C Mietka

On the well-posedness of a quasi-linear Korteweg-de Vries equation

Keywords: quasilinear dispersive equation, energy estimates, gauging technique, parabolic regularization, Bona-Smith technique

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Introduction and main result

More than a century ago, D.J. Korteweg and G. de Vries proposed a model for unidirectional long water waves propagating in a channel. The so-called Korteweg-de Vries equation (KdV) v t + vv x + v xxx = 0 , in fact derived earlier by Boussinesq, drew a lot of attention in the 1960's, when it turned out that it was completely integrable, see e.g. one of the seminal papers by Gardner, Green, Kruskal & Miura [START_REF] Gardner | Method for Solving the Korteweg-deVries Equation[END_REF][START_REF] Gardner | Korteweg-deVries equation and generalization. VI. Methods for exact solution[END_REF] or the book by Ablowitz [START_REF] Mark | Nonlinear dispersive waves[END_REF] for a modern overview. It was soon considered in generalized forms (gKdV) v t + p(v) x + v xxx = 0 .

In particular, the modified KdV equation

(mKdV) v t + v 2 v x + v xxx = 0 .
is also completely integrable. This is not the case for more general nonlinearities. Nevertheless, (KdV) and (gKdV) have been studied by analysts for about 50 years. The state of the art regarding (KdV) is mainly due to Bona & Smith [START_REF] Bona | The initial-value problem for the Korteweg-de Vries equation[END_REF], Kato [START_REF] Kato | Quasi-linear equations of evolution, with applications to partial differential equations[END_REF], Kenig, Ponce & Vega [START_REF] Kenig | The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices[END_REF][START_REF] Kenig | A bilinear estimate with applications to the KdV equation[END_REF] and Christ, Colliander & Tao [START_REF] Christ | Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations[END_REF]. For (gKdV), it is due mainly to Kenig, Ponce & Vega [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF] and Colliander, Keel, Staffilani, Takaoka & Tao [START_REF] Colliander | Sharp global wellposedness for KdV and modified KdV on R and T[END_REF].

In this article, we consider a quasi-linear version of the Korteweg-de Vries equation, in which the dispersive term is not reduced to v xxx . This equation is the most natural generalization of the abstract, Hamiltonian form of (gKdV), which reads (1)

v t = (δH [v]) x with (2) H [v] = 1 2 v 2 x + f (v) ,
and f (v) = -p(v). Our motivation for considering this generalization comes from the so-called Euler-Korteweg system, which involves an energy of the form (2) where κ is not necessarily a constant. The link with the Euler-Korteweg system is that their travelling waves share the same governing ODE, with additional connections in the stability of their periodic waves, see [START_REF] Benzoni-Gavage | Co-periodic stability of periodic waves in some Hamiltonian PDEs[END_REF].

Our qKdV equation [START_REF] Mark | Nonlinear dispersive waves[END_REF] (2) reads in a more explicit form

(3)

∂ t v + ∂ x (p(v)) + ∂ x κ(v)∂ x ( κ(v)∂ x v) = 0 .
Up to our knowledge, the Cauchy problem regarding this quasilinear equation has never been investigated. An apparently more general nonlinear KdV equation

v t + f (v xxx , v xx , v x , v) = 0 ,
was studied by Craig, Kappeler, & Strauss in [START_REF] Craig | Gain of regularity for equations of kdv type[END_REF] and more recently by Linares, Ponce, & Smith in [START_REF] Linares | On the regularity of solutions to a class of nonlinear dispersive equations[END_REF]. However, they use a monotonicity assumption on the nonlinearity, ∂ vxx f (v xxx , v xx , v x , v) ≤ 0, which reads ∂ x κ(v) ≤ 0 for [START_REF] Benzoni-Gavage | Well-posedness of one-dimensional Korteweg models[END_REF]. Unless κ is constant, their results are thus hardly applicable to [START_REF] Benzoni-Gavage | Well-posedness of one-dimensional Korteweg models[END_REF]. Our approach is to use the structure of the equation instead of a monotonicity argument. Of course, Eq. ( 3), includes the semi-linear generalized KdV equations, where κ(v) = 1 and the nonlinearity p is polynomial. In particular p(v) = 1 2 v 2 corresponds to the classical KdV equation and p(v) = 1 3 v 3 to the modified KdV equation. As said before, these two cases are known to fall into the class of integrable equations. Remarkably enough, there is a non constant κ for which (qKdV) is completely integrable, namely κ(v) = ε 2 12 (v + a) -3 , with a ∈ R, ε > 0 and p(v) = v 2 2 , see [START_REF] Dubrovin | Hamiltonian partial differential equations and Painlevé transcendents[END_REF]. We focus on the local-in-time well-posedness of (qKdV) in Sobolev spaces, that is existence and uniqueness of a smooth solution v given smooth initial data v 0 , with continuity of the mapping v 0 → v. The most important part of the work is based on obtaining a priori estimates, using the skew-symmetric form of the leading order term and gauging techniques to control subprincipal remainders. The idea of using gauges for dispersive PDEs, introduced by Lim & Ponce in [START_REF] Wee | On the initial value problem for the one dimensional quasi-linear Schrödinger equations[END_REF] and developed by Kenig, Ponce & Vega in [START_REF] Kenig | The Cauchy problem for quasi-linear Schrödinger equations[END_REF] and later by S. Benzoni-Gavage, R. Danchin & S. Descombes in [START_REF] Benzoni-Gavage | Well-posedness of one-dimensional Korteweg models[END_REF][START_REF] Benzoni-Gavage | On the wellposedness for the Euler-Korteweg model in several space dimensions[END_REF], is a fairly general method to deal with subprincipal terms.

In what follows, we consider an interval I ⊂ R, and solutions of (3) are sought with values in a compact subset J of I. The functions κ : I → R + * and p : I → R are supposed to be smooth. We consider an integer k and denote by H k (R) the classical Sobolev space constructed on L 2 (R) as

H k (R) = {v ∈ L 2 (R) , ∂ l x v ∈ L 2 (R) f or all |l| ≤ k} .
The inner product in L 2 (R) will be denoted for all u and v by u|v . Our main result is the following.

Theorem 1. Assume that k ≥ 4. If p = -f : I ⊂ R → R is C k+1 and κ : I → R + * is C k+2 , then for all v 0 ∈ H k (R), the image of v 0 being in J ⊂⊂ I, there exists a time T > 0 and a unique v ∈ C (0, T ; H k (R)) ∩ C 1 (0, T ; H k-3 (R)) solution to (3) with initial data v 0 . Moreover, v 0 → v maps continuously H k (R) into C (0, T ; H k (R)) ∩ C 1 (0, T ; H k-3 (R)).
Remark 1. From the above statement, by standard arguments one may then build maximal solutions and a close inspection of our estimates shows that the maximal lifespan is finite if and only if the H 4 norm of the solution blows up at the final time.

The proof of this theorem is based on the derivation of a priori estimates for a regularized parabolic problem. We establish these a priori estimates on smooth solutions by taking advantage of the structure of Eq. ( 3). More precisely, we use the skew-symmetry of its leading order term and then gauging techniques to deal with subprincipal remainders. Regarding the existence of solutions, we shall use a fourth order parabolic regularization of (qKdV) and pass to the limit. Uniqueness and continuity of the mapping v 0 → v are proved by means of a priori estimates and a technique adapted from Bona & Smith [START_REF] Bona | The initial-value problem for the Korteweg-de Vries equation[END_REF].

We introduce the notations a = p = -f and α = √ κ. For convenience, we use the same notation for both functions v → α(v) and (t, x) → (α • v)(t, x). In particular, α stands for the derivative of α(v) with respect to v and ∂ t α and ∂ x α for the time and space derivatives of α • v. We use the same convention for the function a and all nonlinear functions of the dependent variable v, unless otherwise specified. To keep notations compact, and hopefully easier to read, we also omit all parentheses in operators. For example, the expression

∂ x (α(v)∂ x (α(v)∂ x v)) ,
will just be denoted by

∂ x α∂ x α∂ x v .
With these conventions, Eq. ( 3) becomes (4)

v t + av x + ∂ x α∂ x α∂ x v = 0 ,
as far as smooth solutions are concerned.

A priori energy estimates in Sobolev spaces

In this section, we investigate a priori bounds in H k (R) for smooth solutions of (4) with k ≥ 4. Following ideas from [START_REF] Wee | On the initial value problem for the one dimensional quasi-linear Schrödinger equations[END_REF][START_REF] Benzoni-Gavage | Well-posedness of one-dimensional Korteweg models[END_REF][START_REF] Benzoni-Gavage | On the wellposedness for the Euler-Korteweg model in several space dimensions[END_REF], we shall make use of the structure of the equation and of gauges, in order to cancel out bad commutators. An ideal structure that would allow a direct computation of energy estimates is v t = skew-symmetric terms + zero th order terms .

Taking the inner product of this kind of equation with v, we see that the skew-symmetric terms cancel out and we readily find

d dt v L 2 v L 2 .
Even if Eq. ( 4) has not exactly this ideal structure, the presence of first order terms can also be handled, up to an integration by parts, not directly in L 2 (R) but in higher order Sobolev spaces. If we intend to get energy estimates in higher order Sobolev spaces than L 2 (R), we had better be sure that this structure is preserved when differentiating the equation to avoid derivatives loss. In what follows, we adapt a method from Lim & Ponce [START_REF] Wee | On the initial value problem for the one dimensional quasi-linear Schrödinger equations[END_REF], and use weighted Sobolev spaces, with weights also called gauges, to transform our equation into a convenient structure and derive a priori estimates without loss of derivatives.

Conservation of the skew-symmetric structure

Let us first focus on the leading order term in [START_REF] Benzoni-Gavage | On the wellposedness for the Euler-Korteweg model in several space dimensions[END_REF]. When looking for an a priori estimate in L 2 (R), we compute the time derivative of v L 2 by taking the inner product of the equation with v. On the skew-symmetric leading order term, an integration by parts yields

∂ x α∂ x α∂ x v|v = -∂ x (α∂ x v) |α∂ x v = 0 .
If we apply the differential operator ∂ x to Eq. ( 4), we lose the skew-symmetry property and the corresponding cancellation. Our aim is to preserve this cancellation throughout the entire differentiation process. For this purpose, we consider the weighted quantity

(5) ∀k ≥ 0 , v k = (α(v)∂ x ) k v , instead of ∂ k x v.
This quantity is well defined if v is smooth enough. If we apply the formal operator (∂ x (α • )) k to Eq. ( 4), we see that the higher order terms have the same form as in the original equation. They read ∂ x α∂ x α∂ x v k , and thus cancel out in the inner product with v k .

