2 L 2 (R d ,d γ) - 1 2 R d u ε 2 log ⎛ ⎝ u ε 2 u ε 2 L 2 (R d ) ⎞ ⎠ d γ = 1 2 ε 4
+ O ε 6 , while, using the test function w a ε ,c ε ∈ M where a ε = 2 ε and c ε = e -a 2 ε 4 , we obtain

d (u ε , 1) 2 = u ′ ε 2 L 2 (R,dγ) = ε 2 and inf w∈M d(u ε , w) α ≤ d (u ε , w a ε ,c ε ) 2 = 1 2 ε 4 + O ε 6 .
In practice we will consider only the case

w u = 1
in (⋆) and the above example shows that the best we can hope for without additional restriction is α ≥ 4. Similar examples in higher dimensions can be obtained by considering for an arbitrary given ν ∈ S d -1 the functions u ε (x) = 1 + ε x ⋅ ν in the limit as ε → 0. This is not a surprise in view of [START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF][START_REF]Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities[END_REF][START_REF] Indrei | Sharp stability for LSI[END_REF], and also of the detailed Taylor expansions of [START_REF] Frank | Degenerate stability of some Sobolev inequalities[END_REF][START_REF] Brigati | Logarithmic Sobolev and interpolation inequalities on the sphere: constructive stability results[END_REF][START_REF]On Gaussian interpolation inequalities[END_REF]. Still with w u = 1, we can expect to have α = 2 in (⋆) under additional conditions, including for d(u, w) = ∇u -∇w L 2 (R d ,d γ) , while it is otherwise banned as shown for instance from [39, Theorem 1.2 (2)], or simply from considering the above example. Before entering the details, let us mention a recent stability result for (S ) with α = 2 involving a constructive although very delicate expression for β > 0 and d(u, w) = uw L 2 (R d ,d γ) that appeared in [START_REF] Dolbeault | Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence[END_REF]. Here we aim at stronger estimates under additional constraints, with w u = 1, which is a different point of view. Let us start by a first stability result.

Proposition 1. For all u ∈ H 1 (R d , d γ) such that u L 2 (R d ) = 1 and x u 2 L 2 (R d ) ≤ d , we have ∇u 2 L 2 (R d ,d γ) - 1 2 R d u 2 log u 2 d γ ≥ 1 2 d R d u 2 log u 2 d γ 2 (2)
and, with ψ(s) ∶= sd 4 log 1 + 4 d s , we also have the stronger estimate

∇u 2 L 2 (R d ,d γ) - 1 2 R d u 2 log u 2 d γ ≥ ψ ∇u 2 L 2 (R d ,d γ) . (3)
Similar results are already known in the literature (see for instance [START_REF] Bobkov | Bounds on the deficit in the logarithmic Sobolev inequality[END_REF][START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF][START_REF]Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities[END_REF][START_REF] Indrei | Sharp stability for LSI[END_REF]) and we claim no originality for the the results. Also see references to earlier proofs at the end of the introduction. Our method is based on the carré du champ method.

Even if some ideas go back to [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF], it is elementary, new as far as we know, and of some use for our other results.

Coming back to (⋆), we may notice that there is no loss of generality in imposing

the condition u L 2 (R d ) = 1, as we can always replace u by u u L 2 (R d ) . Because of the Csiszár-Kullback-Pinsker inequality R d u 2 log u 2 d γ ≥ 1 4 u 2 -1 2 L 1 (R d ,d γ) (4) and u -1 = u 2 -1 u + 1 ≤ u 2 -1 , we find that (2) implies (⋆) type with d(u, w) = u -w L 1 (R d ,d γ)
for nonnegative functions u, α = 4, and β = 1 (32d). For functions far away from the optimal functions, say such that ∇u L 2 (R d ,d γ) ≥ A under the conditions of Proposition 1, Inequality (3) provides us with an even stronger stability result of (⋆) type with α = 2 and d(u, w) = ∇u -∇w L 2 (R d ,d γ) , but with a positive constant β which depends on A > 0. Again, notice that (⋆) with such a distance cannot hold without constraints.

