# Zero Diffusion-Dispersion-Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function

Authors: H. Holden; K. H. Karlsen; D. Mitrovic
Journal: International Journal of Differential Equations
 (2009)

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## Abstract

We consider multidimensional conservation laws with discontinuous
flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities.
We use the approach ofH-measures to investigate the zero diffusion-dispersion-smoothing limit.


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## Body

## 1. Introduction

We consider the convergence of smooth solutionsu=uε(t,x) with (t,x)∈R+×Rd of the nonlinear partial differential equation(1.1)∂tu+divxfϱ(t,x,u)=εdivxb(∇u)+δ∑j=1d∂xjxjxj3u
as ε→0 and δ=δ(ε),ϱ=ϱ(ε)→0. Here f∈C(R;BV(Rt+×Rxd)) is the Caratheodory flux vector such that(1.2)max|u|≤l|fϱ(t,x,u)-f(t,x,u)|→0,ϱ→0,inLlocp(R+×Rd),
for p>2 and every l>0. The aim is to show convergence to a weak solution of the corresponding hyperbolic conservation law:(1.3)∂tu+divxf(t,x,u)=0,u=u(t,x),x∈Rd,t≥0.
We refer to this problem as the zero diffusion-dispersion-smoothing limit.In the case when the fluxf is at least Lipschitz continuous, it is well known that the Cauchy problem corresponding to (1.3) has a unique admissible entropy solution in the sense of Kružhkov [1] (or measure valued solution in the sense of DiPerna [2]). The situation is more complicated when the flux is discontinuous and it has been the subject of intensive investigations in the recent years (see, e.g., [3] and references therein). The one-dimensional case of the problem is widely investigated using several approaches (numerical techniques [3, 4], compensated compactness [5, 6], and kinetic approach [7, 8]). In the multidimensional case there are only a few results concerning existence of a weak solution. In [9] existence is obtained by a two-dimensional variant of compensated compactness, while in [10] the approach of H-measures [11, 12] is used for the case of arbitrary space dimensions. Still, many open questions remain such as the uniqueness and stability of solutions.A problem that has not yet been studied in the context of conservation laws with discontinuous flux, and which is the topic of the present paper, is that of zero diffusion-dispersion limits. When the flux is independent of the spatial and temporal positions, the study of zero diffusion-dispersion limits was initiated in [13] and further addressed in numerous works by LeFloch et al. (e.g., [14–17]). The compensated compactness method is the basic tool used in the one-dimensional situation for the so-called limiting case in which the diffusion and dispersion parameters are in an appropriate balance. On the other hand, when diffusion dominates dispersion, the notion of measure valued solutions [2, 18] is used. More recently, in [19] the limiting case has also been analyzed using the kinetic approach and velocity averaging [20].The remaining part of this paper is organized as follows. In Section2 we collect some basic a priori estimates for smooth solutions of (1.1). In Section 3 we look into the diffusion-dispersion-smoothing limit for multidimensional conservation laws with a flux vector which is discontinuous with respect to spatial variable. In doing so we rely on the a priori estimates from the previous section in combination with Panov's H-measures approach [10]. Finally, in Section 4 we restrict ourselves to the one-dimensional case for which we obtain slightly stronger results using the compensated compactness method.


## 2. A priori Inequalities

Assume that the fluxf in (1.1) is smooth in all variables. Consider a sequence (uε,δ)ε,δ of solutions of(2.1)∂tu+divxf(t,x,u)=εdivxb(∇u)+δ∑j=1d∂xjxjxj3u,u(x,0)=u0(x),x∈Rd.We assume that(uε,δ)ε,δ has enough regularity so that all formal computations below are correct. So, following Schonbek [13], we assume that for every ε,δ>0 we have uε,δ∈L∞([0,T];H4(Rd)).Later on, we will assume that the initial datau0 depends on ε. In this section, we will determine a priori inequalities for the solutions of problem (2.1).To simplify the notation we will writeuε instead of uε,δ.We will need the following assumptions on the diffusion termb(λ)=(b1(λ),…,bn(λ)).(H1) For some positive constantsC1,C2 we have(2.2)C1|λ|2≤λ·b(λ)≤C2|λ|2∀λ∈Rd.(H2) The gradient matrixDb(λ) is a positive definite matrix, uniformly in λ∈Rd, that is, for every λ,ϱ∈Rd, there exists a positive constant C3 such that we have(2.3)ϱTDb(λ)ϱ≥C3|ϱ|2.We use the following notation:(2.4)|D2u|2=∑i,k=1d|∂xixk2u|2.
In the sequel, for a vector valued function g=(g1,…,gd) defined on R+×Rd×R, we denote(2.5)|g|2=∑i=1d|gi|2.
The partial derivative ∂xi in the point (t,x,u), where u possibly depends on (t,x), is defined by the formula(2.6)∂xig(t,x,u(t,x))=(Dxig(t,x,λ))∣λ=u(t,x).
In particular, the total derivative Dxi and the partial derivative ∂xi are connected by the identity(2.7)Dxig(t,x,u)=∂xig(t,x,u)+∂ug(t,x,u)∂xiu.
Finally we use(2.8)divxg(t,x,u)=∑i=1dDxigi(t,x,u),g=(g1,…,gd),Δxq(t,x,u)=∑i=1dDxixi2q(t,x,u),q∈C2(R+×Rd×R).With the previous conventions, we introduce the following assumption on the flux vectorf.(H3) The growth of the velocity variableu and the spatial derivative of the flux f are such that for some C,α>0, p≥1, and every l>0, we have(2.9)max|λ|<l|fi(t,x,λ)|∈Lp(R+×Rd),i=1,…,d,∑i=1d|∂ufi(t,x,u)|≤C,∑i,j=1d|∂xifj(t,x,u)|≤μ(t,x)1+|u|1+α,
where μ∈ℳ(R+×Rd) is a bounded measure (and, accordingly, the above inequality is understood in the sense of measures).Now, we can prove the following theorem.Theorem 2.1.
Suppose that the flux functionf=f(t,x,u) satisfies (H3) and that it is Lipschitz continuous on R+×Rd×R. Assume also that initial data u0 belongs to L2(Rd). Under conditions (H1)-(H2) the sequence of solutions (uε)ε>0 of (2.1) for every t∈[0,T] satisfies the following inequalities:
(2.10)∫Rd|uε(t,x)|2dx+ε∫0t∫Rd|∇uε(t′,x)|2dxdt′≤C4(∫Rd|u0(x)|2dx-∫0t∫Rd∫0uε(t′,x)divxf(t′,x,v)dvdxdt′),(2.11)ε2∫Rd|∇uε(t,x)|2dx+ε3∫0t∫Rd|D2uε(t′,x)|2dxdt′≤C5(ε2∫Rd|∇u0(x)|2dx+ε∫0t∫Rd∑k=1d|∂xkf(t′,x,uε(t′,x))|2dxdt′+∥∂uf∥L∞(R+×Rd×R)2∑k=1d),
for some constants C4 and C5.Proof.
We follow the procedure from [19]. Given a smooth function η=η(u), u∈R, we define
(2.12)qi(t,x,u)=∫0uη′(v)∂vfi(t,x,v)dv,i=1,…,d.
If we multiply (2.1) by η′(u), it becomes
(2.13)∂tη(uε)+∑i=1d∂xiqi(t,x,uε)-∑i=1d∫0uε∂xiv2fi(t,x,v)η′(v)dv+∑i=1dη′(uε)∂xifi(t,x,uε)=ε∑i=1d∂xi(η′(uε)bi(∇uε))-εη′′(uε)∑i=1dbi(∇uε)∂xiuε+δ∑i=1d∂xi(η′(uε)∂xixi2uε)-δ2η′′(uε)∑i=1d∂xi(∂xiuε)2.