Following a definition in [START_REF] Serre | Systems of conservation laws[END_REF] ( §3.6), we call weight the total number of space derivatives in a monomial expression involving a function and its own derivatives. As we will see, when we compute derivatives by applying repeatedly the operator ∂ x (α•) to Eq. ( 4), the coefficients of the remainders we will have to deal with will be products of polynomial functions of v and its derivatives with functions of α(v) or a(v) and their derivatives. As a result, they are polynomials in the variables v k with terms multiplied by functions of v but we extend the definition of weight to this kind of non-polynomial functions. The weight is merely the total number of derivatives.

Proposition 1. For a smooth solution of Eq. (4), if we denote

v k = (α(v)∂ x ) k v, then the equation satisfied by v k is of the form (6) ∂ t v k + a∂ x v k + ∂ x α∂ x α∂ x v k = f k ∂ 2 x v k + g k ∂ x v k + h k ,
where

• f k = f k (v, v 1
) is of (homogeneous) weight 1.

• g k = g k (v, v 1 , v 2 ) is of (homogeneous) weight 2. • h k = h k (v, v 1 , • • • , v k
) consists of two terms, one of (homogeneous) weight k + 1 and another of (homogeneous) weight k + 3.

In Eq. ( 6), we kept a skew-symmetric leading order term in the left hand side on purpose and we gathered commutator terms in the right hand side. These are subprincipal terms because commutator terms between two differential operators of order respectively p 1 and p 2 are of order p 1 + p 2 -1.

Proof. Eq. ( 6) is obtained by repeatedly applying the differential operator ∂ x (α•), which preserves the form of the higher order terms. Indeed,

∂ x α∂ x α∂ x α∂ x v k = ∂ x α∂ x α∂ x v k+1 ,
where the parentheses are omitted. We apply the differential operator ∂ x (α•) and find the equation satisfied by v 1 = α∂ x v. We have

∂ x (α∂ t v) = ∂ t v 1 , ∂ x (αa∂ x v) = a∂ x v 1 + a α (∂ x v) 2 ,
so that the equation satisfied by v 1 is

∂ t v 1 + a∂ x v 1 + ∂ x α∂ x α∂ x v 1 = - a α v 2 1 , hence f 1 = 0, g 1 = 0 and h 1 = -a (v) α(v) v 2 1 .
For k = 2, we use the equation on v 1 and apply again the operator ∂ x (α•). Then, we find

∂ t v 2 +a∂ x v 2 +∂ x α∂ x α∂ x v 2 = -α v 1 ∂ 2 x v 2 + α α v 2 - α 2 α 2 v 2 1 ∂ x v 2 + α a α - a α v 3 1 - 3a α v 1 v 2 , hence f 2 = α v 1 , g 2 = α α v 2 -α 2 α 2 v 2 1 and h 2 = α a α 2 - a α v 3 1 - 3a α v 1 v 2 .
More generally, f k , g k , h k are computed by induction for k ≥ 2. We have

∂ x (α∂ t v k ) = ∂ t v k+1 -α [v t ∂ x v k -v x (v k ) t ] ,
where we can use Eq. ( 4) and ( 6) to write

∂ t v k = - a α v k+1 -α∂ 2 x v k+1 - α α v 1 ∂ x v k+1 + f k α ∂ x v k+1 - α α 2 f k v 1 v k+1 + g k α v k+1 + h k , and v t = - a α v 1 -∂ x v 2 .
Using that

∂ x (αa∂ x v k ) = a∂ x v k+1 + a α∂ x v∂ x v k = a∂ x v k+1 + a α v 1 v k+1 , and
∂ x (α [f k ∂ 2 x v k + g k ∂ x v k + h k ]) = f k ∂ 2 x v k+1 + g k + ∂ x f k -α α 2 f k v 1 ∂ x v k+1 + ∂ x g k -∂ x α α 2 f k v 1 v k+1 + ∂ x (αh k ) . we find that ∂ t v k+1 + a∂ x v k+1 + ∂ x α∂ x α∂ x v k+1 = f k+1 ∂ 2 x v k+1 + g k+1 ∂ x v k+1 + h k+1 , with (7)
f k+1 = f k + α v 1 g k+1 = g k + ∂ x f k -2α α 2 f k v 1 + α 2 α 2 v 2 1 h k+1 = α∂ x h k + ∂ x g k -∂ x α α 2 f k v 1 v k+1 -a α v 1 v k+1 + α α v k+1 α α 2 f k v 2 1 -g k α v 1 -∂ x v 2
Note that these induction relations are valid for k ≥ 2. Indeed, the last term in [START_REF] Christ | Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations[END_REF] in the formula for h k , namely -α α v k+1 ∂ x v 2 involves three derivatives on v at least. For k = 1, this term should be gathered with the ones which define g k+1 = g 2 . This is the reason why we detailed the cases k = 1 and k = 2 in the beginning of the proof. From the first induction, we deduce that for all k ≥ 1 (8)

f k = (k -1)α v 1 = (k -1)α∂ x α .
This explicit expression will be useful in what follows and justifies that the coefficient f k depends only on v and v 1 and is a polynomial in v 1 with weight one multiplied by a bounded function of v, namely v → (k -1)α (v). We do not actually need the exact expression for g k and h k . Instead, we analyse the number of derivatives they contain, that is their weight, and their general form in terms of the variables

v, v 1 , • • • , v k . For k = 2, we have the explicit expression of g 2 which is of weight 2. For k ≥ 2 if g k is of weight 2 then so is g k+1 because g k+1 -g k = ∂ x f k - 2α α 2 f k v 1 + α 2 α 2 v 2 1 is of weight 2, since f k is of weight 1.
Finally, we shall prove by induction that h k has the general form

(9) h k (v, v 1 , • • • , v k ) = i β i (v) k j=1 ∂ γ j x v ,
where the β i involves derivatives of both functions v → α(v) and v → a(v) and for all j, γ j ≤ k and the weight of each of the terms, namely j γ j is either k + 1 or k + 3. Note that in the general form [START_REF] Craig | Gain of regularity for equations of kdv type[END_REF], the γ j are not necessarily distinct. For k = 2 we have

h 2 = α a α 2 - a α v 3 1 - 3a α v 1 v 2 ,
which is of weight 3. It is compatible with the general form [START_REF] Craig | Gain of regularity for equations of kdv type[END_REF]. For k ≥ 2, we have from Eq. ( 7)

h k+1 = α∂ x h k - a α v 1 v k+1 + r k ,
where r k gathers terms of weight k + 4, as g k is of weight 2 and f k is of weight 1. The term -a α v 1 v k+1 is of weight k + 2. Now, from the form (9) and using the chain rule and the product rule, we write

α∂ x h k = α(v)∂ x i β i (v) k j=1 ∂ γ j x v = i βi (v) k j=1 ∂ γj x v ,
where the βi also involve derivatives of both functions v → α(v) and v → a(v) and for all j, γj ≤ k + 1 and the weight of each of the terms, namely j γj is either k + 2 or k + 4.

We see that the term α∂ x h k has the same general form as h k , but with terms of weight k + 2 or k + 4. Then, if h k has the general form ( 9), so it is for h k+1 , with weight raised by one.

Remark 2. Another way to see that h k has the general form (9) is to see that it is made of terms of the form

((α • v)∂ x ) k (f • v) ,
the function f being a combination of the functions α or a and their derivatives. Combining the Faà di Bruno formula, which generalizes the chain rule, and the product rule, we see that the term h k consists of a polynomial expression on v and its derivatives up to v k , each term of the polynomial being multiplied by a bounded function of v.

The structure of Eq. ( 6) is

∂ t v k = skew-symmetric terms + terms of order at most 2 .
In what follows, we will show how we can use gauges to reduce the order of these remainders. We shall start with the control of the zero order term. Then we will find an estimate for the first order term, without gauge fortunately. Finally, we will show how we can use a suitable gauge to control the second order term.

Remark 3. We will have to prove a norm equivalence between the Sobolev norm v H k and the weighted norm v k L 2 . This property will be checked at the end of this section.

Remark 4. To obtain Theorem 1 with k = 4, one may think about a slightly different strategy. It is based on estimating not (α•v∂ x ) 4 v -which we do by introducing some gauge -but

(α • v∂ x )∂ t v. Since (α • v∂ x )∂ t v 2 L 2 is the leading order part of ∂ t v, δ 2 H [v]∂ t v
, the subprincipal terms appearing in the latter computation may dealt with in a gaugeless way. See Remark 5 for a continuation of this comparison.

Gauge estimates for subprincipal remainders

Let us first focus on the zero order term h k . We first prove the following lemma, which is an extension of Lemma 3.6.2 in [START_REF] Serre | Systems of conservation laws[END_REF] to expression of order higher than one and adapted to our particular functions, that is polynomial functions with terms multiplied by bounded functions of v.