Next we aim at explicit results with α = 2, under other constraints. Let

C ⋆ = 1 + 1 1728 ≈ 1.0005787 . Theorem 2. For all u ∈ H 1 (R d , d γ) such that u 2 γ is log-concave and such that R d (1, x) u 2 d γ = (1,0) and R d x 2 u 2 d γ ≤ d , ( 5
)
we have

∇u 2 L 2 (R d ,d γ) - C ⋆ 2 R d u 2 log u 2 d γ ≥ 0 . ( 6
)
The condition ∫ R d x 2 u 2 d γ ≤ d in ( 5) is a simplifying assumption. A result like (6) also holds if ∫ R d x 2 u 2 d γ > d , but with a constant that differs from C ⋆ and actually depends on ∫ R d x 2 u 2 d γ. We refer to Section 3.5: see Proposition 7 for an exten- sion of Theorem 2, and also for further comments on the extension of Proposition 1.

The constant C ⋆ in (8) relies on an estimate of [START_REF] Bobkov | Isoperimetric and analytic inequalities for log-concave probability measures[END_REF]. Inequality [START_REF] Ledoux | Lévy-Gromov's isoperimetric inequality for an infinitedimensional diffusion generator[END_REF] with improved constant C ⋆ > 1 compared to (1) can be recast in the form of a stability inequality of type (⋆) around the normalised Gaussian as

∇u 2 L 2 (R d ,d γ) - 1 2 R d u 2 log u 2 d γ ≥ 1 2 (C ⋆ -1) R d u 2 log u 2 d γ for all functions u ∈ H 1 (R d , d γ) such that u L 2 (R d ) = 1, which covers the case α = 2, β = (C ⋆ -1) 8 and d(u, w) = u -w L 1 (R d ,d γ) in (⋆) for nonnegative functions by (4),
or even in the stronger Ḣ1 (R d , d γ) semi-norm, as

∇u 2 L 2 (R d ,d γ) - 1 2 R d u 2 log u 2 d γ ≥ C ⋆ -1 C ⋆ ∇u 2 L 2 (R d ,d γ) for all functions u ∈ H 1 (R d , d γ) such that u L 2 (R d ) = 1, which corresponds to α = 2, β = (C ⋆ -1) C ⋆ and d(u, w) = ∇(u -w) L 2 (R d ,d γ) in (⋆)
. By the Gaussian Poincaré inequality, notice that the case of (⋆) with

α = 2, β = (C ⋆ -1) C ⋆ and the standard distance d(u, w) = u -w L 2 (R d ,d γ) is also covered.
Log-concavity might appear as a rather restrictive assumption, but this is useful because a function which is compactly supported at time t = 0 evolves through the diffusion flow into a logarithmically concave function after some finite time that can be estimated by the heat flow estimates of [START_REF] Lee | Geometrical properties of solutions of the porous medium equation for large times[END_REF]. This is enough to produce a stability result with an explicit constant. Compact support is in fact a too restrictive condition and we have the following result.

Theorem 3. Let d ≥ 1. For any ε > 0, there is some explicit C > 1 depending only on ε such that, for any u ∈ H 1 (R d , d γ) satisfying (5) and

R d u 2 e ε x 2 d γ < ∞, (7)
then we have

∇u 2 L 2 (R d ,d γ) ≥ C 2 R d u 2 log u 2 d γ . ( 8
)
Additionally, if u is compactly supported in a ball of radius R > 0, then (8) holds with

C = 1 + C ⋆ -1 1 + C ⋆ R 2 .
This expression of the constant C in ( 8) is given in the proof, in Section 3.4. The simpler estimate in terms of R relies on Theorem 2.

