# Asymptotic Limit to Shocks for Scalar Balance Laws Using Relative Entropy

Authors: Young-Sam Kwon
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

We consider the following balance law in one-dimensional spaceR:
(1)∂tU+∂xA(U)=g(U)+ϵ∂xx2U,111111111111U(0,x)=U0(x),111111111111111t>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
(2)S0(x)={CLifx<0,CRifx≥0,
with two constants CL>CR, where the source term g is defined as follows:
(3)g∈C1(R)∩L∞(R),g(CL)=g(CR)=0.Then, the Rankine-Hugoniot condition ensures that the function(4)S0(x-σt)withσ:=A(CL)-A(CR)CL-CR
is a solution to (1) with ϵ=0. Notice that the condition CL>CR implies that they verify the entropy conditions; that is,
(5)∂tη(U)+∂xG(U)-η′(U)g(U)≤0,t>0,x∈R,
for any convex functions η, and G′=η′A′. An easy dimensional analysis shows that, because of those layers, we may have in general
(6)∥U(t)-S(·-σt)∥L22≥Cɛ,
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.
LetCL>CR, T>0 be any number, and let U0∈L∞(R) be such that
(7)(U0-S0)∈L2(R),(ddxU0)+∈L2(R).
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,
(8)∥U(t)-S(t)∥L2(R)2≤C(∥U0-S0∥L2(R)2+ϵlog⁡1ϵ),1111111111111111111111111111111t∈(0,T),
where S(t,x):=S0(x-X(t)), and S0 is defined by (2). Moreover, this curve satisfies
(9)|X˙(t)|≤C,|X(t)-σt|2≤Ct1/4(∥U0-S0∥L2(R)2+ϵlog⁡1ϵ).This isL2 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
(10)∫R‍|U(t)-U¯(t)|2dx≤C∫R‍|U(0)-U¯(0)|2dx,
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 functionX(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
(11)f(x,y):=F(x,y)η(x∣y),
where the associated relative entropy functional η(·∣·) is given by
(12)η(x∣y):=η(x)-η(y)-η′(y)(x-y)
and the flux of the relative entropy F(·,·) is defined by
(13)F(x,y):=G(x)-G(y)-η′(y)(A(x)-A(y)).
Note that for any fixed y and any weak entropic solution U of (1), we have
(14)∂tη(U∣y)+∂xF(U,y)=(η′(U)-η′(y))(ϵ∂xx2U+g(U)).
Hence, f can be seen as a typical velocity associated to the relative entropy η(·,y).Using the strict convexity of the functionη, Leger showed in [1] the following lemma.Lemma 2.
Letx,y∈[-L,L] for any L>0. There exists a constant Λ>0, such that one has(i)
1/Λ≤η′′(x)≤Λ,(ii)
(1/2Λ)(x-y)2≤η(x∣y)≤(1/2)Λ(x-y)2,(iii)
|F(x,y)|≤Λ(x-y)2,(iv)
0≤(∂xf)(x,y)≤Λ,(v)
1/Λ≤(∂yf)(x,y).In the spirit of Leger [1], we consider the solution of the following differential equation in order to define the shift function X:
(15)X˙(t)=f(U(t,X(t)),CL+CR2),X(0)=0.
Note that the existence and uniqueness of X come from the Cauchy-Lipschitz theorem.First,X is Lipschitz, since we have from Lemma 2(16)|X˙(t)|≤|F(U(t,X(t)),(CL+CR)/2)|η(U(t,X(t))∣(CL+CR)/2)≤2Λ2,
