Error Estimates for Scalar Conservation Laws by a Kinetic Approach

We use the kinetic approach of Perthame and Tadmor (1991) to calculate the error estimates for general scalar conservation laws governing problems in gas dynamics or fluid mechanics in general. The Kružkov and Kuznetsov techniques are generalized to this method, and an error bound of order √ ε (where ε is the mean free path) is obtained.


Introduction
The kinetic approaches for scalar conservation laws have attracted attention for the last few years.Much work has been done in this field, however, important developments such as error estimates have not been totally investigated.The Perthame-Tadmor approach [16] is here used to derive the error bound for the following Cauchy problem of conservation laws: where (t,x) ∈ [0,+∞[×R N , u = u(t,x) ∈ R, and where the functions A i (•) = a i (•) are locally Lipschitz in R.
It is well known that there is a lack of regularity of the solutions for t > 0, so that we generally look for weak solutions for (1.1)-(1.2).Moreover, there is a unique weak solution of physical interest, satisfying the additional Kružkov [10] entropy condition: ∈ R being any real scalar.For more details, see for example the classical book of Smoller [17] and the references therein.Now, the problem (1.1)- (1.3), which can be seen as a macroscopic interpretation of gas dynamics, is generally approached by the viscosity method or by methods of numerical purpose.
In this work, we are concerned with a simple microscopic model which is not only adapted to solve the problems of existence, uniqueness, and convergence for scalar conservation laws, but adapted to calculate the error of convergence as well.
Our method consists in using the Perthame-Tadmor model which is analogous to the models of the kinetic theory of gases.This is a BGK model derived from Bhatnagar et al. [1], and contains most of the basic properties of fluid dynamics, mainly conservation of mass, momentum, and energy (see the book by Bouchut et al. [2], and/or the book by Cercignani [6]).A function f (t,x,v) where v is "velocity" is considered.It represents the distribution of "molecules" located at x, having a speed v, at time t.
Let us recall that a discrete-time version of the Perthame-Tadmor approach [16] was previously and simultaneously proposed by Brenier [4,5], by Giga and Miyakawa [9], and by Miyakawa [14].In [16], the authors prove the convergence towards the unique solution of (1.1)- (1.3).We are concerned here with the exact error of convergence.
The paper is organized as follows.In Section 2, we introduce the method of Perthame and Tadmor, giving the main definitions and results.In Section 3, we develop an approximate entropy inequality with a precise "entropy" remainder.Then we give the error estimates for the corresponding conservation law based on the Kružkov-Kuznetsov techniques in Section 4. Section 5 concerns some parabolic case.

Around the kinetic model
In this section, we give the main definitions and results due to Perthame and Tadmor.In 1991, they proposed the following BGK (say pseudo-BGK) model: ) as a kinetic interpretation of the conservation law (1.1)-(1.2) when ε → 0 (see [16] and also Lions et al. [13]).The scalar function is a signature function (the pseudo-Maxwellian), and ε > 0 the mean free path in comparison with the Boltzmann theory.If such a solution f ε is defined, the approximate solution for (1.1) is simply the function while the signature χ w is given by (2.4) A. Omrane 3 Remark 2.1.We immediately deduce the following formulae: and |w| = v∈R |χ w (v)|dv for any w ∈ R.Moreover, 2.1.The well-posedness of the kinetic problem.Consider the linear version of (2.1) (without the term χ uε at the right-hand side).The classical method of characteristics involves solving the equations (2.7) With the varying constant method, the solution of (2.1) is of the following form: (see Omrane [15]).We also have the L 1 contraction (here w ε = v g ε ) where f ε and g ε are two solutions of (2.1).This gives (2.10) , then there is a unique solution to (2.1).Moreover, by (2.10), the kinetic solution is continuously dependent on the initial data.We also have the contraction inequality , and the inequality of contraction (2.11) holds true.
Proof.See the article of Perthame and Tadmor [16] for the proof.

Finite speed of propagation.
In [16], we find the following.We first notice that for any t ∈ [0,T], with the help of Jensen's inequality.Denoting Consequently, the term a(v

and the elements |a
because of the decay of the initial data in (2.8).

The kinetic entropies H( f ε
).The notion of "kinetic" entropy has the same sense as the classical entropy for conservation laws.It is just a microscopic version of it, and the name is due to the presence of the speed vector a(v).
Definition 2.4.The kinetic entropies are the functions H( f ε ) such that for any positive test function ϕ ∈ Ᏸ(R t (according to the H-theorem in Boltzmann theory).
Multiplying (2.1) by sgn( f ε − χ ), ∈ R, and integrating by parts, we have ) Notice that is any given scalar (it may also be any function uniformly bounded in t and x).

The approximate entropy solutions
In this section, we show that the Kružkov method can be applied.Let (s, y) ∈ (0,T) × R N be the new variables, we then define Q = (0,T) 2 × R N x × R N y and ϕ = ϕ(t,s,x, y).Proposition 3.1.Let u = u(s, y) be the unique entropy solution of (1.1)- (1.3), and ) the approximate kinetic solution given by (2.1).Then for any test function ϕ ∈ Ᏸ(Q).
Proof.Recall that ∈ R can be chosen as any = (s, y) for example, uniformly bounded with respect to s and y.We write With the definition on the kinetic entropies and the following triangle inequality: we obtain Using Remark 2.1, and the finite speed of propagation property (2.14), we deduce From another side, the exact entropy solution u(s, y) of (1.1)- If we replace (s, y) by u(s, y) (the exact solution) in (3.5), and (t,x) by u ε (t,x) in (3.6), then we sum up to obtain (3.1).