Let us conclude this introduction with a review of the literature. The logarithmic Sobolev inequality historically appeared in [START_REF] Stam | Some inequalities satisfied by the quantities of information of Fisher and Shannon[END_REF][START_REF] Blachman | The convolution inequality for entropy powers[END_REF], in a form that was later rediscovered as an equivalent scale-invariant form of the Euclidean version of the inequality in [START_REF] Weissler | Logarithmic Sobolev inequalities for the heat-diffusion semigroup[END_REF]. We refer to [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF] for the Gaussian version (1) of the inequality, and also to [START_REF] Federbush | Partially alternate derivation of a result of Nelson[END_REF] for an equivalent result. The optimality case in the inequality has been characterized in [START_REF] Carlen | Superadditivity of Fisher's information and logarithmic Sobolev inequalities[END_REF] but can also be deduced from [START_REF] Émery | Diffusions hypercontractives[END_REF]. Also see [START_REF] Villani | A short proof of the "concavity of entropy power[END_REF] for a short introductory review with an emphasis on information theoretical aspects. The logarithmic Sobolev inequality can be viewed as a limit case of a family of the Gagliardo-Nirenberg-Sobolev (GNS) as observed in [START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], in the Euclidean setting, or as a large dimension limit of the Sobolev inequality according to [START_REF] Beckner | Sobolev inequalities, the Poisson semigroup, and analysis on the sphere S n[END_REF]. See [START_REF] Dolbeault | Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence[END_REF][START_REF]On Gaussian interpolation inequalities[END_REF] for recent developments and further references. We refer to [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF][START_REF] Guionnet | Lectures on logarithmic Sobolev inequalities[END_REF][START_REF] Royer | An initiation to logarithmic Sobolev inequalities[END_REF][START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] to reference books on the topic. In a classical result on stability in functional inequalities, Bianchi and Egnell proved in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] that the deficit in the Sobolev inequality measures the Ḣ1 (R d , d x) distance to the manifold of the Aubin-Talenti functions. The estimate has been made constructive in [START_REF] Dolbeault | Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence[END_REF] where a new L 2 (R d , d x) stability result for the logarithmic Sobolev inequality is also established (also see [START_REF] Indrei | Sharp stability for LSI[END_REF] for furrther results in strong norms). Still in the Euclidean setting a first stability result in strong norms for the logarithmic Sobolev inequality appears in [START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF], where the authors give deficit estimates in various distances for functions inducing a Poincaré inequality. Under the condition x u L 2 (R d ,d γ) = √ d , a stability result measured by an entropy is given in [START_REF]Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities[END_REF]. For sequential stability results in strong norms, we refer to [START_REF] Indrei | Deficit estimates for the logarithmic Sobolev inequality[END_REF] when assuming a bound on u in L 4 (R d , d γ) and to [START_REF] Indrei | Sharp stability for LSI[END_REF] when assuming a bound on x 2 u in L 2 (R d , d γ). Stability according to other notions of distance has been studied in [START_REF]A Stein deficit for the logarithmic Sobolev inequality[END_REF][START_REF] Ledoux | Stein's method, logarithmic Sobolev and transport inequalities[END_REF][START_REF] Feo | Some remarks on the stability of the log-Sobolev inequality for the Gaussian measure[END_REF].

To our knowledge, the first result of stability for the logarithmic Sobolev inequality is a reinforcement of the inequality due to E. Carlen in [START_REF] Carlen | Superadditivity of Fisher's information and logarithmic Sobolev inequalities[END_REF] where he introduces an additional term involving the Wiener transform. Stability in logarithmic Sobolev inequality is related to deficit in Gaussian isoperimetry and we refer to [START_REF] Bobkov | Bounds on the deficit in the logarithmic Sobolev inequality[END_REF] for an introduction to early results in this direction, [START_REF] Barchiesi | Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality[END_REF] for a sharp, dimension-free quantitative Gaussian isoperimetric inequality, and [START_REF] Eldan | Stability of the logarithmic Sobolev inequality via the Föllmer process[END_REF] for recent results and further references. Results of Proposition 1 are known from [14, Theorem 1.1] where it is deduced from the HWI inequality due to F. Otto and C. Villani [START_REF] Otto | Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality[END_REF]. Such estimates have even been refined in [START_REF] Eldan | Stability of the logarithmic Sobolev inequality via the Föllmer process[END_REF]. There are several other proofs. In [START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF], M. Fathi, E. Indrei and M. Ledoux use a Mehler formula for the Ornstein-Uhlenbeck semigroup and Poincaré inequalities. The proof in [START_REF]Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities[END_REF] is based on simple scaling properties of the Euclidean form of the logarithmic Sobolev inequality, which also apply to Gagliardo-Nirenberg inequalities. Various stability results have been proved in Wasserstein's distance: we refer to [START_REF] Indrei | A quantitative log-Sobolev inequality for a two parameter family of functions[END_REF][START_REF] Bobkov | Bounds on the deficit in the logarithmic Sobolev inequality[END_REF][START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF][START_REF] Indrei | Deficit estimates for the logarithmic Sobolev inequality[END_REF][START_REF] Kim | Instability results for the logarithmic Sobolev inequality and its application to related inequalities[END_REF][START_REF] Bolley | Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities[END_REF][START_REF] Eldan | Stability of the logarithmic Sobolev inequality via the Föllmer process[END_REF][START_REF] Indrei | Sharp stability for LSI[END_REF]. A key argument for Theorem 2 is the fact that the heat flow preserves log-concavity according, e.g., [START_REF] Saumard | Log-concavity and strong log-concavity: a review[END_REF], which is a pretty natural property in this framework: see for instance [START_REF] Courtade | Quantitative stability of the entropy power inequality[END_REF].