Lemma 1. Let Q(∂ x v, • • • , ∂ k x v
) be an homogeneous polynomial of weight q ∈ [k, 2k). There exists a smooth function C q,k such that, for all v ∈ H k (R),

Q(∂ x v, • • • , ∂ k x v) L 2 ≤ C q,k ( ∂ x v L ∞ , • • • , ∂ (q-k) x v L ∞ ) v H k Proof. The proof is based on the Gagliardo-Nirenberg inequality. First, note that if v ∈ H k (R) and q < 2k, then for all p ≤ q -k < k, ∂ p x v ∈ H 1 (R) → L ∞ (R)
. By the triangular inequality, it is sufficient to focus on a monomial expression of the form

Q(∂ x v, • • • , ∂ k x v) = k j=1 ∂ γ j x v ,
with for all 1 ≤ j ≤ k, γ j ≤ k satisfy k j=1 γ j = q. If there are some j such that γ j ≤ q -k, then we can estimate the corresponding factors in L ∞ (R). Now, if all the remaining factors satisfy q -k < γ j < k, we can conclude by using the Gagliardo-Nirenberg inequality. We choose a particular l and write, for j = l

∂ γ j x v L ∞ ∂ k x v θ j L 2 ∂ (q-k) x v 1-θ j L ∞ , where θ j = γ j -(q -k) k -(q -k) -1/2
,

and for j = l ∂ γ l x v L 2 ∂ k x v θ l L 2 ∂ (q-k) x v 1-θ l L ∞ , where θ l = γ l -(q -k) -1/2 k -(q -k) -1/2 .
With these relations, we have

θ = k j=1 q-k<γ j ≤k θ j ≤ 1 ,
and then we are able to write

Q(∂ x v, • • • , ∂ k x v) L 2 ≤ C q,k ( ∂ x v L ∞ , • • • , ∂ (q-k) x v L ∞ ) ∂ k x v θ L 2 ∂ (q-k) x v 1-θ L ∞ .
Finally, if there exists j 0 such that γ j 0 = k, the corresponding factor belongs to L 2 (R) and all other j satisfies γ j ≤ q -k. Then, for all j = j 0 , we have ∂

γ j x v ∈ L ∞ (R)
. Combining all previous cases, and using the Sobolev embedding H 1 (R) → L ∞ (R), we can write

Q(∂ x v, • • • , ∂ k x v) L 2 ≤ C q,k ( ∂ x v L ∞ , • • • , ∂ (q-k) x v L ∞ ) v H k .
Recall that h k contains two terms of weight q = k + 1 and q = k + 3. We deduced in Proposition 1 the general form

h k (v, v 1 , • • • , v k ) = i β i (v) k j=1 ∂ γ j x v ,
where the β i are bounded functions of v on J. Then, Lemma 1 gives us the following estimate

h k (v, v 1 , • • • , v k ) L 2 ≤ C k ( v L ∞ , • • • , ∂ 3 x v L ∞ ) v H k .
Let us now focus on the first order terms g k ∂ x v k . This term brings no trouble at all as we can regroup it with the first order term in Eq. ( 6)

[a(v) -g k (v, v 1 , v 2 )] ∂ x v k .
When taking the inner product with v k , we can estimate

(a -g k ) ∂ x v k |v k ≤ ∂ x (a -g k ) L ∞ v k 2 L 2 ≤ C k ( v L ∞ , v 1 L ∞ , v 2 L ∞ , v 3 L ∞ ) v k 2 L 2
This leaves us with the only one remaining term f k ∂ 2

x v k . This one cannot be estimated as the previous ones because it contains too many derivatives. The method we present here consists in using gauges in the equation, following ideas from [START_REF] Wee | On the initial value problem for the one dimensional quasi-linear Schrödinger equations[END_REF].

Formally and in all generality, what we call a gauge is a general differential operator with unknown variable coefficients that cancels out 'bad' commutator terms when applied to the equation. The key property is that the commutator of two differential operators of order respectively p 1 and p 2 is of order p 1 + p 2 -1. In our case, the equation is of leading order three, with a priori two subprincipal terms we wish to reduce to order zero. We could define a gauge of the type φ = zero th term + order (-1) term and apply it to our equation. Doing so, the two commutators with the leading order term would be of second and first order. These two new terms can be gathered with the existing ones, and, in practice, one can choose the coefficients of the gauge as solutions of ODEs, to cancel (or at least control) the subprincipal terms.

In our situation, we have already shown that the first order term can be controlled without this technique. Using this fact, we will define a particular gauge as a function [START_REF] Dubrovin | Hamiltonian partial differential equations and Painlevé transcendents[END_REF] φ

k : (v, • • • , v k ) → φ k (v, • • • , v k ) ,
to cancel the second order term f k ∂ 2 x v k . As we will see in the computation though, we will have to check that the arising first order term can be bounded. Now, we multiply Eq. ( 6) by

φ 2 k φ k ∂ t (φ k v k ) + φ k a∂ x (φ k v k ) + ∂ x φ k α∂ x φ k α∂ x v k + R k = φ k φ k f k ∂ 2 x v k + φ k φ k g k ∂ x v k + φ k φ k h k ,
where the four remaining terms are gathered in

R k = φ k a [φ k , ∂ x ] v k + φ k [φ k , ∂ t ] v k + [φ k φ k , ∂ x (α•)] ∂ x α∂ x v k + ∂ x αφ k [φ k , ∂ x ] α∂ x v k .
We expect the first two terms to be bounded in L 2 (R) because they are of order zero. This will be checked when we find bounds on the function φ k . First, we compute the commutators

(11) [φ 2 k , ∂ x (α•)] = -2αφ k ∂ x φ k , [φ k , ∂ x ] = -∂ x φ k , [φ k , ∂ t ] = -∂ t φ k .
With these relations we are able to compute the last two terms in R k

[φ * k φ k , ∂ x (α•)] ∂ x α∂ x v k = -2α 2 φ k (∂ x φ k )∂ 2 x v k -2α(∂ x α)φ k (∂ x φ k )∂ x vk , ∂ x αφ * k [φ k , ∂ x ] α∂ x v k = -α 2 φ k (∂ x φ k )∂ 2 x v k -∂ x (α 2 φ k (∂ x φ k )) ∂ x v k .
As expected, these terms coming from the commutators are also subprincipal terms. We gather the coefficients of the second order terms and find an ODE on the function φ k .

(

) 3α 2 (∂ x φ k ) + φ k f k = 0 . 12
Remark 5. To proceed with the comment of Remark 4, we observe that

φ 4 • v(α • v∂ x ) 4 v L 2 = ∂ x (α • v∂ x ) 3 v L 2 which differs from (α • v∂ x )∂ t v L 2 -that is essentially α • v∂ 2 x (α • v∂ x ) 2 v L 2
-only in some immaterial way. The advantage of using the gauge strategy to determine a correct functional -instead of deducing it directly from the Hamiltonian structure -is that it naturally generalizes to differentiation by any number of derivative, as we have just shown.

We recall that this process added some commutator terms of first order terms we have to control a posteriori. If we can prove the existence of φ k satisfying the previous ODE and belonging to a suitable space, here say W 3,∞ (R) for example, then we can get our a priori energy estimate by the same argument on first order terms as the one presented above.

Remark 6. We also need to prove a norm equivalence between the usual L 2 norm and some L 2 norm involving φ k . This will be done at the end of this section.

From Eq. 8, we come back to the ODE [START_REF] Gardner | Korteweg-deVries equation and generalization. VI. Methods for exact solution[END_REF] and easily find [START_REF] Kato | Quasi-linear equations of evolution, with applications to partial differential equations[END_REF] φ

k (v) = α(v) -k-1 3 .
Now, with the regularity properties of the function v → α(v), we directly get that φ k is bounded from above and away from zero. Moreover, for all 0 ≤ l ≤ 3, there exists a constant C k such that

∂ l x φ k (v) L ∞ ≤ C k ( v W 3,∞ ) , and
∂ t φ k (v) L ∞ ≤ C k ( v W 3,∞ ) .
Coming back to the first order terms, we check that

∂ x a(v) -g k -2α(∂ x α)φ k (∂ x φ k ) -∂ x α 2 φ k (∂ x φ k ) L ∞ ≤ C k ( v W 3,∞ ) .
Let us sum up formally what we obtained until now. At order k > 0, v k satisfies Eq. ( 6). We define a gauge

φ k (v) = α(v) -(k-1)/3 ,
and multiply Eq. ( 6) by φ 2 k . This operation yields the equation satisfied by the quantity

φ k v k φ k ∂ t (φ k v k ) + ∂ x φ k α∂ x φ k α∂ x v k = φ 2 k R k (v)
, where the function R k gathers all the subprincipal terms we are now able to estimate

φ k R k (v) L 2 ≤ C k ( v W 3,∞ ) φ k v k L 2 .
This is a formal computation and we do not really get this ideal last estimate. In practice, it is not the norm φ k v k L 2 that appears but a combination of it and some norms v H k with or without the gauge φ k . We need to prove norm equivalences between the weighted norms we introduced up to this point.

Weighted norms equivalences

As mentioned before, the final energy estimate cannot be obtained if we do not have some norm equivalence on the quantities we are working with. More precisely, we prove the following lemma Lemma 2. Consider an integer k ≥ 1, let J be a compact subset of I ⊂ R. Let φ k = α -(k-1)/3 with α : I → R + * of class C k+2 . On the one hand, there exists a constant c k depending only J such that, for all function v ∈ H k (R) satisfying v(t, x) ∈ J for all

(t, x) ∈ R + × R, then (14) 1 c k v L 2 ≤ φ k (v)v L 2 ≤ c k v L 2 .
On the other hand, if we denote

v k = (α(v)∂ x ) k v, there exist constants c k and C k-1
depending only on a constant ρ > 0 and J, such that for all function

v ∈ H k (R) with v(t, x) ∈ J for all (t, x) ∈ R + × R and v W 1,∞ ≤ ρ, then (15) 1 c k v 2 H k ≤ C k-1 v 2 H k-1 + φ k v k 2 L 2 ≤ c k v 2 H k .
Proof. The first inequalities are a direct consequence of the fact that the function α is bounded from above and below as J is compact. Regarding the second norm equivalence, we use the same scheme of proof as in Lemma 1 with a weight q = k. Using the definition of v k , we can write

v k -α k (v)∂ k x v = i β i (v) k j=1 ∂ γ j x v ,
where the β i are bounded functions on J, k j=1 γ j = k and for all j, γ j ≤ k -1. The leading order term in v k satisfies

1 c k ∂ k x v L 2 ≤ α k (v)∂ k x v L 2 ≤ c k ∂ k x v L 2 .
By Lemma 1, the remaining terms are bounded by

v k -α k ∂ k x v 2 L 2 ≤ 2qC k-1 ( v W 1,∞ ) v 2 H k-1 .
Then, we get

1 c k ∂ k x v 2 L 2 -C k-1 v 2 H k-1 ≤ v k 2 L 2 ≤ c k ∂ k x v 2 L 2 + C k-1 v 2 H k-1 , rearranged as 1 c k ∂ k x v 2 L 2 ≤ C k-1 v 2 H k-1 + v k 2 L 2 ≤ c k v 2 H k .
Using the inequality [START_REF] Kenig | The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices[END_REF] and without renaming the constants already written for convenience, we finally obtain

1 c k v 2 H k ≤ C k-1 v 2 H k-1 + φ k v k 2 L 2 ≤ c k v 2 H k .
Then for all v ∈ H k (R), we consider the weighted norm | • | k defined recursively by ( 16)

|v| 2 k = φ k v k 2 L 2 + C k-1 |v| 2 k-1 , for k ≥ 1 , |v| 0 = v L 2 .
where the constant C k-1 is determined by induction. This definition leads to the following proposition.

