Asymptotic Limit to Shocks for Scalar Balance Laws Using Relative Entropy

and Applied Analysis 3 Lemma 3. LetU be a solution of (1).Then, for every t ∈ (0, T), one has ‖U(t, ⋅)‖ L ∞ (R) ≤ 󵄩󵄩󵄩󵄩u0 󵄩󵄩󵄩󵄩L∞(R) + 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩L∞(R)t. (17) Proof. From the scalar balance law in (1), we get ∂ t U + ∂ x A (U) − ε∂ 2 xx U ≤ 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩L∞(R). (18) Since ‖u 0 ‖ L ∞ (R) + t‖g‖L∞(R) satisfies (18) and |u0(x)| ≤ ‖u 0 ‖ L ∞ (R) for all x ∈ R, the comparison principle for parabolic equations provides U (t, x) ≤ 󵄩󵄩󵄩󵄩u0 󵄩󵄩󵄩󵄩L∞(R) + t 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩L∞(R) (19) for all (t, x) ∈ (0, T) ×R. In the same method, we also get U (t, x) ≥ − ( 󵄩󵄩󵄩󵄩u0 󵄩󵄩󵄩󵄩L∞(R) + t 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩L∞(R)) (20) for all (t, x) ∈ (0, T) ×R. The idea of the proof is to study the evolution of the relative entropy of the solution with respect to the shock, outside of a small region centered at X(t) (this small region corresponds to the layer localization):

where (, ) :=  0 ( − ()), and  0 is defined by (2) This is  2 stability result to a shock for balance laws up to a shift function.The main point is how to construct a shift function () such that the time derivative of the relative entropy is smaller than convergence rate.Our method is based on the method developed in Leger and Vasseur [1,2] together with using the relative entropy idea and the result cannot be true without shift (see [1]).
The relative entropy method introduced by Dafermos [3,4] and Diperna [5] provides an efficient tool to study the stability and asymptotic limits among thermomechanical theories, which is related to the second law of thermodynamics.They showed, in particular, that if  is a Lipschitzian solution of a suitable conservation law on a lapse of time [0, ], then for any bounded weak entropic solution  it holds for a constant  depending on  and .Since Dafermos [3] and Diperna [5]'s works, there has been much recent progress as applications of the relative entropy method.Chen et al. [6] have applied the relative entropy method to obtain the stability estimates to shocks for gas dynamics which derive the time asymptotic stability of Riemann solutions with large oscillation for the 3 × 3 system of Euler equations.For incompressible limits, see Bardos et al. [7,8], Lions and Masmoudi [9], and Saint Raymond et al. [10][11][12][13] who have studied incompressible limit problems.
There are also many recent results of the weak-uniqueness for the compressible Navier-Stokes equations together with using relative entropy by Germain [14] and Feireisl and Novotný [15].For the relaxation there is an application for compressible models by Lattanzio and Tzavaras [16,17] and we can also see Berthelin et al. [18,19] as some applications of hydrodynamical limit problems.However, in all those cases, the method works as long as the limit solution has a good regularity such that the solution is Lipschitz.This is due to the fact that strong stability as (10) is not true when  has a discontinuity.It has been proven in [1,2], however, that some shocks are strongly stable up to a shift.Choi and Vasseur [20] have recently used this stability property to study sharp estimates for the inviscid limit of viscous scalar conservation laws to a shock.With the same idea, Kwon and Vasseur [21] develop sharp estimates of hydrodynamical limits to shocks for BGK models.For this paper, we derive the optimal rate of convergence to shocks for scalar balance laws up to a shift function ().Thus, it generalizes Choi and Vasseur's work [20].The outline of this paper is as follows.In Section 2 we introduce relative entropy and some properties used in Leger [1].In Section 3 we will derive some estimates of the hyperbolic and parabolic part of relative entropy.In Section 4, we will give the proof of Theorem 1 together with combining the estimates in Section 3. Finally, in the Appendix section, we will add the appendix to give the proof of Proposition 7.

Relative Entropy and Some Properties
In this section we introduce a special drift function (),  ∈ (0, ), defined in Leger [1] and relative entropy.To begin with we need some notations and properties provided in Leger [1].Fix any strictly convex function  ∈  2 ; we first define the normalized relative entropy flux (⋅, ⋅) by where the associated relative entropy functional (⋅ | ⋅) is given by and the flux of the relative entropy (⋅, ⋅) is defined by Note that for any fixed  and any weak entropic solution  of (1), we have Hence,  can be seen as a typical velocity associated to the relative entropy (⋅, ).
Using the strict convexity of the function , Leger showed in [1] the following lemma.

Lemma 2. Let 𝑥, 𝑦 ∈ [−𝐿, 𝐿] for any 𝐿 > 0.
There exists a constant Λ > 0, such that one has In the spirit of Leger [1], we consider the solution of the following differential equation in order to define the shift function : Note that the existence and uniqueness of  come from the Cauchy-Lipschitz theorem.
where we used the fact ‖()‖  ∞ ≤  for  > 0 in the following.
The idea of the proof is to study the evolution of the relative entropy of the solution with respect to the shock, outside of a small region centered at () (this small region corresponds to the layer localization): for any  > 0, where the constant  depends on .From now on we will take a reasonable  > 0 and it will be mentioned in (40) later.
For the rigorous proof, we define the evolution of the integration in (21) as follows: for any fixed  > 0 and  ∈  1 ([0, ]), where an increasing function   is defined by From now on we delete  in   .Thus, the derivative of E   () implies the following lemma.
The proof is provided in Choi and Vasseur [20].We next need a regularity to control hyperbolic part the lemma is as follows.

Estimates on the Hyperbolic and Parabolic Terms
In this section, we prove that the hyperbolic part  1 +  1 in equality (25) is strictly negative and the parabolic part  2 + 2 has a small rate of convergence.Applying Lemmas 2 and 5, we are able to show the main proposition for this section.Proposition 6.Let  1 and  1 be as in Lemma 4.Then, there exists a constant  > 0 such that, for any , , and satisfying we have Proof.Let us start with proving that  1 is strictly negative.The proof of  1 is similar.With the definition of (), we write  1 as where (, ) := [((, ()), (  +  )/2)−((, ),   )].
Consequently, combining the two last inequalities gives the desired result.
We are now going to introduce the parabolic term,  2 + 2 , and the proof is provided in [20] To get good estimate, we take a specific   .For any  ≥ 1/, we now fix the function   in the following explicit way: We use the computation: For the proof of (I), we integrate the estimate of Proposition 7 between 0 and  ∈ (0, ) such that, for any ,  with 1/ ≤  and  ≤ , where  is the constant from Proposition 7, it follows that for any  ≤  0 , which proves (6) by taking (V) = V 2 and applying Gronwall's inequlity.
To end with the proof, it remains to prove (9).Let us define the function  by (48)