In this paper, we carefully distinguish stability results of type (S ) where stability is measured w.r.t. M , and of type (⋆) where the distance to a given function is estimated. Even if this function is normalized and centered, this is not enough as shown in [START_REF] Indrei | Sharp stability for LSI[END_REF]. Many counter-examples to stability are known, involving Wasserstein's distance for instance in [START_REF] Indrei | A quantitative log-Sobolev inequality for a two parameter family of functions[END_REF][START_REF] Bobkov | Bounds on the deficit in the logarithmic Sobolev inequality[END_REF][START_REF] Fathi | Quantitative logarithmic Sobolev inequalities and stability estimates[END_REF][START_REF] Indrei | Deficit estimates for the logarithmic Sobolev inequality[END_REF][START_REF] Kim | Instability results for the logarithmic Sobolev inequality and its application to related inequalities[END_REF], weaker distances like p-Wasserstein, or stronger norms like L p or H 1 : see for instance [START_REF] Indrei | Deficit estimates for the logarithmic Sobolev inequality[END_REF][START_REF] Indrei | Sharp stability for LSI[END_REF]. The main counter-examples which we might try to apply to our setting are [START_REF] Kim | Instability results for the logarithmic Sobolev inequality and its application to related inequalities[END_REF]Theorem 1.3] and [START_REF] Eldan | Stability of the logarithmic Sobolev inequality via the Föllmer process[END_REF]Theorem 4] but, as already noted in [START_REF] Indrei | Deficit estimates for the logarithmic Sobolev inequality[END_REF], they are based on the fact that the second moment diverges along a sequence of test functions, which is forbidden in our assumptions.

This paper is organized as follows. Section 2 is devoted to the standard carré du champ method and a proof of Proposition 1. Theorem 2 is proved in Section 3.3, under a log-concavity assumption. Using properties of the heat flow, the method is extended to the larger class of functions of Theorem 3 in Section 3.4.

Entropy methods and entropy -entropy production stability estimates

This section is devoted to the proof of Proposition 1.

Definitions and preliminary results

Consider the

Ornstein-Uhlenbeck equation on R d ∂h ∂t = L h , h(t = 0, ⋅) = h 0 , (t, x) ∈ R + × R d , (9)
where L h ∶= ∆h -x ⋅ ∇h denotes the Ornstein-Uhlenbeck operator.

Let us recall some classical results. If

h 0 ∈ L 1 (R d , d γ)
is nonnegative, then there exists a unique nonnegative weak solution to (9) (see for instance [START_REF] Evans | Partial differential equations[END_REF]). The two key properties of the Ornstein-Uhlenbeck operator are

R d v 1 (L v 2 )dγ = - R d ∇v 1 ⋅ ∇v 2 d γ and [∇,L ] v = -∇v .
As a consequence, we obtain the two identities

R d (L v) 2 d γ = R d Hess v 2 d γ + R d ∇v 2 d γ (10)
and

R d L v ∇v 2 v d γ = -2 R d Hess v ∶ ∇v ⊗ ∇v v d γ + R d ∇v 4 v 2 d γ , (11)
where

= L v + ∇v 2 v . ( 12
)
Let us fix v L 2 (R d ) = 1, then the entropy and the Fisher information, respectively defined by

E [v] ∶= R d v 2 log v 2 d γ and I [v] ∶= R d ∇v 2 d γ ,
evolve along the flow according to [START_REF] Bobkov | Isoperimetric and analytic inequalities for log-concave probability measures[END_REF]. Using [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] and [START_REF] Blachman | The convolution inequality for entropy powers[END_REF], we obtain the classical expression of the carré du champ method

d d t E [v] = -4I [v] and d d t I [v] = -2 R d (L v) 2 + L v ∇v 2 v d γ if v solves
d d t I [v] + 2 I [v] = -2 R d Hess v - ∇v ⊗ ∇v v 2 d γ ( 13
)
as for instance in [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF][START_REF] Dolbeault | On the Bakry-Emery criterion for linear diffusions and weighted porous media equations[END_REF][START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF]. By writing that

d d t I [v] - 1 2 E [v] ≤ 0 and lim t →+∞ I [v(t,⋅)] - 1 2 E [v(t,⋅)] = 0 ,
we recover the standard proof of the entropy -entropy production inequality