Proposition 2. For all integer s ≥ 1, the weighted norm |•| s defined by ( 16) is equivalent to the H s norm. More precisely, there exists a constant c s depending only on ρ > 0 and J such that for all v ∈ H s (R) with, for all (t,

x) ∈ R + × R, v(t, x) ∈ J and v W 1,∞ ≤ ρ, then 1 c s v 2 H s ≤ |v| 2 s ≤ c s v 2 H s .
Proof. The proof is done by induction. For s = 1, the result is given by the relation ( 15) from Lemma 2. For any s > 1, we have from [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF] and the definition of

|•| s 1 c s v 2 H s ≤ C s-1 v 2 H s-1 -C s-1 |v| 2 s-1 + |v| 2 s ≤ c s v 2 H s .
Then, we get using the induction property ( 16)

1 c s v 2 H s + C s-1 c s-1 -C s-1 v 2 H s-1 ≤ |v| 2 s ≤ c s v 2 H s + C s-1 c s-1 -C s-1 v 2 H s-1 .
Finally, choosing the constant

C s-1 such that C s-1 /c s-1 -C s-1 > 0, we obtain with new constants 1 c s v 2 H s ≤ |v| 2 s ≤ c s v 2 H s .
We are now able to give an a priori bound on a smooth solution of (qKdV).

Proposition 3. For any integer s ≥ 4, a smooth solution v of (qKdV) associated with the initial condition v 0 ∈ H s (R) satisfies

(17) v H s ≤ C s ( v W 3,∞ ) v 0 H s Proof.
Let us come back to the equation satisfied by φ k v k . Using what we have done previously, the second order terms cancel out and we can rearrange the remaining terms in the following way

φ k ∂ t (φ k v k ) + ∂ x φ k α∂ x φ k α∂ x v k = φ 2 k R k (v) ,
where the function R k gathers all the subprincipal terms we are now able to estimate

R(v) L 2 ≤ C k ( v W 3,∞ ) ( φ k v k L 2 + v H k ) .
Taking the inner product with v k , we obtain 1 2

d dt φ k v k 2 L 2 = φ k R k α |φ k v k ,
and then, using Lemma 2, we find some constant C k such that

d dt φ k v k 2 L 2 ≤ C k ( v W 3,∞ ) ( φ k v k L 2 + v H k ) φ k v k L 2 .
Now, from the Proposition 2, without renaming the constants for convenience and by summing the last inequalities for s ≥ k ≥ 1, each one being multiplied by the suitable constant, we get

(18) d dt |v| 2 s ≤ max k {C k ( v W 3,∞ )} v 2 H s
Integrating and calling the maximal constant C, we get

|v(t, •)| 2 s ≤ |v 0 | 2 s + t 0 C( v(τ, •) W 3,∞ ) v(τ, •) 2 H s dτ .
Finally, using Proposition 2, we write

v(t, •) 2 H s ≤ c s |v 0 | 2 s + t 0 C( v(τ, •) W 3,∞ ) v(τ, •) 2 H s dτ .
We finish the proof by Gronwall's lemma.

Existence of a smooth solution

This section is devoted to the proof of local well-posedness for (qKdV). We first state the existence and uniqueness of a smooth solution to a parabolic regularized equation with regularized initial data. Then, using uniform a priori bounds in large norms on this smooth solution, we take a limit to prove the existence of solutions to (qKdV). Here, we shall adapt a method by Bona & Smith [START_REF] Bona | The initial-value problem for the Korteweg-de Vries equation[END_REF] and prove directly that the convergence occurs in the very space C (0, T ; H s (R)) and check uniqueness and continuity of the solution map with respect to the initial data.

Study of a regularized equation

Let us introduce a small parameter ε > 0. For now, we consider the regularized parabolic equation

(19) v t + av x + ∂ x α∂ x α∂ x v + ε 4 ∂ 4 x v = 0 .
Let χ be a function of class C ∞ such that its Fourier transform is compactly supported and equals 1 in a neighborhood of the origin and η ∈ C ∞ (R * + ) a non decreasing function with limit 0 at 0 that will be specified later in the proof. Denote

χ ε = 1 η(ε) χ • η(ε)
.

Given v 0 ∈ H s (R), we define a regularized initial data by

(20) v 0,ε = χ ε * v 0 .
Regarding the existence of a unique solution to [START_REF] Linares | On the regularity of solutions to a class of nonlinear dispersive equations[END_REF] with initial data v 0,ε , we refer to results on analytic semigroups of semi-linear PDEs in [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF] ( §7.3.2) and [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] ( §8.4) combined with semigroup techniques in [START_REF] Benzoni-Gavage | Well-posedness of one-dimensional Korteweg models[END_REF][START_REF] Benzoni-Gavage | On the wellposedness for the Euler-Korteweg model in several space dimensions[END_REF]. More precisely, in their work on the Euler-Korteweg system, S. Benzoni-Gavage, R. Danchin, S. Descombes used a similar fourth order regularization to prove the local existence of solutions of linear problems with variable coefficients with a time of existence independent of ε. They use properties of the analytic semigroup generated by ∂ 4

x and the Duhamel formula to prove the existence and uniqueness by a fixed point method. Their technique can be directly applied to our semi-linear regularized problem [START_REF] Linares | On the regularity of solutions to a class of nonlinear dispersive equations[END_REF]. Thus Eq. ( 19) has a unique solution belonging to C (0, T ; H ∞ (R)), with a time of existence T > 0 depending on the initial data v 0 and ε.

Then, we consider a sequence of smooth solutions (v ε ) ε>0 . To take a limit when ε tends to zero, we have to justify that the time of existence of the solution may be bounded from below independently of ε. We look for uniform a priori bounds on the solution v ε using the techniques presented in the above section. We differentiate Eq. ( 19) by respecting the skew-symmetry of the third order term and use a gauge to cancel remainders. After those two operations, we get that, for all k ≥ 0 ( 21)

φ k ∂ t (φ k v k ) + ∂ x φ k α∂ x φ k α∂ x v k + φ k ε 4 ∂ 4 x (φ k v k ) = φ 2 k R k + ε 4 φ 2 k R ε ,
where the term R k contains all previous zero order remainders and the term R ε is a commutator term arising from the fourth order regularization. To deal with this new commutator term, we first prove the following lemma

Lemma 3. Let Q(v, ∂ x v, • • • , ∂ k+2p x v) be an homogeneous polynomial of weight k + 2p ∈ [k, 2k
) with terms multiplied by bounded functions of v. There exists a constant µ > 0 such that for all v ∈ H k (R) and for all small ε > 0,

ε 2p Q(v, ∂ x v, • • • , ∂ k+2p x v)|∂ k x v ≤ C µ v 2 H k + ε 2p µ ∂ k+p x v 2 L 2 .
Proof. The proof uses that of Lemma 1. A general form of Q is

Q(v, ∂ x v, • • • , ∂ k+2p x v) = i β i (v) k j=1 ∂ γ j x v ,
where the β i are bounded on J and for all 1 ≤ j ≤ k, γ j ≤ k + 2p satisfy k j=1 γ j = k + 2p. First, we take the inner product with ∂ k x v and use an integration by parts to get an expression of the form

P (v, ∂ x v, • • • , ∂ k+p x v)|∂ k+p
x v Now, thanks to the triangular inequality, it is sufficient to work on a monomial expression. Without changing notations for convenience, we consider

P (v, ∂ x v, • • • , ∂ k+p x v) = k j=1 ∂ γ j x v .
where for all 1 ≤ j ≤ k, γ j ≤ k + p satisfy k j=1 γ j = k + p. If for all j, γ j ≤ k, we proceed exactly as in Lemma 1 to get

P |∂ k+p x v v H k ∂ k+p x v L 2 .
Now, if there is a factor with k < γ l ≤ k + p, we use the again Gagliardo-Nirenberg inequality. We know there could be only one because p ≤ k/2.

∂ γ l x v L 2 ∂ k+p x v θ L 2 ∂ k x v 1-θ L 2 , where θ = γ l -k p < 1 .
Finally, combining with previous terms, we have by Young's inequality

P (v, ∂ x v, • • • , ∂ k+p x v) L 2 v 1-θ H k ∂ k+p x v θ L 2 C µ v H k + µ ∂ k+p x v L 2 .
Now, returning to our first polynomial expression

P (v, ∂ x v, • • • , ∂ k+2p x v)|∂ k x v ≤ C µ v H k + µ ∂ k+p x v L 2 ∂ k+p x v L 2 .
Using another time Young's inequality, we find new constants µ and C µ such that

P (v, ∂ x v, • • • , ∂ k+2p x v)|∂ k x v ≤ C µ v 2 H k + µ ∂ k+p x v 2 L 2 .
To conclude the proof, we multiply by the bounded factor ε 2p and find the constant µ to be the maximum of the µ obtained for the various monomials P .

Let us now come back to the regularized equation ( 19), and prove the following proposition.