where we used the fact ∥U(t)∥L∞≤L for t>0 in the following.Lemma 3.
LetU be a solution of (1). Then, for every t∈(0,T), one has
(17)∥U(t,·)∥L∞(R)≤∥u0∥L∞(R)+∥g∥L∞(R)t.Proof.
From the scalar balance law in (1), we get
(18)∂tU+∂xA(U)-ϵ∂xx2U≤∥g∥L∞(R).
Since ∥u0∥L∞(R)+t∥g∥L∞(R) satisfies (18) and |u0(x)|≤∥u0∥L∞(R) for all x∈R, the comparison principle for parabolic equations provides
(19)U(t,x)≤∥u0∥L∞(R)+t∥g∥L∞(R)
for all (t,x)∈(0,T)×R. In the same method, we also get
(20)U(t,x)≥-(∥u0∥L∞(R)+t∥g∥L∞(R))
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 atX(t) (this small region corresponds to the layer localization):
(21)∫-∞X(t)-δɛ‍η(U(t,x)∣CL)dx+∫X(t)+δɛ∞‍η(U(t,x)∣CR)dx.
Indeed, for η(x∣y)=(x-y)2, the following holds:
(22)∫{|x-X(t)|≤Cη}‍η(U(t,x)∣S(t,x))dx≤C|{|x-X(t)|≤Cη}|≤Cη,
for any η>0, where the constant C depends on T. From now on we will take a reasonable δ>0 and it will be mentioned in (40) later.For the rigorous proof, we define the evolution of the integration in (21) as follows:
(23)Eϵδ(t):=∫-∞∞‍[ϕδ(|x-X(t)|ɛ)]2η(U(t,x)∣S(t,x))dx
for any fixed δ>0 and X∈C1([0,T]), where an increasing function ϕδ is defined by
(24)ϕδ(x)={0ifx≤0,1ifx≥δ.
From now on we delete δ in ϕδ. Thus, the derivative of Eϵδ(t) implies the following lemma.Lemma 4.
The functionEϵδ(t), defined in (23), satisfies the following on (0,T):(25)ddtEϵδ(t)=∫X(t)-δɛX(t)2ɛϕ′(-x+X(t)ɛ)ϕ(-x+X(t)ɛ)×[X˙(t)η(U∣CL)-F(U,CL)]dx+∫-∞X(t)[ϕ(-x+X(t)ɛ)]2ϵ∂xx2U(t,x)×(η′(U(t,x))-η′(CL))dx+∫-∞X(t)[ϕ(-x+X(t)ɛ)]2g(U(t,x))×(η′(U(t,x))-η′(CL))dx-∫X(t)X(t)+δɛ2ɛϕ(x-X(t)ɛ)ϕ′(x-X(t)ɛ)×[X˙(t)η(U∣CR)-F(U,CR)]dx+∫x(t)∞[ϕ(x-X(t)ɛ)]2ϵ∂xx2U(t,x)‍×(η′(U(t,x))-η′(CR))dx+∫X(t)∞[ϕ(x-X(t)ɛ)]2g(U(t,x))‍×(η′(U(t,x))-η′(CR))dx:=L1+L2+L3+R1+R2+R3.The proof is provided in Choi and Vasseur [20]. We next need a regularity to control hyperbolic part the lemma is as follows.Lemma 5.
For anyt∈(0,T), there exists C:=C(∥g′∥L∞(R),T)>0 such that one gets
(26)∥(∂xU)+∥L2(R)≤C∥(ddxU0)+∥L2(R).Proof.
Taking derivative to (1) for variable x, multiplying (∂xU)+, and integrating for variable x imply
(27)∫R‍(∂xU)+(∂t∂xU+A′′(U)|∂xU|2+A′(U)∂xx2U-∂xxx3U)dx-∫R‍g′(U)(∂xU)+2dx=12∂t∫R‍(∂xU)+2dx+∫R‍[A′′(U)(∂xU)+3+A′(U)∂x((∂xU)+22)+ϵ∣∂x(∂xU)+2((∂xU)+22)]dx-∫R‍g′(U)(∂xU)+2dx=0.
We apply the integration by parts to obtain the following regularity (26):
(28)12∂t∫R(∂xU)+2dx‍≤∥g′∥L∞(R)∫R(∂xU)+2dx‍.
Thus, integrating (28) for time variable and using Gronwall’s inequality provide the result (26).