I [v] - 1 2 E [v] ≥ 0 , (14)
i.e., of (1) by the method of [START_REF] Émery | Diffusions hypercontractives[END_REF]. Several of the above expression can be rephrased in terms of the pressure variable P ∶= -log h = -2 log v using the following elementary identities ∇v = - ∇P ⊗ ∇P , so that, by taking into account v ∇P = -2∇v and h = v 2 , we have

I [v] = 1 4 R d ∇P 2 h d γ and R d Hess v - ∇v ⊗ ∇v v 2 d γ = 1 4 R d Hess P 2 h d γ .

Improvements under moment constraints

In standard computations based on the carré du champ method, one usually drops the right-hand side in [START_REF] Bobkov | KLS-type isoperimetric bounds for logconcave probability measures[END_REF] which results in the standard exponential decay of [START_REF] Bobkov | Isoperimetric and analytic inequalities for log-concave probability measures[END_REF] and, after integration on t ∈ R + , proves [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF]. Keeping track of the right-hand side in [START_REF] Bobkov | KLS-type isoperimetric bounds for logconcave probability measures[END_REF] provides us with improvements as shown in [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF][START_REF] Demange | Des équations à diffusion rapide aux inégalités de Sobolev sur les modèles de la géométrie[END_REF][START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF] in various interpolation inequalities but generically fails in the case of the logarithmic Sobolev inequality. We remedy to this issue by introducing moment constraints. This is not a very difficult result but, as far as we know, it is new in the framework of the carré du champ method.

I [v(t,⋅)] if v solves

Lemma 4. With the notations of Section

2.1, if v ∈ H 2 (R d , d γ) is a positive function such that ∫ R d x 2 v 2 d γ ≤ d , then 4 I [v] ≤ R d (∆P)h d γ ≤ d R d Hess P 2 h d γ . Proof. Using h ∇P = -∇h, we obtain 4 I [v] = R d ∇P 2 h d γ = - R d ∇P ⋅ ∇h d γ = R d h (L P )dγ.
After recalling that L P = ∆P -x ⋅ ∇P, using an integration by parts we deduce that -

R d h x ⋅ ∇P d γ = R d x ⋅ ∇h d γ = R d h x 2 -d d γ = R d v 2 x 2 -d d γ ≤ 0
which proves the first inequality. The second inequality follows from a Cauchy-Schwarz inequality and the arithmetic-geometric inequality

(∆P) 2 ≤ d Hess P 2 .
Proof of Proposition 1. Let h = v 2 be the solution of ( 9) with initial datum

h 0 = u 2 . Since x ↦ x 2 -d is an eigenfunction of L with corresponding eigenvalue -2 and L is self-adjoint on L 2 (R d , d γ), we have d d t R d x 2 -d h d γ = R d x 2 -d (L h) d γ = R d h L x 2 -d d γ = -2 R d x 2 -d h d γ . ( 15
)
The sign of t ↦ ∫ R d x 2 -d h(t , x) d γ is conserved and in particular we have that

∫ R d x 2 v 2 d γ ≤ d for any t ≥ 0.
For any i = 1, 2. . . d , we also notice that x ↦ x i is also an eigenfunction of L with corresponding eigenvalue -1 so that

d d t R d x h d γ = - R d x h d γ
and, as a consequence ∫ R d x h(t , ⋅)dγ = 0 for all t ≥ 0 because ∫ R d x h 0 d γ = 0.

For smooth enough solutions, we deduce from Lemma 4, ( 13) and ( 14) that

d d t I [v] + 2 I [v] ≤ - 8 d I 2 [v] ≤ 1 2 d d d t (E [v])
2 and obtain by considering the limit as t → +∞ that

I [v] ≥ 1 2 E [v] + 1 2 d (E [v]) 2 .
This provides us with (2). In the general case, one can get rid of the H 2 (R d , d γ)

regularity of Lemma 4 by a standard approximation scheme, which is classical and will not be detailed here.