Proposition 4. Let v 0 ∈ H q (R) with q ≥ 4. For s ≥ q, the unique solution of [START_REF] Linares | On the regularity of solutions to a class of nonlinear dispersive equations[END_REF] with regularized initial data v 0,ε defined in [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF]

satisfies v(t, •) H s v 0 H q η(ε) s-q .
Proof. We follow the steps of proof of Proposition 3. We first take the inner product of (25) with v k to obtain 1 2

d dt φ k v k 2 L 2 + ε 2 ∂ 2 x (φ k v k ) 2 L 2 = φ k R|φ k v k + ε 4 φ k R ε |φ k v k .
Using the techniques of the previous section and the result of lemma 3, we obtain by choosing the Young's inequality constant such that µ < 1/2, 1 2

d dt φ k v k 2 L 2 + 1 2 ε 2 ∂ 2 x (φ k v k ) 2 L 2 ≤ C ∞ φ k v k 2 L 2 + C k ( v L ∞ , v 1 L ∞ ) v H k φ k v k L 2 .
Now, summing on 1 ≤ k ≤ s and multiplying by the suitable constants C k at each step,

d dt |v| 2 s + c s k=1 ε 2 ∂ 2 x (φ k v k ) 2 L 2 ≤ max k C k v 2 H s .
By integration with respect to time

|v(t, •)| 2 s + t 0 c s k=1 ε 2 ∂ 2 x (φ k v k ) 2 L 2 dτ ≤ |v 0 | 2 s + t 0 C v 2 H s dτ .
Finally,

v(t, •) 2 H s + c s t 0 c s k=1 ε 2 ∂ 2 x (φ k v k ) 2 L 2 dτ ≤ c s |v 0 | 2 s + t 0 C v 2 H s dτ .
This last estimate and the norm equivalence give in particular

v(t, •) 2 H s v 0 H s + t 0 C v 2 H s dτ .
A classical mollifier property from appendix C in [START_REF] Benzoni-Gavage | On the wellposedness for the Euler-Korteweg model in several space dimensions[END_REF] gives us

v 0 H s ≤ C v 0 H q η(ε) s-q .
We finish the proof by Gronwall's lemma.

In particular, for initial data v 0 ∈ H s (R), we get a uniform bound of the solution in H s (R) for any s ≥ 4, that is independent of ε. In the proof by S. Benzoni-Gavage, R. Danchin, S. Descombes in [START_REF] Benzoni-Gavage | Well-posedness of one-dimensional Korteweg models[END_REF], this uniform estimate is actually used directly in the fixed point argument to justify that the time of existence of the solution v ε is independent of the regularization parameter ε. Here we obtain this uniformity a posteriori. From now on, we denote by T > 0 the minimal common time of existence of all the solutions in the sequence (v ε ) ε>0 depending only on v 0 .

Convergence to a solution of (qKdV)

From our regularized equations, we have a sequence of solutions (v ε ) ε>0 belonging to C (0, T ; H ∞ (R)) for some T > 0 given the same initial data v 0,ε regularized from v 0 ∈ H q (R), q ≥ 4 for all the sequence. We shall prove that this sequence is a Cauchy sequence in C (0, T ; H s (R)) for any s ≥ q ≥ 4.

For 0 < δ ≤ ε, we denote by v ε and v δ the two corresponding solutions of ( 19) and we look for estimates on z = v ε -v δ . Then, our goal is to prove that z L ∞ (0,T ;H s (R)) goes to zero when ε and δ go to zero. We compute the difference between the two equations on v ε and v δ to find

z t + a(v δ )z x + ∂ x α(v δ )∂ x α(v δ )∂ x z + δ 4 ∂ 4 x z = (ε 4 -δ 4 ) ∂ 4 x v ε + (a(v δ ) -a(v ε )) ∂ x v ε + ∂ x α(v δ )∂ x α(v δ )∂ x v ε -∂ x α(v ε )∂ x α(v ε )∂ x v ε .
We rewrite it in a more compact way [START_REF] Rodrigues | Periodic-coefficient damping estimates, and stability of large-amplitude roll waves in inclined thin film flow[END_REF] z t + a ε,δ z

x + ∂ x 1 2 α 2 ε z xx + ∂ x α δ ∂ x α δ ∂ x z + δ 4 ∂ 4 x z = ε 4 -δ 4 ∂ 4 x v ε + F ε,δ (z) ,
with obvious notations. F ε,δ is a linear function with respect to z of homogeneous weight 3 and

a ε,δ = a ε,δ (v ε , ∂ x v ε , ∂ 2 x v ε , v δ , ∂ x v δ , ∂ 2 x v δ ) .
In this last formulation, we gathered all the subprincipal terms. Again, the first order one can be estimated by a direct computation since ∂ x a ε,δ ∈ L ∞ (R) according to the estimate on the solutions v ε and v δ . Moreover, to take the limit, we will need estimates on low derivatives of the difference z to compensate the loss involved by the high derivatives in v ε . More precisely, the arising of terms with too many derivatives on the coefficients v ε forces us to use estimates given by Proposition 4 and then concede an inverse factor of ε.

To recover this factor we shall prove that a low number of derivatives on the difference z = v ε -v δ can compensate this loss in ε. Then, we shall prove the following lemma Lemma 4. For 0 < δ ≤ ε, let v ε (respectively v δ ) denote the smooth solutions of [START_REF] Linares | On the regularity of solutions to a class of nonlinear dispersive equations[END_REF] with parameter ε (respectively δ) and regularized initial data v 0,ε (respectively v 0,δ ) with v 0 ∈ H q (R), q ≥ 4. Then, for all 0 ≤ p ≤ q,

∂ p x (v ε -v δ ) L ∞ (0,T ;L 2 (R)) = o(η(ε) q-p )
when ε goes to zero.

Proof. To get these new estimates, we start by looking for an estimate in L 2 (R) and then in H q (R). The structure of Eq. ( 22) is obviously different from the one we have worked with previously. In fact, it is principally the arising of the second order term ∂ x 1 2 α 2 ε z xx which causes troubles. Again, we use a gauge φ ε,δ to deal with this term and multiply the equation by φ 2 ε,δ . Exactly as before, by computing commutators we find

(23) φ ε,δ ∂ t (φ ε,δ z) + φ ε,δ a ε,δ ∂ x (φ ε,δ z) + ∂ x α δ φ ε,δ ∂ x α δ φ ε,δ ∂ x z + δ 4 φ ε,δ ∂ 4 x (φ ε,δ z) = φ 2 ε,δ (ε 4 -δ 4 ) ∂ 4 x v ε + φ 2 ε,δ F ε,δ (z) + δ 4 φ ε,δ [φ ε,δ , ∂ 4 x ]z .
In the previous expression, we have already used the cancellation due to our gauge and the remainders of order zero are gathered in the term F ε,δ . As before, the ODE the gauge has to satisfy is

3α 2 δ φ ε,δ ∂ x φ ε,δ = -φ 2 ε,δ ∂ x 1 2 α 2 ε .
To solve this last equation, we rewrite it as

(24) ∂ x φ ε,δ φ ε,δ = - 1 3 ∂ x α ε α ε - ∂ x (α 2 ε ) 6 1 α 2 δ - 1 α 2 ε .
The first part of the right hand side is directly integrable and causes no trouble and the second part belongs to L 1 (R). This yields that φ ε,δ exists and belongs to L ∞ (R) and thus, from Eq. ( 24), we can ensure that for 0 ≤ l ≤ 3

∂ l x φ ε,δ ∈ L ∞ (R) .
Moreover, using the equations satisfied by v ε and v δ and the estimates we have for both of them, we find by a Gagliardo-Nirenberg inequality

∂ t φ ε,δ L ∞ C + (ε 4 ∂ 5 x v ε L ∞ + δ 4 ∂ 5 x v δ L ∞ ) , C + ε 4 η(ε) 3/2 + δ 4 η(δ) 3/2 , ≤ C ,
for a well chosen function η. Those properties justify that all the remainders from commutators gathered in F ε,δ are bounded in L 2 (R) as the gauge and all its derivatives arising in the computation are bounded in L ∞ (R). So the adding of this gauge does not change anything from what was done in the previous section, especially regarding the norm equivalences.

There are two terms left to control. The first one

δ 4 φ ε,δ [φ ε,δ , ∂ 4 x ]z
is treated using the same scheme of proof as in Lemma 3. By the Gagliardo-Nirenberg inequality and the estimates on (v ε ) ε>0 , we get constants µ > 0 and C µ such that

δ 4 φ ε,δ [φ ε,δ , ∂ 4 x ]z|z ≤ δ 4 C µ v n 2 L 2 + δ 4 µ ∂ 2 x z 2 L 2
For the second one, we write

φ ε,δ ε 4 -δ 4 ∂ 4 x v ε |φ ε,δ z ε 4 ∂ 4 x v ε L 2 φ ε,δ z L 2 ε 4 η(ε) 4-q v 0 H q φ ε,δ z L 2 .
Finally, choosing Young's inequality constant such that µ < 1/2, the estimate we get by taking the inner product of eq. ( 23) with z is 1 2

d dt φ ε,δ z 2 L 2 + δ 4 2 ∂ 2 x (φ ε,δ z) 2 L 2 ≤ C φ ε,δ z 2 L 2 + ε 4 η(ε) 4-q v 0 H q φ ε,δ z L 2 .
An integration in time and Gronwall's lemma yield

φ ε,δ z(t) L 2 ≤ φ ε,δ z(0) L 2 + ε 4 η(ε) 4-q v 0 H q .
A classical property on mollifiers, again from appendix C of [START_REF] Benzoni-Gavage | On the wellposedness for the Euler-Korteweg model in several space dimensions[END_REF], and norm equivalences finally give

z L ∞ (0,T ;L 2 (R)) = o(η(ε) q ) + ε 4 η(ε) 4-q = o(η(ε) q ) ,
for a suitable choice of η, namely

η(ε) = ε β with β < 1.
This last relation yields the desired estimate in L 2 (R). Let us now focus on the property in H q (R) To do this, we need to differentiate the equation q times using again our differential operator α(v δ )∂ x and the corresponding gauge φ q (v δ ) from the previous section. We recall that z satisfies

z t + a ε,δ z x + ∂ x 1 2 α 2 ε z xx + ∂ x α δ ∂ x α δ ∂ x z -δ 4 ∂ 4 x z = ε 4 -δ 4 ∂ 4 x v ε + F ε,δ (z) ,
The total gauge we will use is Φ q = φ q (v δ )φ ε,δ . Then, exactly as before, differentiating q times, multiplying by Φ 2 q and computing commutators (25)

Φ q ∂ t (Φ q z q ) + Φ q a ε,δ ∂ x (Φ q z q ) + ∂ x α δ Φ q ∂ x α δ Φ q ∂ x z q + δ 4 Φ q ∂ 4 x Φ q z q = Φ 2 q F ε,δ (z, • • • , z q ) + δ 4 Φ q [Φ q , ∂ 4 x ]z q + δ 4 (Φ q ) 2 [(∂ x (α δ •)) q , ∂ 4 x ]z + Φ 2 q (ε 4 -δ 4 ) (∂ x (α δ •)) q ∂ 4 x v ε .
In this last expression, we have already used cancellations from the gauges and the term F ε,δ contains all the remainders of commutators we have encountered before. This term involves up to q + 3 derivatives on v ε and up to q derivatives on v δ and will have to be estimated again to make sure it remains bounded uniformly in ε. In the following we deal with the four terms in the right hand side.

