Asymptotic Limit to Shocks for Scalar Balance Laws Using Relative Entropy

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 690801, 7 pages http://dx.doi.org/10.1155/2014/690801 Research Article Asymptotic Limit to Shocks for Scalar Balance Laws Using Relative Entropy Young-Sam Kwon Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea Correspondence should be addressed to Young-Sam Kwon; Received 22 April 2014; Accepted 4 July 2014; Published 16 July 2014 Academic Editor: Milan Pokorny Copyright © 2014 Young-Sam Kwon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider a scalar balance law with a strict convex flux. In this paper, we study inviscid limit to shocks for scalar balance laws up to a shift function, which is based on the relative entropy. 1. Introduction We consider the following balance law in one-dimensional space R: 2 𝑈, 𝜕𝑡 𝑈 + 𝜕𝑥 𝐴 (𝑈) = 𝑔 (𝑈) + 𝜖𝜕𝑥𝑥 𝑈 (0, 𝑥) = 𝑈0 (𝑥) , (1) 𝑡 > 0, 𝑥 ∈ R, where the flux 𝐴󸀠󸀠 (V) := 𝑎󸀠 (V) ≥ 𝑐 for some constant 𝑐 > 0 and 𝑈0 ∈ 𝐿∞ (R). The existence of global unique weak solutions of (1) has been studied by Kruzkov. In this paper, we are interested in getting the optimal rate of convergence linked to a layer. Let us consider the shock solutions of the scalar conservation laws with the given source term (1) with the initial data 𝐶 𝑆0 (𝑥) = { 𝐿 𝐶𝑅 if 𝑥 < 0, if 𝑥 ≥ 0, (2) with two constants 𝐶𝐿 > 𝐶𝑅 , where the source term 𝑔 is defined as follows: 1 ∞ 𝑔 ∈ 𝐶 (R) ∩ 𝐿 (R) , 𝑔 (𝐶𝐿 ) = 𝑔 (𝐶𝑅 ) = 0. (3) Then, the Rankine-Hugoniot condition ensures that the function 𝑆0 (𝑥 − 𝜎𝑡) with 𝜎 := 𝐴 (𝐶𝐿 ) − 𝐴 (𝐶𝑅 ) 𝐶𝐿 − 𝐶𝑅 (4) is a solution to (1) with 𝜖 = 0. Notice that the condition 𝐶𝐿 > 𝐶𝑅 implies that they verify the entropy conditions; that is, 𝜕𝑡 𝜂 (𝑈) + 𝜕𝑥 𝐺 (𝑈) − 𝜂󸀠 (𝑈) 𝑔 (𝑈) ≤ 0, 𝑡 > 0, 𝑥 ∈ R, (5) for any convex functions 𝜂, and 𝐺󸀠 = 𝜂󸀠 𝐴󸀠 . An easy dimensional analysis shows that, because of those layers, we may have in general ‖𝑈 (𝑡) − 𝑆 (⋅ − 𝜎𝑡)‖2𝐿2 ≥ 𝐶𝜀, (6) for some 𝜖 > 0 which means that the 𝐿2 stability for two solutions 𝑈, 𝑆 does not hold. We are interested in deriving the extremal 𝐿2 stability up to a shift function. The main result is as follows. Theorem 1. Let 𝐶𝐿 > 𝐶𝑅 , 𝑇 > 0 be any number, and let 𝑈0 ∈ 𝐿∞ (R) be such that (𝑈0 − 𝑆0 ) ∈ 𝐿2 (R) , ( 𝑑 𝑈 ) ∈ 𝐿2 (R) . 𝑑𝑥 0 + (7) Suppose that 𝑈 is a solution of (1). Then there exists a Lipschitz curve 𝑋 ∈ 𝐿∞ (0, 𝑇), 𝐶 := 𝐶(‖𝜂󸀠󸀠 ‖𝐿∞ , ‖𝑔󸀠 ‖𝐿∞ , 𝑇), and 𝜖0 > 0 such that 𝑋(0) = 0 and for any 0 < 𝜖 < 𝜖0 , 1 󵄩2 󵄩 ‖𝑈 (𝑡) − 𝑆 (𝑡)‖2𝐿2 (R) ≤ 𝐶 (󵄩󵄩󵄩𝑈0 − 𝑆0 󵄩󵄩󵄩𝐿2 (R) + 𝜖 log ) , 𝜖 𝑡 ∈ (0, 𝑇) , (8) 2 Abstract and Applied Analysis where 𝑆(𝑡, 𝑥) := 𝑆0 (𝑥 − 𝑋(𝑡)), and 𝑆0 is defined by (2). Moreover, this curve satisfies 󵄨󵄨 ̇ 󵄨󵄨 󵄨󵄨𝑋 (𝑡)󵄨󵄨 ≤ 𝐶, 󵄨 󵄨 (9) 1 󵄩2 󵄩 |𝑋 (𝑡) − 𝜎𝑡|2 ≤ 𝐶𝑡1/4 (󵄩󵄩󵄩𝑈0 − 𝑆0 󵄩󵄩󵄩𝐿2 (R) + 𝜖 log ) . 𝜖 2 This is 𝐿 stability result to a shock for balance laws up to a shift function. The main point is how to construct a shift function 𝑋(𝑡) 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 𝑈 is a Lipschitzian solution of a suitable conservation law on a lapse of time [0, 𝑇], then for any bounded weak entropic solution 𝑈 it holds 󵄨2 󵄨2 󵄨 󵄨 ∫ 󵄨󵄨󵄨󵄨𝑈 (𝑡) − 𝑈 (𝑡)󵄨󵄨󵄨󵄨 𝑑𝑥 ≤ 𝐶 ∫ 󵄨󵄨󵄨󵄨𝑈 (0) − 𝑈 (0)󵄨󵄨󵄨󵄨 𝑑𝑥, R R (10) for a constant 𝐶 depending on 𝑈 and 𝑇. 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. [10–13] 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 𝑈 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 𝑋(𝑡). 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. 2. Relative Entropy and Some Properties In this section we introduce a special drift function 𝑋(𝑡), 𝑡 ∈ (0, 𝑇), 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 𝜂 ∈ 𝐶2 ; we first define the normalized relative entropy flux 𝑔(⋅, ⋅) by 𝑓 (𝑥, 𝑦) := 𝐹 (𝑥, 𝑦) , 𝜂 (𝑥 | 𝑦) (11) where the associated relative entropy functional 𝜂(⋅ | ⋅) is given by 𝜂 (𝑥 | 𝑦) := 𝜂 (𝑥) − 𝜂 (𝑦) − 𝜂󸀠 (𝑦) (𝑥 − 𝑦) (12) and the flux of the relative entropy 𝐹(⋅, ⋅) is defined by 𝐹 (𝑥, 𝑦) := 𝐺 (𝑥) − 𝐺 (𝑦) − 𝜂󸀠 (𝑦) (𝐴 (𝑥) − 𝐴 (𝑦)) . (13) Note that for any fixed 𝑦 and any weak entropic solution 𝑈 of (1), we have 2 𝑈 + 𝑔 (𝑈)) . 𝜕𝑡 𝜂 (𝑈 | 𝑦)+𝜕𝑥 𝐹 (𝑈, 𝑦) = (𝜂󸀠 (𝑈) − 𝜂󸀠 (𝑦)) (𝜖𝜕𝑥𝑥 (14) Hence, 𝑓 can be seen as a typical velocity associated to the relative entropy 𝜂(⋅, 𝑦). Using the strict convexity of the function 𝜂, 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 (...truncated)