As in [START_REF] Brigati | Logarithmic Sobolev and interpolation inequalities on the sphere: constructive stability results[END_REF], a better estimate is achieved as follows. Let

φ(s) ∶= d 4 e 2 d s -1 ∀ s ≥ 0 . Using d d t E [v] = -4I [v], we notice that d d t I [v] -φ E [v] = - 8 d I [v] -φ E [v] .
Since lim t →+∞ I [v(t,⋅)] = 0 as can be deduced from a Gronwall estimate relying on

d d t I [v] ≤ -2I [v]
and lim t →+∞ E [v(t,⋅)] = 0 as a consequence of (1), one knows that lim

t →+∞ I [v(t,⋅)] -φ E [v(t,⋅)] = 0 . Moreover, Gronwall estimates show that I [v(t,⋅)] -φ E [v(t,⋅)] cannot change
sign and an asymptotic expansion as t → +∞ as in [START_REF] Brigati | Logarithmic Sobolev and interpolation inequalities on the sphere: constructive stability results[END_REF]Appendix B.4] is enough to

obtain that I [v(t,⋅)]-φ E [v(t,⋅)
] takes nonnegative values for t > 0 large enough.

Altogether, we conclude that

I [v(t,⋅)] -φ E [v(t,⋅)] ≥ 0
for any t ≥ 0 and, as a particular case, at t = 0 for v(0, ⋅) = u. The function φ is convex increasing and, as such, invertible, so that we can also write

ϕ I [u] -1 -φ E [u] ≥ 0 .
his completes the proof of (3) with the convex monotone increasing function

ψ(s) ∶= s - 1 2 φ -1
(s).

Stability results

Log-concave measures and Poincaré inequality

According to [START_REF] Ledoux | Lévy-Gromov's isoperimetric inequality for an infinitedimensional diffusion generator[END_REF], given a Borel probability measure µ on R d , its isoperimetric constant is defined as

h(µ) ∶= inf A P µ (A) min µ(A), 1 -µ(A)
where the infimum is taken on the set of arbitrary Borel subset R d with µ-perimeter

or surface measure P µ (A) ∶= lim ε→0 + µ(A ε ) -µ(A) ε and A ε ∶= {x ∈ R d ∶ x -a < ε for some a ∈ A}.
Here and in what follows, we shall say that a measure µ with density e -ψ with respect to Lebesgue's measure is a log-concave probability measure if ψ is a convex function, and denote by λ 1 (µ) the first positive eigenvalue of -L ψ where L ψ is the Ornstein-Uhlenbeck operator L ψ ∶= ∆-∇ψ⋅∇. In that case, we learn from [START_REF] Ledoux | Spectral gap, logarithmic Sobolev constant, and geometric bounds[END_REF]Ineq. (5.8)] that

1 4 h(µ) 2 ≤ λ 1 (µ) ≤ 36 h(µ) 2
where the lower bound is J. Cheeger's inequality that goes back to [START_REF] Cheeger | A lower bound for the smallest eigenvalue of the Laplacian[END_REF] for Riemannian manifolds and also to earlier works by V.G. Maz'ya [START_REF] Maz'ya | The negative spectrum of the n-dimensional Schrödinger operator[END_REF][START_REF]On the solvability of the Neumann problem[END_REF]. This bound was later improved in [START_REF] Buser | A note on the isoperimetric constant[END_REF][START_REF] Ledoux | A simple analytic proof of an inequality by P. Buser[END_REF]. The characterization of h(µ) has been actively studied, but it is out of the scope of the present paper. We learn from [12, Theorem 1.2] and [START_REF] Bobkov | Isoperimetric and analytic inequalities for log-concave probability measures[END_REF]Ineq. (3.4)] that

h(µ) ≥ 1 6 3 ∫ R d x -x µ 2 d µ where x µ = R d x d µ
for any log-concave probability measure µ. This estimate is closely related with the results by R. Kannan, L. Lovász and M. Simonovits in [START_REF] Kannan | Isoperimetric problems for convex bodies and a localization lemma[END_REF] and their conjecture, which again lies out of the scope of the present paper (see for instance [START_REF] Bobkov | KLS-type isoperimetric bounds for logconcave probability measures[END_REF] for a recent work on the topic).

Altogether, if µ is a log-concave probability measure with d µ = e -ψ d x such that

∫ R d x 2 d µ ≤ d , then we have the Poincaré inequality R d ∇f 2 d µ ≥ 1 432 R d f 2 d µ ∀ f ∈ H 1 (R d , d µ) such that R d f d µ = 0 . ( 16
)
We refer to [START_REF] Cattiaux | Functional inequalities for perturbed measures with applications to log-concave measures and to some Bayesian problems[END_REF] and references therein for further estimates on λ 1 (µ).