Let us deal first with the second one. This term is the same as in the L 2 case treated previously and we will later set Young's inequality constant to control it.

For the third one, we use directly the lemma 3 with k = 4 and p = 2 to get that there exists constants µ > 0 and C µ such that

δ 4 [(∂ x (α δ •)) q , ∂ 4 x ]z|z q C µ z 2 H q + δ 4 µ ∂ 2 x z q 2 L 2 .
We rewrite the last term as

Φ 2 q ε 4 -δ 4 (∂ x ((α δ -α ε )•)) q ∂ 4 x v ε + Φ 2 q ε 4 -δ 4 (∂ x (α ε •)) q ∂ 4 x v ε .
In the first part, it appears at most order q derivatives of z and order q + 4 derivatives of v ε . We rewrite it as a sum of terms of the general form

ε 4 -δ 4 ∂ q x z ∂ q+4-l x v ε ,
with 0 ≤ l ≤ q. Using a bootstrap argument, we get the estimate

ε 4 -δ 4 ∂ l x z ∂ q+4-l x v ε L 2 ε 4 ∂ q x z L 2 ∂ q+4-l x v ε L ∞ = o(η(ε) q-l ) ε 4 η(ε) (4-l)/2+(5-l)/2 = o(1) .
For the second part of this last term, the term which has the worst possible loss of derivative is

ε 4 -δ 4 v ε ∂ q+4 x v ε . 21
In this case, we estimate as before

ε 4 -δ 4 v ε ∂ q+4 x v ε L 2 ε 4 η(ε) 4 = o(1) ,
with our definition of the function η. This final estimate proves that the last term in eq. ( 25) goes to zero when ε goes to zero.

We are left with F ε,δ , which we rewrite in general form

i β i (v ε , v δ , z) j,l,m ∂ γ j x v ε ∂ γ l x v δ ∂ γm x z ,
where for all j, l and m, γ j ≤ q + 3, γ l ≤ q, γ m ≤ q and j,l,m γ j + γ l + γ m = q + 3 .

If γ j ≤ q then the Gagliardo-Nirenberg inequality gives us our estimate as 3 ≤ γ m ≤ q and there is no loss in 1/η(ε) involved. The issue occurs when there is more than q derivatives on either v ε . In this case, γ l ≤ 3 and the corresponding factors are bounded in L ∞ . Then, we write for 1 ≤ l ≤ 3

∂ q+3-l x v ε L ∞ ∂ l x z L 2 = o(η(ε) q-l ) η(ε) 3+1/2-l = o(1) .
Finally, let us gather all we have done before. We take the inner product of Eq. (25) by z q and, gathering all Young's inequalities constants such that their sum is less than 1/2, we can write 1 2

d dt Φ q z q 2 L 2 + δ 4 2 ∂ 2 x (Φ q z q ) 2 L 2 Φ q z q 2 L 2 + F (ε) Φ q z q || L 2 ,
where we gathered in F (ε) = o(1) all the previously treated terms. Gronwall's lemma yields

(26) Φ q z q (t, •) L 2 Φ q z q (0, •) L 2 + F (ε) ,
and with mollifiers properties

Φ q z q L ∞ (0,T ;L 2 (R)) = o(1) .
Together with φ ε,δ z 2 L ∞ (0,T ;L 2 (R)) = o(η(ε) q ), we complete the proof by interpolation and norm equivalences.

Corollary 1. The sequence (v ε ) ε>0 is a Cauchy sequence in C (0, T ; H s (R)) for any s > 3 + 1/2. Then, its limit v ∈ C (0, T ; H s (R)) ∩ C 1 (0, T ; H s-3 (R)) is a solution to (qKdV) with initial data v 0 .

As announced, here we adapt a technique originally introduced by Bona & Smith in [START_REF] Bona | The initial-value problem for the Korteweg-de Vries equation[END_REF] and later exploited in many papers, see [START_REF] Benzoni-Gavage | Local well-posedness of nonlocal Burgers equations[END_REF][START_REF] Benzoni-Gavage | Well-posedness of one-dimensional Korteweg models[END_REF][START_REF] Benzoni-Gavage | On the wellposedness for the Euler-Korteweg model in several space dimensions[END_REF] for example. We prove the following theorem Theorem 2. For an integer s ≥ 4, let K be a strictly positive constant. For all v 0 ∈ H s (R) of norm not greater than K, the mapping

H s (R) → C (0, T ; H s (R)) ∩ C 1 (0, T, H s-3 (R)) v 0 → v , solution of (qKdV) with initial data v 0 is continuous.
Proof. We aim at proving that for any sequence of initial conditions (v n 0 ) n≥0 going to v 0 in H s (R), then the corresponding sequence of solutions (v n ) n≥0 goes to v, the solution corresponding to the initial data v 0 . We start by writing

v n -v H s ≤ v n -v n ε H s + v n ε -v ε H s + v ε -v H s .
We first focus on the first and third terms. Let us rewrite what we obtained in (26). For ε ≥ δ > 0, using norm equivalences and taking the limit δ → 0, we get

(27) v ε (t, •) -v(t, •) H s ≤ C K ( v ε,0 -v 0 H s + F (ε)) ,
where F (ε) goes to zero when ε goes to zero. This kind of estimate is also true for the difference between the solutions v n ε and v n (28)

v n ε (t, •) -v n (t, •) H s ≤ C K v n ε,0 -v n 0 H s + F (ε) .
Moreover, we have

(29) v n ε,0 -v n 0 H s ≤ v n ε,0 -v ε,0 H s + v ε,0 -v 0 H s + v 0 -v n 0 H s , ≤ 2 v n 0 -v 0 H s + v ε,0 -v 0 H s .
Now for the second term, we have to revisit the proof of Proposition 4 and more precisely the way we obtained estimate (26). Here, we have to estimate the difference between two solutions with same regularization parameter ε, but which satisfy the same regularized equation with different initial data. We proceed exactly the same way with some cancellations due to the fact that we actually take δ = ε in the computation. The only point where we cannot follow the proof concerns the terms evaluated by

∂ s+3-q x v ε L ∞ ∂ q x (v n ε -v ε ) L 2 ,
for 0 ≤ q ≤ 3. Indeed, we have balanced the coefficients involving ε to actually get uniform estimates with respect to this parameter (it was the purpose of Lemma 4) but here, we cannot do the same. In fact we cannot hope to have a uniform estimate in ε but, by a Gagliardo-Nirenberg inequality we find

∂ s+3-q x v ε L ∞ ∂ q x (v n ε -v ε ) L 2 ≤ C η(ε) 3-q+1/2 v n ε -v ε H s .
Finally, we obtain the following estimate (30)

v n ε -v ε L ∞ H s ≤ C ε v n 0 -v 0 H s ,
with the constant C ε going to +∞ when ε goes to zero. Now, using all previous estimates

v n -v L ∞ H s ≤ C K v n ε,0 -v n 0 H s + F (ε) + C ε v n 0 -v 0 H s + C K ( v ε,0 -v 0 H s + F (ε)) ,
which finally yields

v n -v L ∞ H s ≤ 2C K ( v ε,0 -v 0 H s + F (ε)) + (2C K + C ε ) v n 0 -v 0 H s .
Now we find that lim sup

n→+∞ v n -v L ∞ H s ≤ 2C K ( v ε,0 -v 0 H s + F (ε)) .
This finishes the proof as the right hand side of the last inequality goes to zero when ε goes to zero for any initial condition v 0 .

The proof of uniqueness is a straightforward corollary of the previous one. Instead of considering the difference between v n and v, we consider the difference between two different solutions u and v but with the same initial data u 0 = v 0 . Following the exact same steps of the previous proof, with v ε,0 = u ε,0 , we deduce

u -v L ∞ H s ≤ 2C K ( v ε,0 -v 0 H s + F (ε)) .
Then, we get uniqueness when taking the limit ε goes to zero.

Concluding remarks

The present well-posedness result is set on the real line x ∈ R. Nevertheless, as the proof use only the structure of the qKdV equation and does not uses any dispersion estimate, nothing prevents us from considering the same problem set on the torus R/ΞZ for any period Ξ > 0. Then, we are able to give a similar well-posedness result on a unidimensional torus.

Theorem 3. Assume that k ≥ 4. If p = -f : I ⊂ R → R is C k+1 and κ : I → R + * is C k+2 , then for all Ξ > 0, v 0 ∈ H k (R/ΞZ), there exists a time T > 0 and a unique v ∈ C (0, T ; H k (R/ΞZ))∩C 1 (0, T ; H k-3 (R/ΞZ)) solution to (3) with initial data v 0 . Moreover, v 0 → v maps continuously H k (R/ΞZ) into C (0, T ; H k (R/ΞZ)) ∩ C 1 (0, T ; H k-3 (R/ΞZ)).

Though we have not carried out such task here, it is likely that one may adapt the strategy expounded here to deal with non integer indices of regularity and relax the constraint k ≥ 4 to k > 3 + 1/2.

At first, we were aiming at a well-posedness result compatible with the study of the non-linear stability of a known solution. More precisely, we were initially looking for a solution of the equation as a perturbation around a bounded and infinitely differentiable given solution of (qKdV). Nevertheless, it appears that the gauge technique does not work well in this case. The functions defining the gauge are harder to construct when coefficients depend not only the unknown solution but also on the known profile, especially when both parts have different localization properties. A perspective we did not investigate here would be the study of a perturbations around a periodic state. The periodicity property, used similarly as in [START_REF] Rodrigues | Periodic-coefficient damping estimates, and stability of large-amplitude roll waves in inclined thin film flow[END_REF], should enable us to overcome the previous difficulty and should give us the suitable gauge estimates.