Choosing here η(u)=u2/2 and integrating over [0,t)×Rd, we get
(2.14)∫Rd|uε(t,x)|2dx+ε∫0t∫Rd∇uε(t′,x)·b(∇uε(t′,x))dxdt′=∫Rd|u0(x)|2dx+∑j=1d∫0t∫Rd∫0uε(t′,x)vDxjv2fj(t′,x,v)dvdxdt′-∑i=1d∫0t∫Rduε(t′,x)∂xifi(t′,x,uε(t′,x))dxdt′=∫Rd|u0(x)|2dx-∑i=1d∫0t∫Rd∫0uε(t′,x)∂xifi(t′,x,v)dvdxdt′,
where the second equality sign is justified by the following partial integration:
(2.15)∫0t∫Rd∫0uεvDxjv2fj(t′,x,v)dvdxdt′=∫0t∫Rduε∂xifi(t′,x,uε)dxdt′-∫0t∫Rd∫0uε∂xifi(t′,x,v)dvdxdt′.
Now inequality (2.10) follows from (2.14), using (H1).
As for inequality (2.11), we start by using (2.14), namely,
(2.16)∫Rd|uε(t,x)|2dx+ε∫0t∫Rd∇uε(t′,x)·b(∇uε(t′,x))dxdt′=∫Rd|u0(x)|2dx-∑i=1d∫0t∫Rd∫0uε(t′,x)∂xifi(t′,x,v)dvdxdt′≤∫Rd|u0(x)|2dx+∑i=1d∫0t∫Rd∫0uε(t′,x)|∂xifi(t′,x,v)|dvdxdt′≤∫Rd|u0(x)|2dx+∫0t∫Rd∫Rμ(t′,x)1+|v|1+αdvdxdt′≤∫Rd|u0(x)|2dx+C∫0t∫Rdμ(t′,x)dxdt′,
where C=∫R(dv/(1+|v|1+α)).
From here, using (H3), we conclude in particular that(2.17)ε∫0t∫Rd|∇uε(t′,x)|2dxdt′≤C11,
for some constant C11 independent of ε.
Next, we differentiate (2.1) with respect to xk and multiply the expression by ∂xku. Integrating over Rd, using integration by parts and then summing over k=1,…,d, we get:
(2.18)12∫Rd∂t|∇uε|2dx-∑k=1d∫Rd(∇∂xkuε)·(∂xkfk(t,x,uε)+∂ufk∂xkuε)dx=-ε∑k=1d∫Rd(∇∂xkuε)TDb(∇uε)(∇∂xkuε)dx.
Integrating this over [0,t] and using the Cauchy-Schwarz inequality and condition (H2), we find
(2.19)12∫Rd|∇uε(t,·)|2dx+εC3∑k=1d∫0t∫Rd|∇∂xkuε|2dxdt′≤12∫Rd|∇u0|2dx+∑k=1d∥∇(∂xkuε)∥L2(R+×Rd)∥∂xkfk(·,·,uε)+∂ufk∂xkuε∥L2(R+×Rd),
where C3 is independent of ε. Then, using Young's inequality (the constant C3 is the same as previously mentioned)
(2.20)ab≤C3ε2a2+12C3εb2,a,b∈R,
we obtain
(2.21)12∫Rd|∇uε(t,·)|2dx+εC3∑k=1d∫0t∫Rd|∇∂xkuε|2dxdt′≤12∫Rd|∇u0|2dx+C3ε2∑k=1d∫0t∫Rd|∇∂xkuε|2dxdt′+12C3ε∫0t∫Rd∑k=1d|∂xkfk(t′,x,uε)+∂ufk∂xkuε|2dxdt′.
Multiplying this by ε2, using (a+b)2≤2a2+2b2, and applying (2.17), we conclude
(2.22)ε22∫Rd|∇uε(t,·)|2dx+C3ε32∫Rd∫0t|D2uε|2dxdt′≤ε22∫Rd|∇u0|2dxdt′+εC3∫0t∫Rd∑k=1d|∂xkfk(t′,x,uε(t′,x))|2dxdt′+C11C3∥∂ufk∥L∞(R+×Rd×R)2.
This inequality is actually inequality (2.11) when we take C5=2max{1,1/C3,C11/C3}/min{1,C3}.


## 3. The Multidimensional Case

Consider the following initial-value problem. Findu=u(t,x) such that(3.1)∂tu+divxf(t,x,u)=0,u(x,0)=u0(x),x∈Rd,
where u0∈L2(Rd) is a given initial data.For the fluxf=(f1,…,fd) we need the following assumption, denoted  (H4).(H4a) For the fluxf=f(t,x,u), (t,x,u)∈R+×Rd×R, we assume that f∈C(R;BV(R+×Rd)) and that for every l∈R+ we have maxu∈[-l,l]|f(t,x,u)|∈Lp(R+×Rd), p>2 .(H4b) There exists a sequencefϱ=(f1ϱ,…,fdϱ), ϱ∈(0,1), such that fϱ=fϱ(t,x,u)∈C1(R+×Rd×R), satisfying for some p>2 and every l∈R+:(3.2a)maxz∈[-l,l]|fϱ(·,·,z)-f(·,·,z)|→ϱ→00inLp(R+×Rd)=0,(3.2b)∑i=1d∫R+×Rd|∂xifiϱ(t,x,u)|dxdt≤C̃11+|u|1+α,(3.2c)ϱ∑i=1,kd∫R+×Rd|∂xkfiϱ(t,x,u)|2dxdt≤C̃2,(3.2d)∑i=1d|∂ufiϱ(t,x,u)|≤Cβ(ϱ),(3.2e)∑i=1d∫R+×Rd|∂xiu2fiϱ(t,x,u)|dxdt≤C̃31+|u|1+α,
where C̃i, i=1,2,3, and C are constants, while the function β:R→R is such that limρ→0β(ρ)=0.In the case when we have only vanishing diffusion, it is usually possible to obtain uniformL∞ bound for the corresponding sequence of solutions under relatively mild assumptions on the flux and initial data (see, e.g., [9, 10]). In the case when we have both vanishing diffusion and vanishing dispersion, we must assume more on the flux in order to obtain even much weaker bounds (see Theorem 3.2). We remark that demand on controlling the flux at infinity is rather usual in the case of conservation laws with vanishing diffusion and dispersion (see, e.g., [16, 17, 19]).Remark 3.1.
For an arbitrary compactly supported, nonnegativeφ1∈C0∞(R+×Rd) and φ2∈C0∞(R) with total mass one denote
(3.3)φϱ(z,u)=1ϱd+1φ1(zϱ)1β(ϱ)φ2(uβ(ϱ)),z∈R+×Rd and u∈R, where β is a positive function tending to zero as ϱ→0. In the case when the flux f∈C(R;BV(R+×Rd))∩BV(R×R+×Rd)) is bounded, straightforward computation shows that the sequence fϱ=f⋆φϱ=(f1ϱ,…,fdϱ) satisfies (H4b) with β(ϱ)=ϱ.We also need to assume that the fluxf is genuinely nonlinear, that is, for every (t,x)∈R+×Rd and every ξ∈Rd∖{0}, the mapping(3.4)R∋λ↦∑i=1dfi(t,x,λ)ξi|ξ|
is nonconstant on every nondegenerate interval of the real line.We will analyze the vanishing diffusion-dispersion-smoothing limit of the problem(3.5)∂tu+divxfϱ(t,x,u)=εdivxb(∇u)+δ∑j=1d∂xjxjxj3u,(3.6)u(x,0)=u0,ε(x),x∈Rd,
where the flux fϱ satisfies the conditions (H4b). We denote the solution of (3.5)-(3.6) by uε=uε(t,x). We assume that(3.7)∥u0,ε-u0∥L2(Rd)→0,∥u0,ε∥L2(Rd)+ε∥u0,ε∥H1(Rd)≤C.
We also assume that ϱ=ϱ(ε)→0 and δ=δ(ε)→0 as ε→0. We want to prove that under certain conditions, a sequence of solutions (uε)ε>0 of (3.5)-(3.6) converges to a weak solution of problem (3.1) as ε→0. To do this in the multidimensional case we use the approach of H-measures, introduced in [11] and further developed in [10, 21]. In the one-dimensional case, we use the compensated compactness method, following [13].In order to accomplish the plan we need the following a priori estimates.Theorem 3.2 (a priori inequalities).