## 3. Estimates on the Hyperbolic and Parabolic Terms

In this section, we prove that the hyperbolic partL1+R1 in equality (25) is strictly negative and the parabolic part L2+R2 has a small rate of convergence. Applying Lemmas 2 and 5, we are able to show the main proposition for this section.Proposition 6.
LetL1 and R1 be as in Lemma 4. Then, there exists a constant θ>0 such that, for any ɛ,δ, and satisfying
(29)ɛδ≤θ,
we have
(30)L1+R1≤-θɛ∫X(t)-δɛX(t)+δɛ‍ϕ(|x-X(t)|ɛ)ϕ′(|x-X(t)|ɛ)η(U∣S)dx.Proof.
Let us start with proving thatL1 is strictly negative. The proof of R1 is similar. With the definition of X(t), we write L1 as
(31)L1=∫X(t)-δɛX(t)‍2ɛϕ(-x+X(t)ɛ)ϕ′(-x+X(t)ɛ)·η(U∣CL)·G(t,x)dx,
where G(t,x):=[f(U(t,X(t)),(CL+CR)/2)-f(U(t,x),CL)]. Using Lemma 5 we find
(32)U(t,X(t))-U(t,x)=∫xX(t)‍(∂xU)(t,y)dy≤∫xX(t)‍(∂xU)+(t,y)dy≤|X(t)-x|∥(∂xU)+∥L2(R)≤C|X(t)-x|∥(ddxU0)+∥L2(R).
We next observe that G(t,x), t∈(0,T), x∈R, is strictly negative. To do this, we rewrite the function G as
(33)G(t,x)=f(U(t,X(t)),CL+CR2)-f(U(t,x),CL+CR2)+f(U(t,x),CL+CR2)-f(U(t,x),CL).
For x∈[X(t)-δϵ,X(t)], Lemma 2 and the inequality (32) imply that
(34)G(t,x)≤f(U(t,x)+C∥(ddxU0)+∥L2(R)ϵδ,CL+CR2)-f(U(t,x),CL+CR2)+f(U(t,x),CL+CR2)-f(U(t,x),CL)≤ΛC∥(ddxU0)+∥L2(R)ϵδ-CL-CR2Λ≤-θ2<0,
for ϵδ>0 small enough. Since ϕ(·),ϕ′(·), and η(·∣·)≥0, we get
(35)L1≤-θ∫X(t)-δɛX(t)‍1ɛϕ(-x+X(t)ɛ)×ϕ′(-x+X(t)ɛ)η(U∣CL)dx.
Similarly, we also obtain that
(36)R1≤-θ∫X(t)X(t)+δɛ‍1ɛϕ(x-X(t)ɛ)×ϕ′(x-X(t)ɛ)η(U∣CR)dx.
Consequently, combining the two last inequalities gives the desired result.We are now going to introduce the parabolic term,L2+R2, and the proof is provided in [20] (see Appendix).Proposition 7.
LetL2,R2 be given in Lemma 4. Then, there exists a constant C>0 such that the following inequality holds:
(37)L2+R2≤Cɛ∫X(t)-δɛX(t)+δɛ[ϕ′(|x-X(t)|ɛ)]2dx.