Acknowledgement. The author would like to thank S. Benzoni-Gavage and L.M. Rodrigues for their helpful guidance through many fruitful discussions. This work has been supported by ANR project BoND (ANR-13-BS01-0009-01). The author also wants to thank Frédéric Rousset for several insightful comments made during some of the BoND meetings.


Younes Bouchaala
email: younes.bouchaala@inria.fr

Oyunchimeg Shagdar
email: oyunchimeg.shagdar@inria.fr

Paul Muhlethaler
email: paul.muhlethaler@inria.fr

Eva Team

Analysis of broadcast strategies and network parameters in IEEE 802.11p VANETs using simple analytical models

This paper proposes and analyzes different broadcast strategies in IEEE 802.11p Vehicular Ad-hoc NETworks (VANETs). The first strategy is the default IEEE 802.11p strategy. Using a model derived from the Bianchi model, we provide the network performance in terms of throughput and success rate. The second strategy is to use an acknowledgment technique similar to the acknowledgment with point-to-point traffic. A node will send its broadcast packet as in the default case,but it requires an acknowledgment from a neighbor node. This node may be a random neighbor or may be selected according to precise rules. We analyze this second strategy in terms of throughput and success rate. Somewhat surprisingly, we show that this second strategy improves the delivery ratio of the transmitted packets but reduces the overall throughput. This means that if the CAM messages (Cooperative Awareness Messages) are broadcasted, the total number of packets actually delivered will be greater with the default strategy than with the improved strategy. We propose a third strategy which consists in using the default strategy for normal packets, but we add random redundant transmissions to ensure greater reliability for very important packets. We show that with this simple technique, not only do we obtain suitable reliability, but we also achieve larger global throughput than with the acknowledgment-oriented technique. Another contribution of this paper is to compute network performance in terms of throughput and success rate with respect to the network parameters and to analyze their impact on performances.

I. INTRODUCTION

IEEE 802.11p was proposed as the main communication protocol to offer Wireless Access in Vehicular Environments (WAVE) [START_REF] Team | Wireless Access in Vehicular Environments (WAVE) Draft Standard[END_REF]. IEEE802.11p is required to support Intelligent Transportation Systems (ITS) applications by providing communication between vehicles (V2V), and between vehicles and roadside infrastructure (V2I). Nowadays, one of the main ITS applications expected by vehicle manufacturers is safety applications that rely on the broadcast principle. Therefore, a reliable broadcast scheme is necessary to ensure the reliable reception of critical messages such as priority Cooperative Awareness Messages (CAM) [START_REF]Intelligent Transport Systems (ITS) -Vehicular Communications -Basic Set of Applications -Part 2 : Specification of Cooperative Awareness Basic Service[END_REF] and Decentralized Environmental Notification Messages (DENM) [START_REF]Intelligent Transport Systems (ITS); Vehicular Communications; Basic Set of Applications; Part 3: Specifications of Decentralized Environmental Notification Basic Service[END_REF]. In the contentionbased Medium Access Control (MAC) of IEEE802.11p, several studies have shown a correlation between an increase in the number of connected vehicles and increase in packet loss rate. Several approaches to improve the reliability of broadcasting have been proposed in the literature, as reported in Section II. This paper's main contribution is to propose and to analyze two broadcast strategies and to compare them with the default IEEE 802.11p broadcasting method [START_REF] Xu | Non-saturated throughput analysis of ieee 802.11 ad hoc networks[END_REF], [START_REF] Wang | Characterising the Behaviour of IEEE 802.11 Broadcast Transmissions in Ad Hoc Wireless LANs[END_REF]. We propose a mathematical model derived from the Bianchi [START_REF] Bianchi | Performance Analysis of the IEEE 802.11 Distributed Coordination Function[END_REF] model to analyze the network performances of the default broadcast service of the IEEE 802.11p protocol in terms of throughput and packet delivery ratio, The remainder of this paper is organized as follows. Section II briefly reviews related work; Section III describes the proposed system model and the analytical model. We propose three different broadcast techniques. Simulation results are reported in Section IV. Finally Section V concludes the paper.

II. RELATED WORK

The fundamental Medium Access Control (MAC) technique of the IEEE 802.11 based Wireless Local Area Networks (WLANs) is known as the Distributed Coordination Function (DCF). DCF is a Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) scheme that assumes all packet losses within a WLAN are due to packet collision. To avoid to packet losses, DCF triggers a binary slotted exponential backoff procedure. In the contention-based MAC used in IEEE 802.11, as well as the amendment IEEE 802.11p, packet loss greatly depends on the channel contention, therefore, many studies have been carried out in the literature to evaluate the system throughput for WLANs as well as for Wireless Access in Vehicular Environments. In his performance Analysis of the IEEE 802.11 DCF, Bianchi [START_REF] Bianchi | Performance Analysis of the IEEE 802.11 Distributed Coordination Function[END_REF] provided an analytical model to evaluate the throughput performance of both basic access and Request To Send/Clear To Send (RTS/CTS) access mechanisms as well as a combination of the two assuming a finite number of terminals and ideal channel conditions. Bianchi demonstrated the accuracy of his model for predicting the system throughput.

Unfortunately, most IEEE 802.11 DCF performance evaluation studies proposed in the literature cover the unicast transmission mode. There do, however exist a few studies related to the IEEE 802.11 protocol in broadcast mode. Hafeez et al. [START_REF] Hafeez | Performance analysis and enhancement of the dsrc for vanet's safety applications[END_REF] have proposed an analytical study in which they model each terminal as one two-dimensional Markov chain to calculate the probability of successful transmissions of the periodic status messages (CAM), as well as the priority messages (DENM). They show that their model gives accurate results when estimating the recommended throughput level of IEEE 802.11p for each category of messages. Ghahramani et al. [START_REF] Ghahramani | A new hybrid model for performance evaluation of ieee 802.11p broadcast mode in vehicular ad hoc networks: A numerical analysis[END_REF] start from the assumption that the number of contending vehicles in VANETs varies, enabling them to model the dynamicity of the contending terminals and to add more accuracy to the existing methodology of IEEE 802.11p MAC broadcast mode evaluation. The study in [START_REF] Wang | Characterising the Behaviour of IEEE 802.11 Broadcast Transmissions in Ad Hoc Wireless LANs[END_REF] proposes a simple analysis of IEEE 802.11 in broadcast mode. In this paper we re-use the results of [START_REF] Wang | Characterising the Behaviour of IEEE 802.11 Broadcast Transmissions in Ad Hoc Wireless LANs[END_REF] and we adapt and exploit them in the context of VANETs. Since we also propose to use reliable broadcast, we will use and adapt the study of the point-to-point mode as our broadcast transmissions will require acknowledgment.

III. SYSTEM MODEL

We use models which are very close to the Bianchi model [START_REF] Bianchi | Performance Analysis of the IEEE 802.11 Distributed Coordination Function[END_REF]. In these models M is the number of vehicles in the network within the same carrier sense range; λ is the packet arrival rate in a station assumed to be Poisson and the duration of a packet is denoted by T .

The other parameters are related to the IEEE 802.11p protocol. These parameters are : the duration of a mini-slot σ, and the number of back-off slots W . σ should be greater than the sensing delay of the Carrier Sense Multiple Access (CSMA) scheme of the IEEE 802.11p protocol. In usual implementations, σ is of the same order as the sensing delay of CSMA. W is the greatest back-off window that a node can select. In practice, W is set to the maximum duration of the back-off. When IEEE 802.11 uses an acknowledgment to enhance the transmission success rate, another parameter n is required in order to fix the maximum number of retransmissions before a packet is discarded.

All the models derived from the Bianchi model assume that the channel can be modeled as a succession of slots. Each slot may be a mini-slot (of duration σ) or a slot of duration T . The mini-slots are used to represent the time intervals during which the channel is idle. There is no activity during these intervals and thus the nodes which are in back-off mode just decrement their back-off counters. When the back-off counter reaches zero, the node transmits its packet. The slot of duration T corresponds to the transmission of a packet. This transmission will succeed if only one node transmits in this slot as there will be a collision if several nodes transmit simultaneously.

The models derived from [START_REF] Bianchi | Performance Analysis of the IEEE 802.11 Distributed Coordination Function[END_REF] all introduce τ the node transmission rate at the beginning of a slot. If no node transmits, this occurs with a probability of (1 -τ ) M thus the current slot will be a mini-slot of duration σ. If at least one node is transmitting, the current slot will of duration T , which occurs with a probability of 1 -(1 -τ ) M . Thus, the mean duration between two slots will be

(1-(1-τ ) M )T +(1-τ ) M σ.
This duration is called the duration of a pseudo-slot. It is also possible to compute the throughput t of the system

t = (1 -(1 -τ ) M )T (1 -(1 -τ ) M )T + (1 -τ ) M σ
The successful throughput is t s is given by

t s = M τ (1 -τ ) M -1 T (1 -(1 -τ ) M )T + (1 -τ ) M σ. ( 1
)
t s is also the probability of successful transmission for a randomly transmitted packet.

Thus, the performance of the network is completely defined if we can compute τ . The models derived from [START_REF] Bianchi | Performance Analysis of the IEEE 802.11 Distributed Coordination Function[END_REF] are Markovian models whose states are the value of the backoff counter. When there are retransmissions, the value of the back-off counter is complemented by the number of previous transmissions. The transitions in these models are simple. When the station is in the idle state, the transmission to a backoff state (between 0 and W ) is random with the probability of 1/(W + 1). When the station is in back-off in the state 1 ≤ k + 1 ≤ W , the transition is towards state k with the probability of p e = (1 -τ ) M , this means that there is no transmission. With probability 1 -p e the station with back-off counter k + 1 remains in the same state. When the back-off counter reaches 0, the state transmission rate to the idle state is 1.

A. Pure broadcast

We first consider a model without retransmission, which represents the default operation mode of 802.11p. In the model with no retransmission [START_REF] Wang | Characterising the Behaviour of IEEE 802.11 Broadcast Transmissions in Ad Hoc Wireless LANs[END_REF] the resolution of the steady-state of the Markov chain leads to the following equation:

b 0 = τ = 1 q + 1 + W 2(1 -τ ) M -1
.