Suppose that the fluxf(t,x,u) satisfies (H4). Also assume that the initial data u0 satisfies (3.7). Under these conditions the sequence of smooth solutions (uε)ε>0 of (3.5)-(3.6) satisfies the following inequalities for every t∈[0,T]:
(3.8)∫Rd|uε(t,x)|2dx+ε∫0t∫Rd|∇uε(x,s)|2dxds≤C4(∫Rd|u0,ε(x)|2dx+C10),(3.9)ε2∫Rd|∇uε(t,x)|2dx+ε3∫0t∫Rd|D2uε(t′,x)|2dxdt′≤C5(ε2∫Rd|∇u0,ε(x)|2dx+εϱC11+C12β(ϱ)2),
for some constants C10,C11,C12 (the constants C4,C5 are introduced in Theorem 2.1).Proof.
For every fixedϱ, the function fϱ=(f1ϱ,…,fdϱ) is smooth, and, due to (H4), we see that fϱ satisfies (H3). This means that we can apply Theorem 2.1.
Replacing the fluxf by fϱ from (3.5) and u0 by u0,ε from (3.6) in (2.10) and (2.11), we get
(3.10)∫Rd|uε(t,x)|2dx+ε∫0t∫Rd|∇uε(x,s)|2dxds≤C3(∫Rd|u0,ε(x)|2dx-∫0t∫Rd∫0uε(t′,x)divxfϱ(t′,x,v)dvdxdt′),(3.11)ε2∫Rd|∇uε(t,x)|2dx+ε3∫0t∫Rd|D2uε(t′,x)|2dxdt′≤C4(∑k=1d∑i=1dε2∫Rd|∇u0,ε(x)|2dx+∥∂ufϱ∥L∞(R+×Rd×R)2+ε∫0t∫Rd∑k=1d∑i=1d[∂xkfiϱ(t′,x,uε(t′,x))]2dxdt′).
To proceed, we use assumption (H4). We have
(3.12)∫0t∫Rd∫0uε(t′,x)divfiϱ(t′,x,v)dvdxdt′≤∫0t∫Rd∫R∑i=1d|∂xifiϱ(t′,x,v)|dvdxdt≤∫RC̃11+|v|1+αdv≤C10,
which together with (3.10) immediately gives (3.8).
Similarly, combining (H4) and (3.11), and arguing as in (3.12), we get (3.9).In this section, we will inspect the convergence of a family(uε)ε>0 of solutions to (3.5)-(3.6) in the case when(3.13)b(λ1,…,λd)=(λ1,…,λd)
for the function b appearing in the right-hand side of (3.5). This is not an essential restriction, but we will use it in order to simplify the presentation.Thus, we use the following theorem which can be proved using theH-measures approach (see, e.g., [10, Corollary 2 and Remark 3]). We let θ denote the Heaviside function.Theorem 3.3 (see [10]).
Assume that the vectorf(t,x,u) is genuinely nonlinear in the sense of (3.4). Then each family (vε(t,x))ε>0⊂L∞(R+×Rd) such that for every c∈R the distribution
(3.14)∂t(θ(vε-c)(vε-c))+divx(θ(vε-c)(f(t,x,vε)-f(t,x,c)))
is precompact in Hloc-1 contains a subsequence convergent in Lloc1(R+×Rd).We can now prove the following theorem.Theorem 3.4.
Assume that the flux vectorf is genuinely nonlinear in the sense of (3.4) and that it satisfies (H4). Furthermore, assume that
(3.15)ϱ=ε,δ=ε2ρ2(ε)withρ(ε)=𝒪(β(ε)),
and that u0,ε satisfies (3.7). Then, there exists a subsequence of the family (uε)ε>0 of solutions to (3.5)–(3.6) that converges to a weak solution of problem (3.1).Proof.
We will use Theorem3.3. Since it is well known that the family (uε)ε>0 of solutions of problem (3.5)–(3.6) is not uniformly bounded, we cannot directly apply the conditions of Theorem 3.3.
Take an arbitraryC2 function S=S(u), u∈R, and multiply the regularized equation (3.5) by S′(uε). As usual, put
(3.16)q(t,x,u)=∫0uS′(v)∂ufϱdv,q=(q1,…,qd).
We easily find that(3.17)∂tS(uε)+divxq(t,x,uε)-divxq(t,x,v)∣v=uε+S′(uε)divxfϱ(t,x,v)∣v=uε=εdivx(S′(uε)∇uε)-εS′′(uε)|∇uε|2+δ∑j=1dDxj(S′(uε)∂xjxj2uε)-δ∑j=1dS′′(uε)∂xjuε∂xjxj2uε.
We will apply this formula repeatedly with different choices for S(u).
In order to apply Theorem3.3, we will consider a truncated sequence (Tl(uε))ε>0, where the truncation function Tl is defined for every fixed l∈N as
(3.18)Tl(u)={-l,u≤-l,u,-l≤u≤l,l,u≥l.
We will prove that the sequence (Tl(uε))ε>0 is precompact for every fixed l. Denote by ul a subsequential limit (in Lloc1) of the family (Tl(uε))ε>0, which gives raise to a new sequence (ul)l>1 that we prove converges to a weak solution of (3.1).
To carry out this plan, we must replaceTl by a C2 regularization Tl,σ:R→R. We define Tl,σ:R→R by Tl,σ(0)=0 and
(3.19)Tl,σ′(u)={1,|u|<l,l-|u|+σσ,l<|u|<l+σ,0,|u|>l+σ.
Next, we want to estimate∥Tl,σ′′(uε)∇uε∥L2(R+×Rd). To accomplish this, we insert the functions Tl,σ± for S in (3.17) where Tl,σ± are defined by Tl,σ±(0)=0 and
(3.20)(Tl,σ+)′(u)={1,u<l,l+σ-uσ,l<u<l+σ,0,u>l+σ,(3.21)(Tl,σ-)′(u)={1,u>-l,l+σ+uσ,-l-σ<u<-l,0,u<-l-σ.
Notice that
(3.22)(Tl,σ±(u))′≤1,|Tl,σ±(u)|≤|u|+σ2,Tl,σ+(u)=Tl,σ-(u)for-l≤u≤l.
By inserting S(u)=-Tl,σ+(u), q=q+(t,x,u)=-∫0u(Tl,σ+)′(v)∂ufϱdv in (3.17) and integrating over Πt=[0,t]×Rd, we get
(3.23)-∫RdTl,σ+(uε)dx+∫RdTl,σ+(u0)dx+εσ∬Πt∩{l<uε<l+σ}|∇uε|2dxdt=∬Πtdivxq+(t,x,v)∣v=uεdxdt+∬Πt(Tl,σ+)′(uε)divxfϱ(t,x,v)∣v=uεdxdt-δσ∬Πt∩{l<uε<l+σ}∑j=1d∂xjuε∂xjxj2uεdxdt.
Similarly, for S(u)=Tl,σ-(u), q=q-(t,x,u)=∫0u(Tl,σ-)′(v)∂ufϱdv, we have from (3.17)
(3.24)∫RdTl,σ-(uε)dx-∫RdTl,σ-(u0)dx+εσ∬Πt∩{-l-σ<uε<-l}|∇uε|2dxdt=∬Πtdivxq-(t,x,v)∣v=uεdxdt-∬Πt(Tl,σ-)′(uε)divxfϱ(t,x,v)∣v=uεdxdt+δσ∬Πt∩{-l-σ<uε<-l}∑j=1d∂xjuε∂xjxj2uεdxdt.
Adding (3.23) to (3.24), we get
(3.25)εσ∬Πt∩{l<|uε|<l+σ}|∇uε|2dxdt=-∫Rd(Tl,σ-(uε)-Tl,σ+(uε))dx+∫Rd(Tl,σ-(u0)-Tl,σ+(u0))dx+∬Πtdivxq-(t,x,v)∣v=uεdxdt+∬Πtdivxq+(t,x,v)∣v=uεdxdt-∬Πt(Tl,σ-)′(uε)divxfϱ(t,x,v)∣v=uεdxdt+∬Πt(Tl,σ+)′(uε)divxfϱ(t,x,v)∣v=uεdxdt+δσ∬Πt∩{-l-σ<uε<-l}∑j=1d∂xjuε∂xjxj2uεdxdt-δσ∬Πt∩{l<uε<l+σ}∑j=1d∂xjuε∂xjxj2uεdxdt.