Time evolution, log-concave densities and Poincaré inequality

Lemma 5. Let us consider consider a solution h of (9) with initial datum h 0 = v 2 and assume that µ 0 ∶= h 0 γ is log-concave. Then µ t ∶= h(t , ⋅)γ is log-concave for all t ≥ 0.

Proof. The function g ∶= h γ solves the Fokker-Planck equation

∂g ∂t = ∆g + ∇ ⋅ (x g ).
The function f such that

f (t, x) ∶= g 1 2 log(1 + 2 t ), x √ 1 + 2 t ∀(t, x) ∈ R + × R d
solves the heat equation and can be represented using the heat kernel. According for instance to [START_REF] Saumard | Log-concavity and strong log-concavity: a review[END_REF][START_REF] Bardet | Functional inequalities for Gaussian convolutions of compactly supported measures: Explicit bounds and dimension dependence[END_REF], log-concavity is preserved under convolution, which completes the proof.

The log-concavity property becomes true under the action of the flow of ( 9) after some delay t ⋆ for large classes of initial data. With the notation of Lemma 5, for any R > 0, we read from [48, Theorem 5.1] by K. Lee and J-L. Vázquez that µ t is logconcave for any

t ≥ t ⋆ (R) ∶= log √ R 2 + 1 , ( 17
)
if v is compactly supported in a ball of radius R > 0, by reducing the problem to the heat flow as in the above proof. As a consequence, we know that (16) holds for any

t ≥ t ⋆ (R).
Alternatively, under Assumption [START_REF] Barchiesi | Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality[END_REF], we learn from a recent paper [23, Theorem 2] by H.-B. Chen, S. Chewi, and J. Niles-Weed that the Poincaré inequality

R d ∇ f 2 d µ t ≥ λ 1 (µ t ) R d f 2 d µ t ∀ f ∈ H 1 (R d , d µ t ) such that R d f d µ t = 0 (18)
holds for all t ≥ t ε ⋆ with

t ε ⋆ ∶= log √ 1 + ε -1 , 1 λ 1 (µ t ) ≤ τ ε τ ε τ -1 + A 1 ε τ-1 and τ = 1 2 e 2t
-1 .

Explicit stability results for log-concave densities

Let us start by an elementary observation.

Lemma 6. If h ∈ H 1 (R d , d γ) is such that ∫ R d x h d γ = 0 and P = -log h is the pressure variable, then R d ∇P h d γ = 0 . Proof. The result follows from ∫ R d ∇P h d γ = -∫ R d ∇h d γ = ∫ R d x h d γ = 0.
With this result in hand, we can now prove our first main result.

Proof of Theorem 2. The function h = v 2 is such that ∫ R d x h d γ = 0 and Lemma 6 applies. Since h γ is log-concave, we can apply ( 16) with f = ∂P ∂x i for any i = 1, 2,. . . d and obtain

R d Hess P 2 h d γ ≥ 1 432 R d ∇P 2 h d γ .
It follows from (13) that

d d t R d ∇v 2 d γ + 2 R d ∇v 2 d γ ≤ - 1 864 R d ∇v 2 d γ ,
and the stability result is obtained as in the proof of Proposition 1.

Extension by entropy methods and flows

This section is devoted to the proof of Theorem 3. The key idea is to evolve the function by the Ornstein-Uhlenbeck equation, so that the solution after an initial time layer has the log-concavity property of Theorem 2, at least if the initial datum has compact support. To some extent, the strategy is similar to the one used in [START_REF] Bonforte | Stability in Gagliardo-Nirenberg-Sobolev inequalities: Flows, regularity and the entropy method[END_REF]. During the initial time layer, we use an improved version of the entropy -entropy production inequality which arises as a consequence of the carré du champ method.