(

) 2
with:

q = 1 -e -λ((1-(1-τ ) M )T +(1-τ ) M σ) . (3)
where b 0 denotes the probability that the node has a pending packet whose back-off is 0. q denotes the probability of at least one packet arriving during a pseudo-slot. A pseudo slot is a slot of duration equal to the mean duration of a slot on the channel, i.e. σ weighted by the probability of an idle slot plus T weighted by the probability of a transmission slot. Using (2) and ( 3) we obtain the following fix-point equation in τ :

τ = 1 1 -e -λ((1-(1-τ ) M )T +(1-τ ) M σ) + 1 + W 2(1 -τ ) M -1 (4)
which can be easily solved numerically. If τ is known via (4) then the successful throughput can be easily computed using [START_REF] Team | Wireless Access in Vehicular Environments (WAVE) Draft Standard[END_REF].

B. Broadcast with acknowledgment request

We will also discuss a model with retransmission. The protocol is still broadcast but we will assume that a node requests an acknowledgment from, for example, a random neighbor. If the acknowledgment is sent, the node will transmit another packet. If not, the packet is transmitted again and the collision window is doubled. The collision window reaches 2 n W after n collision. If a packet reaches n + 1 collisions, the packet is dropped. Thus even if the transmission remains in broadcast mode, the protocol operates as in unicast mode with an acknowledgment. We can apply the model derived in [START_REF] Daneshgaran | Unsaturated throughput analysis of IEEE 802.11 in presence of non ideal transmission channel and capture effects[END_REF]. In the model with retransmission (up to n retransmissions) the resolution of the steady-state of the Markov chain leads to the following equation:

b 0 = τ = 2(1 -2P col )q q[(W + 1)(1 -2P col ) + W P coll (1 -(2P coll ) n ] + 2(1 -q)(1 -P col )(1 -2P col ) (5)
with

P coll = 1 -(1 -τ ) n-1 -(n -1)τ (1 -τ ) n-2 . ( 6
)
If we use ( 3) and ( 6) in ( 5) we obtain a fix-point equation in τ . The probability of successful transmission for a packet actually sent is:

1 -(1 -t s ) n .
It is possible to compute the network performance with a similar algorithm. A node still requests an acknowledgment from, for example, a random neighbor but if there is no acknowledgment, the packet is simply retransmitted without any change to the collision window. This is done up to n times, after which the packet is discarded. In such a case, ( 5) is simply changed into the following equation:

τ = b 0,0 1 -P col = 2q q(W + 1) + 2(1 -q)(1 -P col ) (7)
The proof of this equation is given in the appendix. As q and P col depends on τ , we obtain a fix-point equation in τ .

C. Broadcast with n random retransmissions

To improve the probability of successful transmission for a small number of dedicated packets, we propose using random transmissions for these packets. These packets are randomly re-transmitted n -1 times, thus, in total, a packet will be transmitted n times. We assume that the number of additional packets due to these retransmissions is negligible and so the station transmission probability is τ given by (4). The probability of successful transmission for a normal packet is t s which is computed by [START_REF] Team | Wireless Access in Vehicular Environments (WAVE) Draft Standard[END_REF]. The probability of successful transmission for a packet with n retransmissions is

1 -(1 -t s ) n .

D. Parameters of vehicular networks

We assume that the vehicles are randomly located on nb lanes. In one lane, the mean distance between two vehicles is l. We denote by cs the carrier sense1 , when a node transmits at distance up to cs from the current node the channel is sensed busy, see Figure 1. For a distance larger than cs, the channel is sensed idle. Thus, given cs, nb and l we can compute the number of vehicles M within the same carrier sense area.

M = 2 × cs × nb l . (8)
We assume that each vehicle periodically sends messages, and even if it is not completely true, we assume that the traffic is Poisson with a mean rate corresponding to the synchronous traffic. The value of M can be used in ( 3), ( 4), ( 5) and ( 7) to compute τ .

In this paper we assume that the M nodes which are within carrier sense range of the current node are those which can create a collision with this node if the current node and another node within this area transmit nearly simultaneously. In this paper we also assume that the transmission from the current node is local2 so that hidden collisions from nodes outside the carrier sense range are not possible. Thus collisions can only occur when transmissions are simultaneous.

IV. SIMULATION RESULTS

We use the following figures λ = 10, σ = 77 bits. The packet size, including the overhead, is 3998 bits and the data rate is 6 Mbps. We use (if not otherwise defined) l = 25 m for the mean distance between two vehicles on a lane. We vary the carrier sense range cs from 300 m to 1400 m.

A. Effect of the carrier range and the transmission strategy

In this section we vary cs and we study the different transmission strategies.

In Figure 2, we present the percentage of success for each broadcast strategy for a transmitted packet. This means that for a transmitted packet we compute the probability that this packet is successfully received. We observe that the acknowledgment procedure has a significant impact on the success rate, especially when the network load is very high. In Figure 3 we present the percentage of successfully transmitted packets. This is the ratio of the number of successfully received packets divided by the number of generated packets. In this ratio we include the loss due to collision and the loss due to the limitation of the total bandwidth. We observe that the techniques using acknowledgments produce a smaller percentage of received packets. This is because managing retransmissions in the techniques using acknowledgments consumes more bandwidth than the others. In Figure 4 we compare techniques using acknowledgments with and without using the binary exponential back-off. We observe that the percentage of successfully transmitted packets is higher with the binary exponential back-off. This is because this back-off reduces congestion. Without binary exponential back-off, the model does not compute any stable equilibrium for cs ≥ 1128 if W = 16 , for cs ≥ 1179 if W = 32 and for cs ≥ 1304 if W = 64. Actually, we have the same phenomenon with the binary exponential back-off but de-stabilization occurs with larger values of cs. The binary exponential back-off helps reduce congestion when the load increases.

When we have a target for the percentage of successfully transmitted packets, Figures 3 and4 can be used to compute the convenient carrier sense range to reach this goal. For instance, if we must satisfy a percentage of successful transmissions greater than 0.95, then we must use a carrier sense range smaller than 800 m.

Figure 5 shows the effect of the maximum number of retransmissions on the percentage of successfully transmitted packets. We observe that when n increases, the percentage of successfully transmitted packets also increases, but above a small threshold (n=4) the gain obtained with larger values of n becomes small. Above n=8 there is nearly no advantage in increasing n.

B. Effect of the distance between the vehicles

In this section, we vary the distance l between vehicles and we use a simple transmission strategy with a constant back-off window of 32. We consider the following figures cs = 300, 600, 900 or 1200 m for the carrier sense range. In Figure 6, we observe that for nb = 6 and nb = 8 the percentage of successfully transmitted packets is low when cs = 900, 1200 m. Figure 6 can also be used to find the suitable network parameters to ensure given performance thresholds.

C. Effect of the number of lanes

In this section, we vary the number of lanes nb. We use a simple transmission strategy with a constant back-off window of 32. We consider the following figures cs = 300, 600, 900 and 1200 m for the carrier sense range . In Figure 7, we observe that, for nb = 6 and nb = 8, the percentage of successfully transmitted packets is low when cs = 900, 1200 m.

V. CONCLUSION

In this paper, we have shown that simple models allow network performance of IEEE 802.11p to be obtained. Thus if we can estimate the important parameters of a VANET : packet generation rate, packet length, distance between vehicles, number of lanes, carrier sense range, we can easily evaluate the success rate of a random transmission. We have studied various transmission techniques with and without acknowledgments. We have shown that using acknowledgments incurs overhead which degrades the overall performance of the network in terms of packets successfully transmitted while it improves the success rate of actually transmitted packets. A feasible solution could be to use the simple scheme without any acknowledgment but a few blind transmissions of the same packet could be performed when this packet contains very important information. We would obtain the same effect as with a transmission with an acknowledgment request.

VI. APPENDIX

We prove [START_REF] Hafeez | Performance analysis and enhancement of the dsrc for vanet's safety applications[END_REF]. The proof is an adaptation of [START_REF] Daneshgaran | Unsaturated throughput analysis of IEEE 802.11 in presence of non ideal transmission channel and capture effects[END_REF]. We use the same notation3 and use the same transmission state diagram represented in Figure 8. The only modification in [START_REF] Daneshgaran | Unsaturated throughput analysis of IEEE 802.11 in presence of non ideal transmission channel and capture effects[END_REF] is in W

i ; we have 1 ≤ i ≤ n W i = W instead of W i = 2 i W .
b i,k denotes the stationary probability that a node waits for a transmission with a back-off counter k ∈ [1, W -1] for the ith transmission (there have been i -1 previous unsuccessful transmission attempts).

We still have b i,0 = P i col b 0,0 and b n,0 = P n col 1-P col . We also have:

b I = (1 -q)(1 -P col ) n i=0 b i,0 + (1 -q)b I = (1 -q)(1 -P col ) q n i=0 b i,0 = b 0,0 1 -q q .
For k ∈ For i = 0 we have:

P i coll + P n coll 1 -P col + 1 1 -P col + 2(1 -q) q = b 0,0 2 W 1 -P col + 1 1 -P col + 2(1 -q) q
Thus b 0,0 1 -P col = τ = 2q q(W + 1) + 2(1 -q)(1 -P col ) and we remark that the computation does not depend on the maximum number of retransmissions n.

Fig. 1 .

 1 Fig. 1. Number of vehicles in the carrier sense range of a transmitter.

Fig. 2 .

 2 Fig. 2. Percentage of transmitted packets successfully received.

Fig. 3 .

 3 Fig. 3. Percentage of transmitted packets successfully received versus carrier sense range.

Fig. 4 .

 4 Fig. 4. Percentage of transmitted packets successfully received versus carrier sense range.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Percentage of transmitted packets successfully received versus carrier sense range for various values of n.

Fig. 7 .

 7 Fig. 7. Percentage of transmitted packets versus the number of lanes.

[ 1 ,Fig. 8 .

 18 Fig. 8. State diagram of the back-off scheme with retransmission but with a constant back-off window (no binary exponential back-off).

  For i ∈ [1, n -1] we have:

we express the carrier sense in meters but it can also be expressed in Watts or in decibel

for instance the transmission is for the neighbor vehicles

instead of m used in[START_REF] Daneshgaran | Unsaturated throughput analysis of IEEE 802.11 in presence of non ideal transmission channel and capture effects[END_REF] we use n and instead of Peq we use P col i.e m-> n and Peq-> P col