The test functions.
We define ϕ as follows: where ψ η is the characteristic function defined by and where the function θ h is the classical mollifier.For h > 0, we also have We have the following corollary.Corollary 3.2.Under the hypothesis of Proposition 3.1, the following entropy inequality holds: Proof.To get (3.9), it is convenient to choose ϕ as in (3.7) (see [15,18]).With this, we obtain hence we have the left-hand side of (3.9).

Bounds.
Let us begin with some properties of the test function.For t 0 and t 1 in [0,T], we have A. Omrane 7 Indeed, note that Moreover, we easily see that , for i = 0,1. (3.13) In particular (3.11) holds, and we have the lemma.
Let u(s, y) be the entropy solution of (1.1)-( 1.3), and u ε (t,x) the approximate solution given by (2.3), then (3.14) Proof.Using (3.11), we write 8 Error bounds to conservation laws by kinetic approaches and with the same arguments.
We now control the term sup 0≤t≤T f ε − χ uε L 1 (R N x ×Rv ) at the right-hand side of (3.14).
Proposition 3.4.Consider the kinetic initial data for every t > 0, where Proof.Suppose that f ε (0,x,v) BV(R N ;L 1 (Rv)) is uniformly bounded in ε, then using the contraction inequality (2.11), the corresponding solution f ε (t,x,v) satisfies that for every that is for every t ≥ 0 and every ε > 0, It is important to notice that we do not necessarily recover the L 1 regularity of the term and not for a(v) A. Omrane 9 where z = t,x i , for i = 1,...,N.Then we can write Using this definition together with the finite speed of propagation property (2.14), we obtain By (3.22) and (3.24), we also deduce That is for all T > 0 and every 0 ≤ t ≤ T, we have (3.27) Remark 3.5.Denote u ε (0,x) ≡ u ε0 (x) := u ε0 .Then, it is not restrictive to replace lim ε→0 f ε (0,x,v) − χ uε0 (v) L 1 (R N x ×Rv ) = 0 by f ε (0,x,v) = χ uε0 (v) in the above proposition.This is useful for the applications.

.29)
Proof.If we take the simplified initial data form (Maxwellian data) of Remark 3.5, then Proposition 3.4 gives Moreover, noticing that we immediately deduce the result from Lemma 3.3.

Generalization to the Kuznetsov technique
In this section, we will estimate the difference where t 0 and t 1 are in [0,T], T > 0. The techniques we will use are not new (see, e.g., Bouchut and Perthame [3]), and are known as the Kuznetsov techniques (see Kuznetsov [11], and Kuznetsov and Volshin [12]).The new thing here is the generalization of these techniques to the kinetic approach of Perthame and Tadmor, as the approximate solution is obtained from the kinetic model.We prepare by the lemma.
Proof.The proof is classical and relies on the translation property in x and the contraction (2.11).
Proposition 4.2.Suppose that the hypothesis of Proposition 3.6 is satisfied.Then Proof.We split the integral I using (3.12) into I = I 0 + I 1 relatively to t 0 and t 1 , where For t = t 0 , we use the triangle inequality u(s, y) − u ε (t,x) ≤ u(s, y) − u t 0 , y + u t 0 , y − u t 0 ,x which also may be written as A. Omrane 11 We then integrate as follows: θ h x i − y i dx dy dt ds At the left-hand side, we have At the right-hand side, we have the following: using Lemma 4.1.We also have using the change of variables Finally,

.13)
For t = t 1 , we consider the following triangle inequality: With the same techniques, we obtain We sum up by (4.16) A. Omrane 13 Theorem 4.3.Suppose that initially f ε (0,x,v) ∈ L 1 (R v ;BV(R N )), which is compactly supported in v, and that f ε (0,x,v) ≡ f 0 (x,v) = χ u0(x) (v).Let u(s, y) be the entropy solution of (1.1)- (1.3), and u ε (t,x) the approximate solution given by (2.3).Then for any T > 0, and t 0 , t 1 in [0,T], the following L 1 x error bound holds true, with C being a positive constant independent on ε.
Proof.From Proposition 3.6, we have where M is given by (3.29).Hence (4.19) The optimal choice of h is h = √ ε.Then where

The parabolic case
In this short section, we discuss the kinetic approach for the following parabolic case: where Δ = N i=1 is the Laplacian operator.The corresponding kinetic model is which reads as We multiply by sgn( f ε − χ ) and integrate over v, we then easily obtain in the weak sense that We consider here | f ε − χ | as the "kinetic entropies" for (5.2) in the sense that (5.5) We do the same analysis as in Sections 3 and 4. The error found is the same.Notice the following remainder: where x i0 , i 0 ∈ 1,...,N, is the space variable so that |∂x i f | ≤ |∂x i0 f |.
Remark 5.1.It would be interesting to study the kinetic approach to the case of parabolichyperbolic problems in the sense of Chen and Karlsen in [7] and Chen and Perthame in [8].

Remark 4 . 4 .
Notice that the error is global in time, and optimal in the sense that we cannot have more than √ ε, since we use the Kuznetsov techniques.
• ∇ x f ε either.The quantities |∂ f ε /∂t| and |∂ f ε /∂x i | for 1 ≤ i ≤ N are only measures that we can bound by the BV norm.Indeed, for ϕ ∈ Ᏸ, denote by • ᏹ the norm