Proof of Theorem 3. Let h be the solution to [START_REF] Beckner | Sobolev inequalities, the Poisson semigroup, and analysis on the sphere S n[END_REF] with initial datum h 0 = u 2 and define

Q(t) ∶= I h(t , ⋅) E h(t , ⋅) ∀ t ≥ 0 .
Let us assume first that v has compact support. With no loss of generality, we can assume that v is supported in B (0,R) for some R > 0. With t ⋆ = t ⋆ (R) given by ( 17), we know from [48, Theorem 5.1] that µ t is log-concave at t = t ⋆ and Theorem 2 applies:

Q(t ⋆ ) ≥ C ⋆ 2 .
With an estimate similar to [16, Lemma 2.9], we learn from Section 2 that

d Q d t ≤ 2 Q (2Q -1) . (19)
An integration on (0, t ⋆ ) shows that

Q(0) ≥ 1 2 1 + 2 Q(t ⋆ ) -1 1 + 2 Q(t ⋆ )(e 2 t ⋆ -1) ≥ 1 2 1 + C ⋆ -1 1 + C ⋆ R 2 = C 2 .
Under the more general assumption (7), we rely on ( 18) and obtain with same notations as above and

t ⋆ = t ε ⋆ that R d Hess P 2 h(t , ⋅)dγ ≥ λ 1 (µ t ) R d ∇P 2 h(t , ⋅)dγ ∀ t ≥ t ⋆ .
Moreover, for some explicit t 0 = t 0 (ε) > t * , we notice that t ↦ λ 1 (µ t ) is nonincreasing on (t 0 , +∞). Hence we deduce from I [v(t 0 , ⋅)] - Using [START_REF] Buser | A note on the isoperimetric constant[END_REF], we obtain

C = 1 + C 0 -1 1 + C 0 (e 2 t 0 -1)
. This concludes the proof. Proof. We learn from (15) that and the remainder of the proof of Theorem 2 is unchanged.

Normalization issues

A similar extension of Theorem 3 can be done on the same basis. Details are left to the reader. As for Proposition 1, we can make the following observations. The Knowing that z ′ < 0 is an improvement on the decay rate I [v(t,⋅)] ≤ I [u]e -2 t can be rephrased as an improved entropy -entropy production inequality for A > 0.

  Hess v = ∇ ⊗ ∇v is the Hessian matrix of v. Here we use the following notations. If a and b take values in R d , a ⊗ b denotes the matrix (a i b j ) 1≤i , j ≤d . With matrix valued m and n, we define m ∶ n = ∑ d i , j =1 m i , j n i , j and m 2 = m ∶ m. If h is a nonnegative solution of (9), notice that v = √ h solves ∂v ∂t

4 + 4 λ 1

 441 λ 1 (µ s ) E ′ [v(s,⋅)]d s ≥ 0after an integration by parts thatQ(t 0 ) ≥ (µ t 0 ) =∶ C 0 .

Proposition 7 . 2 L 2 (

 722 If we do not assume that uL 2 (R d ,d γ) = 1 and x u L 2 (R d ,d γ) ≤ d , it is still possibleto state the analogue of Theorem 2, but the price to be paid is a dependence onκ[u] ∶= u L 2 (R d ,d γ) max √ d , (x -x 0 )u L 2 (R d ,d γ) where x 0 = R d x h 0 d γ ,which goes as follows. For all u ∈ H 1 R d , (1 + x 2 )dγ such that ∫ R d x u 2 d γ = 0, and u 2 γ is log-concave, we have ∇u R d ,d γ) -

R d x 2 2 L 2 (d x 2 -

 2222 h(t , x) d γ = d u R d ,d γ) + e -2t R d h 0 d γ ∀ t ≥ 0 .Hence[12, Theorem 1.2] and [12, Ineq. (3.4)] apply with h(µ) ≥ κ[u]

u 2 x 2 - 2 -Hess P 2 h d γ ≤ - 1 2 d 4 I

 2214 case ∫ R d x 2 v 2 d γ ≤ d is already covered in Lemma 4. If A ∶= R d d d γis positive, let us consider the solution h of (9) with initial datum h 0 = u 2 . We know from (15) thatR d h(t , x) x d d γ = A e -2tand deduce as in the proof of Lemma 4 that4 I [v] = R d ∇P 2 h d γ = R d h (L P )dγ + A e -2t ≤ d R d Hess P 2 h d γ + A e -2twith P = -logh. Hence by[START_REF] Bobkov | KLS-type isoperimetric bounds for logconcave probability measures[END_REF], we learn that [v] -A e -2t 2 and this estimate can be rephrased with z(t ) ∶= e 2 t I [v(t,⋅)]

© 2023 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Acknowledgements

The authors thank Max Fathi and Pierre Cardaliaguet for fruitful discussions and Emanuel Indrei for stimulating interactions. G.B. has been funded by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 754362.