## 4. Proof of Theorem1

From Lemma4, Proposition 6, and Proposition 7, we get
(38)ddtEϵδ(t)≤Cɛ∫X(t)-δɛX(t)+δɛ[ϕ′(|x-X(t)|ɛ)]2χ{ϕ′-C*ϕ>0}dx+L3+R3.
Applying the change of variables z=(x-X(t))/ɛ and changing θ by inf⁡(θ,C*) if necessary, we find
(39)ddtEϵδ(t)≤Cɛ∫X(t)-δɛX(t)+δɛ‍[(ϕ′)2χ{ϕ′-θϕ>0}](|x-X(t)|ɛ)dx+L3+R3≤C∫0δ‍(ϕ′)2(z)χ{ϕ′-θϕ>0}(z)dz+L3+R3.
To get good estimate, we take a specific ϕδ. For any δ≥1/θ, we now fix the function ϕδ in the following explicit way:
(40)ϕδ(x)={θe1-θδx,forx∈[0,1θ),eθ(x-δ),forx∈[1θ,δ].
We use the computation:
(41)∫0δ(ϕδ′(x))2χ{ϕδ′>θϕδ}dx=Cθ·e-2θδ.For the proof of (I), we integrate the estimate of Proposition7 between 0 and t∈(0,T) such that, for any ɛ, δ with 1/θ≤δ and ɛδ≤θ, where θ is the constant from Proposition 7, it follows that
(42)ddtEϵδ(t)≤Ce-θδ+L3+R3,
which implies that
(43)∫{|x-X(t)|≥δɛ}‍η(U(t,x)∣S(t,x))dx≤Eϵδ(0)+∫0t‍ddtEϵδ(t)ds≤∫R‍η(U0∣S0)dx+∫0t‍L3+R3ds+CTe-θδ.
By taking ɛ0:=θ2, we have, for any ɛ≤β≤ɛ0,
(44)∫{|x-X(t)|≥β/θ}‍η(U(t,x)∣S(t,x))dx≤∫R‍η(U0∣S0)dx+∫0t‍L3+R3ds+CTe-β/ɛ.
Let us observe that
(45)∫R‍η(U∣S)dx=∫{|x-X(t)|≥Cβ}‍η(U∣S)dx+∫{|x-X(t)|<Cβ}‍η(U∣S)dx,(46)∫0tL3+R3ds≤∫0t∫R|g(U(t,x))-g(S(t,x))|‍‍×|η′(U(t,x))-η′(S(t,x))|dxds≤C∫0t∫R‍(U(t,x)-S(t,x))2dxds≤C∫0t∫Rη(U(t,x)∣S(t,x))dxds,
where we have here used the mean value theorem and the definition of source term (3). Consequently, using inequlaities (22), (44), and (45) and taking β=ɛlog⁡(1/ɛ), we get, for any t∈(0,T),
(47)∫R‍η(U∣S)dx≤∫R‍η(U0∣S0)dx+∫0t‍∫R‍η(U∣S)dxds+Cɛlog⁡(1ɛ),
for any ɛ≤ɛ0, which proves (6) by taking η(v)=v2 and applying Gronwall’s inequlity.To end with the proof, it remains to prove (9). Let us define the function ψ by
(48)ψ(x):={0if|x|>2,1if|x|≤12-|x|if1<|x|≤2.
Let s∈(0,t) and let R>0. Multiplying ΨR(s,x):=ψ((x-X(s))/R) to (1) and integrating in x, we get
(49)0=-dds∫‍ΨR·Udx+∫‍∂x(ΨR)A(U)dx+∫‍∂t(ΨR)Udx+∫‍ΨR(ϵ∂xx2U+g(U))dx=-dds∫‍ψ(x-X(s)R)·U(s,x)dx︸(I)+1R∫‍ψ′(x-X(s)R)·(A(U(s,x))-X˙(s)U(s,x))dx︸(II)-ϵR∫‍ψ′(x-X(s)R)∂xU(s,x)dx︸(III)+∫‍ψ(x-X(s)R)g(U)dx︸(IV).
By using the above observation, we have
(50)(σ-X˙(s))=1CL-CR(A(CL)-A(CR)-(CL-CR)X˙(s))=1CL-CR(A(CL)-A(CR)-(CL-CR)X˙(s)-(II)+(I)+(III)-(IV)(CL-CR)X˙(s)).
Then we integrate the above equation in time on [0,t] to get
(51)|σt-X(t)|≤C(t·max⁡s∈(0,t)|A(CL)-A(CR)-(CL-CR)X˙(s)-(II)|︸(II′)+|∫0t‍(I)ds|+t·max⁡s∈(0,t)|(III)|+t·max⁡s∈(0,t)|(IV)||A(CL)-A(CR)-(CL-CR)X˙(s)-(II)|︸(II′)).From the result of Choi and Vasseur [20], we already know the following results:
(52)(II′)2≤CR·∫Rη(U(s)∣S(s))dx,|(III)|≤CϵR,(53)|∫0t(I)ds|2≤CR(∫Rη(U(t)∣S(t))dx+∫Rη(U0∣S0)dx).
We now estimate (IV). This directly follows from the definition of source term (3) and Holder’s inequality:
(54)|(IV)|≤∫-2R+X(s)X(s)‍|g(U)-g(CL)|dx+∫X(s)2R+X(s)‍|g(U)-g(CR)|dx≤CR∥U-S∥L2(R)≤CR∥U0-S0∥L2(R).Finally, by using (54), we combine (52) and (53) together with (51) to get, for any t∈(0,T),
(55)|σt-X(t)|2≤C(t2R2+R+t2R)·(∫R|U0-S0|2dx+ϵlog⁡⁡1ϵ).Consequently, takingR=t1/2 provides the estimate (9).



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