From (3.22) and the definition of q- and q+, it follows
(3.26)εσ∬Πt∩{l<|uε|<l+σ}|∇uε|2dxdt≤∫|uε|>l2|uε|dx+∫|u0|>l2|u0|dx+2∬Πt∫R∑i=1d|Dxiv2fiϱ(t,x,v)|dvdxdt+2∬Πt∑i=1d|∂xifiϱ(t,x,uε)|dxdt+2δσ∬Πt∩{l-σ<|uε|<l}∑j=1d|∂xjuε∂xjxj2uε|dxdt.
Without loss of generality, we can assume that l>1. Having this in mind, we get from (H4) and (3.26)
(3.27)εσ∬Πt∩{l<|uε|<l+σ}|∇uε|2dxdt≤∫|uε|>l2|uε|2dx+∫|u0|>l2|u0|2dx+2∫R∑i=1dC̃31+|v|1+αdv+2∬Πt∑i=1d|∂xifiϱ(t,x,uε)|dxdt+2δσ∬Πt∩{l<|uε|<l+σ}∑j=1d|∂xjuε∂xjxj2uε|dxdt≤∫Rd2(|uε(x,t)|2+|u0(x,t)|2)dx+K1+K2+2δσε2∑i=1d∥ε1/2∂xiuε∥L2(R+×Rd)×∥ε3/2∂xixi2uε∥L2(R+×Rd)≤K5+(δ2σ2ε4(β(ϱ))2+δ2σ2ε4)1/2K3K4,
where Ki, i=1,…,5, are constants such that (cf. (3.8) and (3.9))
(3.28)2∫R∑i=1dC̃31+|v|1+αdv≤K1,2∬Πt∑i=1d|∂xifiϱ(t,x,uε)|dxdt≤K2,∑i=1d∥ε1/2∂xiuε∥L2(R+×Rd)≤K3,∑i=1d∥ε3/2∂xixi2uε∥L2(R+×Rd)≤(1(β(ϱ))2+εϱ)1/2K4,∫Rd2(|uε(x,t)|2+|u0(x,t)|2)dx+K1+K2≤K5.
These estimates follow from (H4) and the a priori estimates (3.8), (3.9). If in addition we use the assumption ε=ϱ from (3.15), we conclude
(3.29)δσε2∥ε1/2∇uε∥L2(R+×Rd)∑i=1d∥ε3/2∂xixi2uε∥L2(R+×Rd)≤(δ2σ2ε4β2(ε)+δ2σ2ε4)1/2K3K4.
Thus, in view of (3.27),
(3.30)εσ∬Πt∩{l<|uε|<l+σ}|∇uε|2dxdt≤K5+(δ2σ2ε4β2(ε)+δ2σ2ε4)1/2K3K4,
which is the sought for estimate for ∥Tl,σ′′(uε)∇uε∥L2(R+×Rd).
Next, take a functionUρ(z) satisfying Uρ(0)=0 and
(3.31)Uρ′(z)={0,z<0,zρ,0<z<ρ,1,z>ρ.
Clearly, Uρ is convex, and Uρ′(z)→θ(z) in Llocp(R) as ρ→0, for any p<∞; as before, θ denotes the Heaviside function.
InsertingS(uε)=Uρ(Tl,σ(uε)-c) in (3.17), we get
(3.32)∂tUρ(Tl,σ(uε)-c)+divx∫uεUρ′(Tl,σ(v)-c)Tl,σ′(v)∂vfϱ(t,x,v)dv=∫uεUρ′(Tl,σ(v)-c)Tl,σ′(v)divx∂vfϱ(t,x,v)dv-Uρ′(Tl,σ(uε)-c)Tl,σ′(uε)divxfϱ(t,x,v)∣v=uε+εΔxUρ(Tl,σ(uε)-c)-εDuu2[Uρ(Tl,σ(uε)-c)]|∇uε|2+δ∑i=1dDxi(Du[Uρ(Tl,σ(uε)-c)]∂xixi2uε)-δ∑i=1dDuu2[Uρ(Tl,σ(uε)-c)]∂xiuε∂xixi2uε.
We rewrite the previous expression in the following manner:
(3.33)∂t(θ(Tl(uε)-c)(Tl(uε)-c))+divx(θ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c)))=Γ1,ε+Γ2,ε+Γ3,ε+Γ4,ε+Γ5,ε+Γ6,ε+Γ7,ε,
where
(3.34)Γ1,ε=∂t(θ(Tl(uε)-c)(Tl(uε)-c)-Uρ(Tl,σ(uε)-c)),Γ2,ε=divx(∫θ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))-∫uεUρ′(Tl,σ(v)-c)Tl,σ′(v)∂vfϱ(t,x,v)dv(f(t,x,Tl(uε))-f(t,x,c))),Γ3,ε=∫uεUρ′(Tl,σ(v)-c)Tl,σ′(v)divx∂vfϱ(t,x,v)dv-Uρ′(Tl,σ(uε)-c)Tl,σ′(uε)divxfϱ(t,x,v)∣v=uε,Γ4,ε=εΔxUρ(Tl,σ(uε)-c)+δ∑i=1dDxi(Du[Uρ(Tl,σ(uε)-c)]∂xixi2uε),Γ5,ε=-εUρ′(Tl,σ(uε)-c)Tl,σ′′(uε)|∇uε|2,Γ6,ε=-δ∑i=1dDuu2[Uρ(Tl,σ(uε)-c)]∂xiuε∂xixi2uε,Γ7,ε=-εUρ′′(Tl,σ(uε)-c)(Tl,σ′(uε))2|∇uε|2.
To continue, we assume thatσ depends on ε in the following way:
(3.35)σ=ρ=𝒪(β(ε)).
From here, we will prove that the sequence(Tl(uε))ε>0 satisfies the assumptions of Theorem 3.3. Accordingly, we need to prove that the left-hand side of (3.33) is precompact in Hloc-1(R+×Rd).
To accomplish this, we use Murat's lemma ([22, Chapter 1, Corollary 1]). More precisely, we have to prove the following.
(i) When the left-hand side of (3.33) is written in the form divQε, we have Qε∈Llocp(R+×Rd) for p>2.
(ii) The right-hand side of (3.33) is of the form ℳloc,B+Hloc,c-1, where ℳloc,B denotes a set of families which are locally bounded in the space of measures, and Hloc,c-1 is a set of families precompact in Hloc-1.
First, sinceTl(uε) is uniformly bounded by l, we see that (i) is satisfied.
To prove (ii), we consider each term on the right-hand side of (3.33). First we prove that
(3.36)Γ1,ε=∂t(θ(Tl(uε)-c)(Tl(uε)-c)-Uρ(Tl,σ(uε)-c))∈Hloc,c-1.
We have
(3.37)θ(Tl(uε)-c)(Tl(uε)-c)-Uρ(Tl,σ(uε)-c)=θ(Tl(uε)-c)(Tl(uε)-c)-θ(Tl,σ(uε)-c)(Tl,σ(uε)-c)+θ(Tl,σ(uε)-c)(Tl,σ(uε)-c)-Uρ(Tl,σ(uε)-c).
Since the function θ(z-c)(z-c) is Lipschitz continuous in z with the Lipschitz constant one, and, according to definition of Uρ, it holds |Uρ(z)-θ(z)z|≤1/2ρ, we conclude from the last expression
(3.38)|θ(Tl(uε)-c)(Tl(uε)-c)-Uρ(Tl,σ(uε)-c)|≤|Tl(uε)-Tl,σ(uε)|+𝒪(ρ)≤𝒪(σ)+𝒪(ρ).
From this and assumptions (3.15) and (3.35) on σ=σ(ε) and ρ=ρ(ε), it follows that as ε→0(3.39)θ(Tl(uε)-c)(Tl(uε)-c)-Uρ(Tl,σ(uε)-c)→0
in Llocp for all p<∞. Thus, (since we can take p=2 as well) we see that Γ1,ε∈Hloc,c-1.
Next, we will prove that(3.40)Γ2,ε=divx(∫uεθ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))-∫uεUρ′(Tl,σ(v)-c)Tl,σ′(v)∂vfϱ(t,x,v)dv)∈Hloc,c-1+ℳloc,B.
Indeed,
(3.41)θ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))-∫uεUρ′(Tl,σ(v)-c)Tl,σ′(v)∂vfϱ(t,x,v)dv=θ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))-θ(Tl,σ(uε)-c)(fϱ(t,x,Tl,σ(uε))-fϱ(t,x,c))+θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))-∫uεUρ′(Tl,σ(v)-c)Tl′(v)∂vfϱ(t,x,v)dv-∫uεUρ′(Tl,σ(v)-c)(Tl,σ′(v)-Tl′(v))∂vfϱ(t,x,v)dv.
Since Tl(u)=u if |u|≤l and Tl′(u)=0 if |u|≥l,
(3.42)∫uεUρ′(Tl,σ(v)-c)Tl′(v)∂vfϱ(t,x,v)dv=∫uεUρ′(Tl,σ(v)-c)Tl′(v)∂vfϱ(t,x,Tl(v))dv,
from which we conclude
(3.43)θ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))-∫uεUρ′(Tl,σ(v)-c)Tl,σ′(v)∂vfϱ(t,x,v)dv=θ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))-θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))+θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))-∫uεUρ′(Tl,σ(v)-c)Tl′(v)∂vfϱ(t,x,Tl(v))dv-∫uεUρ′(Tl,σ(v)-c)(Tl,σ′(v)-Tl′(v))∂vfϱ(t,x,v)dv=θ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))-θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))+θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))-∫uεθ(Tl,σ(v)-c)Dv[fϱ(t,x,Tl(v))]dv-∫uεUρ′(Tl,σ(v)-c)(Tl,σ′(v)-Tl′(v))∂vfϱ(t,x,v)dv-∫uε(Uρ′(Tl,σ(v)-c)-θ(Tl,σ(v)-c))Tl′(v)∂vfϱ(t,x,Tl(v))dv=Γ2,ε1+Γ2,ε2+Γ2,ε3,
with
(3.44)Γ2,ε1=θ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))-θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c)),Γ2,ε2=θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))-∫uεθ(Tl,σ(v)-c)Dv[fϱ(t,x,Tl(v))]dv,Γ2,ε3=-∫uεUρ′(Tl,σ(v)-c)(Tl,σ′(v)-Tl′(v))∂vfϱ(t,x,v)dv-∫uε(Uρ′(Tl,σ(v)-c)-θ(Tl,σ(v)-c))Tl′(v)∂vfϱ(t,x,Tl(v))dv.
Consider now each term on the right-hand side of (3.43). Since Tl is a continuous function and Tl(u)∈[-l,l], the function f(t,x,Tl(u)) is uniformly continuous in u∈R. Therefore, we have pointwise on R+×Rd:
(3.45)|Γ2,ε1|=|θ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))-θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))|→0asσ→0.
Since maxu∈[-l,l]f(t,x,u)∈Lp(R+×Rd), p>2, Lebesgue's dominated convergence theorem yields |Γ2,ε1|=oσ,Llocp(1), where ∫R+×Rd|oσ,Lp(1)|pdxdt→0 as σ→0. Thus, we conclude
(3.46)divxΓ2,ε1∈Hloc-1(R+×Rd).
We pass toΓ2,ε2. We have to distinguish between different cases depending on the relative size of c and l. Consider first the case when |c|≤l, in which case we have Tl(c)=c and Tl,σ(c)=c. Thus,
(3.47)|Γ2,ε2|=|∫uεθ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))-∫uεθ(Tl,σ(v)-c)Dv[fϱ(t,x,Tl(v))]dv|=|∫uεθ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))-θ(Tl,σ(uε)-c)∫cuεDv[fϱ(t,x,Tl(v))]dv|=|θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))-θ(Tl,σ(uε)-c)(fϱ(t,x,Tl(uε))-fϱ(t,x,c))|≤|θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-fϱ(t,x,Tl(uε)))|+|θ(Tl,σ(uε)-c)(f(t,x,c)-fϱ(t,x,c))|≤|θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-fϱ(t,x,Tl,σ(uε)))|+|θ(Tl,σ(uε)-c)(fϱ(t,x,Tl,σ(uε))-fϱ(t,x,Tl(uε)))|+|θ(Tl,σ(uε)-c)(f(t,x,c)-fϱ(t,x,c))|=oϱ,Llocp(1)+𝒪(σβ(ϱ))+oϱ,Llocp(1)=𝒪(1)+oϱ,Llocp(1),
where oϱ,Llocp(1) appears due to (3.2a), and 𝒪(1) comes from (3.35).
Forc>l we have c≥l+σ for a σ small enough, and therefore θ(Tl,σ(uε)-c)≡0. On the other hand, for c<-l we have c≤-l-σ, and so θ(Tl,σ(uε)-c)≡1. Thus, the problematic case is when c<-l. In this case, we have instead of (3.47)
(3.48)Γ2,ε2=θ(Tl,σ(uε)-c)(f(t,x,Tl,σ(uε))-f(t,x,c))-∫uεθ(Tl,σ(v)-c)Dv[fϱ(t,x,Tl(v))]dv=f(t,x,Tl,σ(uε))-fϱ(t,x,Tl(uε))+fϱ(t,x,-l)-f(t,x,c)
implying
(3.49)divxΓ2,ε2∈Hloc,c-1+ℳloc,B,
since f(t,x,Tl,σ(uε))-fϱ(t,x,Tl(uε))→0 in Llocp(R+×Rd) for p≥2, and fϱ(t,x,-l)-f(t,x,c)∈BV(R+×Rd).
It remains to estimateΓ2,ε3. Noticing that |Uρ′|,|Tl,σ′|≤1, we get
(3.50)|∫uεUρ′(Tl,σ(v)-c)(Tl,σ′(v)-Tl′(v))∂vfϱ(t,x,v)dv|≤Cβ(ρ)∫R|Tl,σ′(v)-Tl′(v)|dv=𝒪(σβ(ϱ))=(3.15),(3.35)𝒪(1),
where C is the constant given by (3.2d).
Similarly, from (3.2d) and since |Tl′(v)|≤1, we have
(3.51)|∫uε(Uρ′(Tl,σ(v)-c)-θ(Tl,σ(v)-c))Tl′(v)∂vfϱ(t,x,Tl(v))dv|≤Cβ(ρ)∫-ll|Uρ′(Tl,σ(v)-c)-θ(Tl,σ(v)-c)|dv=𝒪(ϱβ(ϱ))=(3.15)𝒪(1),
from which we conclude that Γ2,ε3 is bounded in Lloc2. From assumptions (3.15) and (3.35), as well as for the estimates (3.46)–(3.51), it follows that the expression from (3.43) is bounded in Lloc2 from which it follows that Γ2,ε∈Hloc,c-1.
The next term is(3.52)Γ3,ε=∫uεUρ′(Tl,σ(v)-c)Tl,σ′(v)divx∂vfϱ(t,x,v)dv-Uρ′(Tl,σ(uε)-c)Tl,σ′(uε)divxfϱ(t,x,v)∣v=uε.
According to (H4), it is clear that Γ3,ε∈ℳloc,B. Indeed, since |Uρ′|,|Tl,σ′|≤1 we have from (3.2b) and (3.2e)
(3.53)∬R+×Rd|Γ3,ε|dxdt≤∫RC̃31+|v|1+αdv+C̃1≤K6
for a constant K6, implying the claim.
Next, we claim that(3.54)Γ4,ε=∑i=1dDxi(εDxiUρ(Tl,σ(uε)-c)+δDu[Uρ(Tl,σ(uε)-c)]∂xixi2uε)∈Hloc,c-1.
Due to a priori estimates (3.8) and (3.9) and, again, the fact that |Tl,σ′|,|Uρ′|≤1, we see that for every i=1,…,d(3.55)εDxiUρ(Tl,σ(uε)-c)+δDu[Uρ(Tl,σ(uε)-c)]∂xixi2uε→0
in L2(R+×Rd). Therefore, Γ4,ε∈Hloc,c-1.
Further, we claim that(3.56)Γ5,ε=εUρ′(Tl,σ(uε)-c)Tl,σ′′(uε)|∇uε|2∈ℳloc,B.
Since |Uρ′|≤1 and |Tl,σ′′|≤1σ we have from (3.30) (recall (3.28) for the definition of the constants Kj for j=3,4,5)
(3.57)ε∫R+×Rd|Uρ′(Tl,σ(uε)-c)Tl,σ′′(uε)||∇uε|2dxdt≤εσ∫l<|uε|<l+σ|∇uε|2dxdt≤(3.30)K5+(δ2σ2ε4+δ2σ2(β(ϱ))2ε4)1/2K3K4≤K6,
for some constant K6, according to assumptions (3.15) and (3.35) on δ=δ(ε), σ=σ(ε), ϱ=ϱ(ε), and β(ϱ)=β(ε3). Thus, we see that Γ5,ε∈ℳloc,B.
Next, we need to show(3.58)Γ6,ε=δ∑i=1dDuu2[Uρ(Tl,σ(uε)-c)]∂xiuε∂xixi2uε∈ℳloc,B.
In view of a priori estimates (3.8) and (3.9), and assumptions (3.15) and (3.35), it holds
(3.59)∬R+×Rdδ|∑i=1dDuu2[Uρ(Tl,σ(uε)-c)]∂xiuε∂xixi2uε|dxdt≤∑i=1d(δσ+δρ)(∬R+×Rd|∂xiuε|2dxdt)1/2(∬R+×Rd|∂xixi2uε|2dxdt)1/2≤(δε2σ+δε2ρ)∑i=1d(ε∫R+×Rd|∂xiuε|2dxdt)1/2(ε3∫R+×Rd|∂xixi2uε|2dxdt)1/2≤K7(δε2σ+δε2ρ)(εϱ+1(β(ϱ))2)1/2≤K8,
for some constants K7 and K8. The second estimate holds since Uρ′′≤1/ρ and Tl,σ′′≤1/σ implying |Duu[Uρ(Tl,σ(uε)-c)]|≤(1/σ+1/ρ). Therefore, Γ6,ε∈ℳloc,B.
Finally, we will prove that(3.60)Γ7ε=-εUρ′′(Tl,σ(uε)-c)(Tl,σ′(uε))2|∇uε|2∈ℳloc,B.
First, notice that suppUρ′′=(0,ϱ), and therefore
(3.61)Uρ′′(Tl,σ(uε)-c)≠0forc≤Tl,σ(uε)≤c+ρ.
Assume first that |c|>l. Since we can choose ρ=σ (see (3.35)) arbitrarily small, we can assume that |c|>l+σ. In that case Uρ′′(Tl,σ(uε)-c)≠0 only if l+σ≤Tl,σ(uε)≤l+σ+ρ which is never fulfilled according to the definition of Tl,σ (see (3.22)). So, in this case,
(3.62)Γ7ε≡0∈ℳloc,B.
Next we assume that|c|<l. As before, we can assume that |c|<l-ρ since we can choose ρ=σ arbitrarily small. From (3.61), we see that Uρ′′(Tl,σ(uε)-c)≠0 if -l≤c≤Tl,σ(uε)≤c+ρ≤l implying that Uρ′′(Tl,σ(uε)-c)=Uρ′′(uε-c). Thus,
(3.63)Γ7ε=-εUρ′′(uε-c)|∇uε|2∈ℳloc,B,
according to (3.30) (we put there l=c).
Finally, assume that|c|=l. From (3.61) and (3.19), we conclude
(3.64)(Tl,σ′(uε))2Uρ′′(Tl,σ(uε)-l)≠0iff(l≤Tl,σ(uε)≤l+ρanduε≤l+σ)iff(l≤uε≤l+σ).
From here and since |Tl,σ′|≤1, it follows (recall that we assume ρ=σ)
(3.65)∬ΠtεUρ′′(Tl,σ(uε)-c)(Tl,σ′(uε))2|∇uε|2dxdt≤εσ∬Πt∩{l<|uε|<l+σ}|∇uε|2dxdt<∞,
according to (3.30). From here, (3.62) and (3.63), we conclude (3.60).
Collecting the previous items, due to the properties ofΓi,ε, i=1,…,7, it follows from (3.33) that
(3.66)∂tθ(Tl(uε)-c)(Tl(uε)-c)+divxθ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))∈ℳloc,B+Hloc,c-1.
Therefore, we see that (ii) is satisfied and we can use Murat's lemma to conclude that
(3.67)∂tθ(Tl(uε)-c)(Tl(uε)-c)+divxθ(Tl(uε)-c)(f(t,x,Tl(uε))-f(t,x,c))∈Hloc,c-1.
Thus we conclude that the conditions of Theorem 3.3 are satisfied, and we find that for every l>0 the sequence (Tl(uε))ε>0 is precompact in Lloc1(R+×R).
Since the sequence(uε)ε>0 is uniformly bounded in L2(R+×Rd), from [23, Lemma 7], we conclude that (uε)ε>0 is precompact in Lloc1(R+×Rd).


## 4. The One-Dimensional Case

We will analyze the convergence of the sequence(uε)ε>0 of solutions to (3.5)–(3.6) in the one dimensional case. Unlike the situation we had in the previous section, we will assume that the flux is continuously differentiable with respect to u. This will enable us to optimize the ratio δ/ε2. We will work under the following assumptions on the flux f=f(t,x,u) denoted  (H4′).(H4a′) For the flux f=f(t,x,u) we assume that f∈C1(R;BV(R+×Rx))∩L∞(R×R+×Rx) and ∂uf∈L∞(R×R+×Rx).(H4b′) There exists a sequence (fϱ)ϱ>0 defined on R+×R×R, smooth in (t,x)∈R+×R, and continuously differentiable in u∈R, satisfying for some p>2 and every l>0:(4.1)max|z|≤l|fϱ(t,x,z)-f(t,x,z)|→0,ϱ→0inLlocp(R+×R),∬R+×R|∂xfϱ(t,x,u)|dxdt≤C̃11+|u|1+α,∬R+×Rϱ|∂xfϱ(t,x,u)|2dxdt≤C̃2(t,x),∬R+×R|∂xu2fϱ(t,x,u)|dxdt≤C̃31+|u|1+α,|∂ufϱ(t,x,u)|≤C,
where C̃i, i=1,2,3, and C are constants independent on (t,x,u)∈R+×Rd×R.Under these assumptions we will prove the following.(i)
Without assuming nondegeneracy of the flux, the sequence(uε)ε>0 converges along a subsequence to a solution of (3.1)–(11) in the distributional sense when δ=𝒪(ε2) and ϱ=𝒪(ε) (less stringent assumptions than in the multidimensional case).(ii)
If, in addition, we assumef∈C2(R;BV(R+×Rx))∩L∞(R×R+×Rx), and that f is genuinely nonlinear in the sense of (4.12), the sequence (uε)ε>0 of solutions of problem (3.5)–(3.6) is strongly precompact in Lloc1(R+×R) when δ=𝒪(ε2).Remark 4.1.
The proof relies on a priori inequalities (3.8) and (3.9). Notice that thanks to (H4a′), we can take β(ϱ)=1 in inequality (3.9).We will need the fundamental theorem of Young measures.Theorem 4.2 (see [24]).
Assume that the sequence(uεk) is uniformly bounded in Llocp(R+×Rd)), p≥1. Then, there exists a subsequence (not relabeled) (uεk) and a sequence of probability measures
(4.2)ν(t,x)∈ℳ(R),(t,x)∈R+×Rd
such that the limit
(4.3)g̅(t,x):=limk→∞g(t,x,uεk(t,x))
exists in the distributional sense for all g measurable with respect to (t,x)∈R+×Rd, continuous in u∈R and satisfying uniformly in (t,x):
(4.4)|g(t,x,u)|≤C(1+|u|q),
for a constant C independent of u, and q such that 0≤q<p. The limit is represented by the expectation value
(4.5)g̅(t,x)=∫R+×Rdg(t,x,λ)dν(t,x)(λ),
for almost all points (t,x)∈R+×Rd.
One refers to such a sequence of measuresν=(ν(t,x)) as the Young measures associated to the sequence (uεk)k∈N.
Furthermore,(4.6)uεk→uinLlocr(R+×Rd),1≤r<p,
if and only if
(4.7)νy=δu(y)a.e.Before we continue, we need to recall the celebrated Div-Curl lemma.Lemma 4.3 (Div-Curl).
LetQ⊂R2 be a bounded domain, and suppose that
(4.8)vε1⇀v¯1,vε2⇀v¯2,wε1⇀w¯1,wε2⇀w¯2,
in L2(Q) as ε↓0. Assume also that the two sequences {div(vε1,vε2)}ε>0 and {curl(wε1,wε2)}ε>0 lie in a (common) compact subset of Hloc-1(Q), where div(vε1,vε2)=∂x1vε1+∂x2vε2 and curl(wε1,wε2)=∂x1wε2-∂x2wε1. Then along a subsequence
(4.9)(vε1,vε2)·(wε1,wε2)→(v¯1,v¯2)·(w¯1,w¯2)in𝒟′(Q)asε↓0.Lemma 4.4.
Assume that(uε)ε>0∈L2(R+×R) converges weakly in L2(R+×R) to a function u∈L2(R+×R). Assume that η(t,x,λ), (t,x,λ)∈R+×R2, is a function satisfying (4.4) with q=2 such that η∈C2(Rλ;L∞∩BV(Rt+×Rx)).
Byηn one denotes the truncation of the function η:
(4.10)ηn(t,x,λ)={η(t,x,λ),|λ|<n,0,|λ|>2n,(t,x)∈R+×R,
and qn(t,x,λ) the corresponding entropy flux.
If for everyn∈N one has
(4.11)div(t,x)(ηn(t,x,uε),qn(t,x,uε))∈Hloc,c-1(R+×R),
then the limit function u is a weak solution of (1.3).
Furthermore, if the flux functionf=f(t,x,λ) is twice differentiable with respect to λ, and is genuinely nonlinear, that is, for every (t,x)∈R+×Rd the mapping
(4.12)R∋λ↦∂λf(t,x,λ)isnonconstant
on nondegenerate intervals, then (uε)ε>0 converges strongly along a subsequence to u in Lloc1(R+×R).Proof.
We will apply the method of compensated compactness as in [13].
First, notice that according to Theorem4.2 there exist a subsequence (uεk)⊂(uε) and a sequence of probability measures
(4.13)ν(t,x)∈ℳ(R),(t,x)∈R+×R
such that the limit
(4.14)g̅(t,x):=limk→∞g(t,x,uεk(t,x))
exists in the distributional sense for all g measurable with respect to (t,x)∈R+×R, continuous in u∈R, and satisfying (4.4) for some q∈R such that 0≤q<p, and is represented by the expectation value
(4.15)g̅(t,x)=∫R+×Rg(t,x,λ)dν(t,x)(λ),
for almost all points (t,x)∈R+×R. Next, notice that the function f satisfies (4.4) for q=1. Indeed, from (H4a′), it follows ∂u|f|=sgn(f)∂uf≤C, and from here |f|≤Ĉ(1+|u|), for a constant Ĉ which depends on the constant C and the L∞ bound of the function f. From this, we conclude that for the flux function f(t,x,v) we have
(4.16)limk→∞f(t,x,uεk(t,x))=∫R+×Rf(t,x,λ)dν(t,x)(λ).
To continue, notice that(4.17)u(t,x)=∫λdν(t,x)(λ).
Takeη(u)=I(u)=u in (4.10), and consider the vector fields (In(uε),fn(t,x,uε)) where ∂λfn(t,x,uε)=In′(v)∂λf(t,x,uε), and (-ψn(t,x,uε),ϕn(uε)), where ϕ∈C1(R) is an arbitrary entropy, and ψn is the entropy flux corresponding to ϕn. Here In and ϕn denote the smooth truncation functions of I and ϕ, respectively (cf. (4.10)).
According to (4.11), we can apply the Div-Curl lemma on the given vector fields. Hence, we get after letting ε→0 along a subsequence:
(4.18)∫(In(λ)ψn(t,x,λ)-ϕn(λ)fn(t,x,λ))dν(t,x)(λ)=∫(u̅n(t,x)ψn(t,x,λ)-f̅n(t,x)ϕn(λ))dν(t,x)(λ),
where
(4.19)f̅n(t,x)=∫fn(t,x,λ)dν(t,x)(λ),u̅n(t,x)=∫In(λ)dν(t,x)(λ).
Then, putϕ(λ)=|λ-u(t,x)|. Notice that for |λ|<n it holds ψn(t,x,λ)=sgn(λ-u(t,x))(f(t,x,λ)-f(t,x,u(t,x))). Therefore, we have from (4.18)
(4.20)∫-nn(λsgn(λ-u(t,x))(f(t,x,λ)-f(t,x,u(t,x)))-|u(t,x)-λ|f(t,x,λ))dν(t,x)(λ)-∫-nn(u(t,x)sgn(λ-u(t,x))(f(t,x,λ)-f(t,x,u(t,x)))-|u(t,x)-λ|f̅n)dν(t,x)(λ)=-(∫-∞-n+∫n∞)(In(λ)ψn(t,x,λ)-ϕn(λ)fn(t,x,λ))dν(t,x)(λ)+(∫-∞-n+∫n∞)(u(t,x)ψn(t,x,λ)-f̅nϕn(λ))dν(t,x)(λ)+(∫-∞-n+∫n∞)(In(λ)-λ)dν(t,x)(λ)∫ψn(t,x,λ)dν(t,x)(λ).
It is clear that for every fixed (t,x)∈R+×Rd the right-hand side of (4.20) tends to zero as n→∞ implying (due to the Lebesgue dominated convergence theorem)
(4.21)∫(λsgn(λ-u(t,x))(f(t,x,λ)-f(t,x,u(t,x)))-|u(t,x)-λ|f(t,x,λ))dν(t,x)(λ)-∫(u(t,x)sgn(λ-u(t,x))(f(t,x,λ)-f(t,x,u(t,x)))-|u(t,x)-λ|f̅(t,x))dν(t,x)(λ)=0in𝒟′(R+×Rd).
Now, a standard procedure gives (see, e.g., [6, Remark 2.3])
(4.22)(f(t,x,u(t,x))-f̅(t,x))∫|λ-u(t,x)|dν(t,x)(λ)=0,
where f̅(t,x)=∫f(t,x,λ)dν(t,x)(λ). From here, it follows that u is a weak solution to (3.1). This concludes the first part of the lemma. For the details of the procedure one should consult, for example, [13].
Now, assume thatf∈C2(R;BV(R+×Rx))∩L∞(R×R+×Rx), and that it is genuinely nonlinear in the sense of (4.12).
Then, take arbitrary entropiesη1(t,x,u)∈C1((R;L∞∩BV(Rt+×Rx))) and η2∈C1(R), and denote by q1(t,x,u) and q2(u), respectively, their corresponding entropy fluxes. Assume that ηi, qi, ηiqi, i=1,2, satisfy (4.4) for q<2. Notice that ∂uη1 depends explicitly on (t,x), while Duη2 does not. Denote by η1,n and η2,n the appropriate smooth truncations (cf. (4.10)) and by q1,n and q2,n the corresponding entropy fluxes, that is,
(4.23)q1,n(t,x,λ)=∫λ∂zη1,n(t,x,z)∂zf(t,x,z)dz,q1,n(t,x,λ)=∫λ∂zη2,n(z)∂zf(t,x,z)dz.
Due to (4.11) and the Div-Curl lemma the following commutation relation holds:
(4.24)∫R(η1,n(t,x,λ)q2,n(t,x,λ)-η2(λ)q1,n(t,x,λ))dν(t,x)=∫Rη1,n(t,x,λ)dν(t,x)∫Rq2,n(t,x,λ)dν(t,x)-∫Rη2,n(λ)dν(t,x)∫Rq1,n(t,x,λ)dν(t,x).
Letting n→∞ as in (4.20), we get
(4.25)∫R(η1(t,x,λ)q2(t,x,λ)-η2(λ)q1(t,x,λ))dν(t,x)=∫Rη1(t,x,λ)dν(t,x)∫Rq2(t,x,λ)dν(t,x)-∫Rη2(λ)dν(t,x)∫Rq1(t,x,λ)dν(t,x).
Next, recall that the functionf satisfies (4.4) for q=1. Therefore, the following entropy-entropy fluxes are admissible:
(4.26)η1(t,x,λ)=f(t,x,λ)-f(t,x,u(t,x)),q1(t,x,λ)=∫u(t,x)λ(∂vf(t,x,v))2dv,η2(λ)=λ-u(t,x),q2(t,x,λ)=f(t,x,λ)-f(t,x,u(t,x)).
Then, following [6], we insert the last quantities in (4.25) which yields the following relation:
(4.27)(∫R(f(t,x,λ)-f(t,x,u(t,x)))dν(t,x))2+∫R((λ-u(t,x))∫u(t,x)λ(∂ϱf(t,x,ϱ))2dϱ-(f(t,x,λ)-f(t,x,u))2)dν(t,x)(λ)=0.
By the Cauchy–Schwarz inequality
(4.28)(f(t,x,λ)-f(t,x,u))2=(∫u(t,x)λ∂ϱf(t,x,ϱ)dϱ)2≤(λ-u(t,x))∫u(t,x)λ[∂ϱf(t,x,ϱ)]2dϱdν(t,x)(λ),
with the equality only if f(t,x,ϱ) is constant for all ϱ between u(t,x) and λ. Still, this is not possible according to the genuine nonlinearity condition (4.12). Thus, from this and (4.27), we conclude that
(4.29)(λ-u(t,x))∫u(t,x)λ[∂ϱf(t,x,ϱ)]2dϱdν(t,x)(λ)=0,
that is, that ν(t,x)=δu(t,x) a.e. on R+×R implying strong Lloc1 convergence of (uε)ε>0 along a subsequence (see Theorem 4.2).Now we are ready to prove the main theorem of the section.Theorem 4.5.
Assume that(4.30)δ=δ(ε)=𝒪(ε2),ϱ=𝒪(ε),ε→0,
and u0∈H1(R).
Assume that the flux functionf from (3.1) with d=1 satisfies (H4′). Assume also that the function b from (3.5) satisfies (H1) and (H2). Then a subsequence of solutions (uεk)⊂(uε) of problem (3.5)–(3.6) converges in the sense of distributions to a weak solution of problem (3.1).
If the flux functionf∈C2(R;BV(R+×Rx))∩L∞(R×R+×Rx), and if it is genuinely nonlinear in the sense of (4.12), then a subsequence of solutions (uεk)⊂(uε) of problem (3.5)–(3.6) converges strongly in L1(R+×R) to a weak solution of (3.1).Proof.
Assume thatη(t,x,λ), (t,x,λ)∈R+×R2 is a function such that η∈C2(R;L∞∩BV(Rt+×Rx)). As usual, denote by ηn the truncation given by (4.10), and let the entropy flux corresponding to ηn and f be
(4.31)qn(t,x,u)=∫u∂vηn(t,x,v)∂vf(t,x,v)dv.
According to Lemma4.4, it is enough to prove that for every fixed n∈N the expression div(ηn(t,x,uε(t,x)),qn(t,x,uε(t,x))) is precompact in Hloc-1(R+×R).
In order to prove the latter, take the following mollifierηn,ε(t,x,u)=ηn(·,·,u)⋆(1/ε1/2)ω(t/ε1/4)ω(x/ε1/4), where ω is a nonnegative real function with unit mass. Denote the entropy flux corresponding to ηn and f by
(4.32)qn,ε(t,x,u)=∫u∂vηn,ε(t,x,v)∂vfϱ(t,x,v)dv.
Recall that here (and in the sequel) we assume that ϱ=𝒪(ε). Actually, we can take ϱ=ε without loss of generality.
Notice that according to the assumptions onη and the choice of the mollifier ηn,ε we have
(4.33)∬R+×R(|∂tηn,ε(t,x,u)|+|∂xηn,ε(t,x,u)|+|∂xvηn,ε(t,x,u)|)dxdt≤C̃4,∬R+×R(|∂xηn,ε(t,x,u)|2+|∂xv2ηn,ε(t,x,u)|2)dxdt≤C̃4ε,
for a constant C̃4.
Then, applying (3.17) with S replaced by ηn,ε, we find
(4.34)Dtηn(t,x,uε)+Dxqn(t,x,uε)=∫uε(∂xv2fϱ(t,x,v)∂vηn,ε(t,x,v)+∂vfϱ(t,x,v)∂xv2ηn,ε(t,x,v))dv-∂vηn,ε(t,x,uε)∂xfϱ(t,x,uε)+∂tηn,ε(t,x,uε)+εDx(∂vηn,ε(t,x,uε)b(∂xuε))-ε∂vv2ηn,ε(t,x,uε)b(∂xuε)∂xuε-εb(∂xuε)∂xv2ηn,ε(t,x,uε)-δ∂xx2uε∂xv2ηn,ε(t,x,uε)+δDx(∂vηn,ε(t,x,uε)∂xx2uε)-δ2∂vv2ηn,ε(t,x,uε)Dx(∂xuε)2+Dx(-qn,ε(t,x,uε)+qn(t,x,uε))+Dt(-ηn,ε(t,x,uε)+ηn(t,x,uε)).
Now, we apply a similar procedure as in the multidimensional case.
Combining (H4b') and (4.33), we get
(4.35)∬R+×R|∫uε(∂xv2fϱ(t,x,v)∂vηn,ε(t,x,v)+∂vfϱ(t,x,v)∂xv2ηn,ε(t,x,v))dv|dxdt≤C̃6,
for a constant C̃6, implying boundedness of the subintegral expression in the sense of measures.
Similarly, for a constantC̃7(4.36)∬R+×R|-∂vηn,ε(t,x,uε)∂xfϱ(t,x,uε)-∂tηn,ε(t,x,uε)|dxdt≤C̃7,
implying boundedness of the subintegral expression in the sense of measures.
Then, combining (4.33) with (3.8) and (3.9) we infer (see estimation of Γ6ε) that
(4.37)-εb(∂xuε)∂xv2ηn,ε(t,x,uε)-δ∂xx2uε∂xv2ηn,ε(t,x,uε)
is bounded in ℳ(R+×R).
Next,(4.38)Dx(ε∂vηn(t,x,uε)b(∂xuε)+δ∂vηn(t,x,uεk)∂xx2uεk)
is precompact in H-1(R+×R) since |ηn′|<C, δ=𝒪(ε2), ϱ=𝒪(ε), and from (3.8) and (3.9) (see also Remark 4.1) we have
(4.39)εb(∂xuε)+δ∂xx2uε→0asε→0inL2(R+×R).
Similarly, by (3.8) and (3.9) (see the estimation of Γ6ε again)
(4.40)ε∂vvηn,ε(t,x,uε)b(∂xuε)∂xuε+δ2∂vvηn,ε(t,x,uε)Dx(∂xuε)2∈ℳloc,B(R+×R).
Next, due to (H4b′) and the well-known properties of the convolution, it holds for a constant C independent on ε:
(4.41)|qn,ε(t,x,uε)-qn(t,x,uε)|≤Cε→0inLloc2(R+×R)asε→0
for arbitrary p>0 implying
(4.42)Dx(qn,ϱ(t,x,uε)-qn(t,x,uε))∈Hloc,c-1.
Similarly, it is easy to see that(4.43)max-2n<v<2n(-ηn,ε(t,x,uε)+ηn(t,x,uε))→0inL2(R+×R),
and thus
(4.44)Dt(-ηn,ε(t,x,uε)+ηn(t,x,uε))∈Hc-1(R+×R).
From (4.35)–(4.44) and the fact that (ηn(t,x,uε),qn(t,x,uε))∈L∞(R+×R), we conclude using Murat's lemma that
(4.45)div(ηn(t,x,uε),qn(t,x,uε))∈Hloc,c-1(R+×R).
Finally, relying on Lemma4.4 we conclude the proof of the theorem.



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*Source: 279818-2009-12-03.xml*