doi:10.1155/2009/279818

*Research Article*

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

**H. Holden,**

^{1, 2}**K. H. Karlsen,**

^{3}**and D. Mitrovic**

^{1}*1**Department of Mathematical Sciences, Norwegian University of Science and Technology,*
*Alfred Getz vei 1, 7491 Trondheim, Norway*

*2**Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway*

*3**Department of Mathematics, Centre of Mathematics for Applications, University of Oslo,*
*P.O. Box 1053, Blindern, 0316 Oslo, Norway*

Correspondence should be addressed to H. Holden,holden@math.ntnu.no Received 2 April 2009; Revised 24 August 2009; Accepted 24 September 2009 Recommended by Philippe G. LeFloch

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 of*H-measures to investigate the zero diﬀusion-dispersion-smoothing limit.*

Copyrightq2009 H. Holden et al. 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.

**1. Introduction**

We consider the convergence of smooth solutions*u* *u**ε*t, xwith t, x ∈ **R**^{}×**R*** ^{d}* of the
nonlinear partial diﬀerential equation

*∂*_{t}*u*div_{x}*f** _{}*t, x, u

*εdiv*

_{x}*b∇u δ*

*d*

*j1*

*∂*^{3}_{x}_{j}_{x}_{j}_{x}_{j}*u* 1.1

as*ε* → 0 and*δδε, ε* → 0. Here*f* ∈*CR;BV*R^{}* _{t}* ×

**R**

^{d}*is the Caratheodory flux vector such that*

_{x}max|u|≤l*f** _{}*t, x, u−

*ft, x, u*−→0, −→0, in

*L*

^{p}_{loc}

**R**^{}×**R**^{d}

*,* 1.2

for *p >* 2 and every *l >* 0. The aim is to show convergence to a weak solution of the
corresponding hyperbolic conservation law:

*∂**t**u*div*x**ft, x, u *0, *uut, x, x*∈**R**^{d}*, t*≥0. 1.3

We refer to this problem as the zero diﬀusion-dispersion-smoothing limit.

In the case when the flux*f*is at least Lipschitz continuous, it is well known that the
Cauchy problem corresponding to1.3has a unique admissible entropy solution in the sense
of Kruˇzhkov1 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 yearssee, e.g.,3and references therein. The one-dimensional
case of the problem is widely investigated using several approachesnumerical techniques
3,4, compensated compactness5,6, and kinetic approach7,8. 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 solutions2,18is used.

More recently, in19the limiting case has also been analyzed using the kinetic approach and velocity averaging20.

The remaining part of this paper is organized as follows. InSection 2we collect some basic a priori estimates for smooth solutions of1.1. InSection 3we 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 approach10. Finally, inSection 4we 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 flux *f* in 1.1is smooth in all variables. Consider a sequence u*ε,δ** _{ε,δ}* of
solutions of

*∂*_{t}*u*div_{x}*ft, x, u εdiv*_{x}*b∇u δ*
*d*
*j1*

*∂*^{3}_{x}_{j}_{x}_{j}_{x}_{j}*u,*

*ux,*0 *u*0x, *x*∈**R**^{d}*.*

2.1

We assume thatu*ε,δ** _{ε,δ}*has enough regularity so that all formal computations below
are correct. So, following Schonbek13, we assume that for every

*ε, δ >*0 we have

*u*

*∈*

_{ε,δ}*L*

^{∞}0, T;

*H*

^{4}R

*.*

^{d}Later on, we will assume that the initial data*u*0depends on*ε. In this section, we will*
determine a priori inequalities for the solutions of problem2.1.

To simplify the notation we will write*u** _{ε}*instead of

*u*

*.*

_{ε,δ}We will need the following assumptions on the diﬀusion term*bλ b*1λ, . . . , b*n*λ.

H1For some positive constants*C*1*, C*2we have

*C*1|λ|^{2} ≤*λ*·*bλ*≤*C*2|λ|^{2} ∀ *λ*∈**R**^{d}*.* 2.2
H2The gradient matrix*Dbλ*is a positive definite matrix, uniformly in*λ*∈**R*** ^{d}*, that
is, for every

*λ,*∈

**R**

*, there exists a positive constant*

^{d}*C*

_{3}such that we have

^{T}*Dbλ*≥*C*3^{2}*.* 2.3

We use the following notation:

D^{2}*u*^{2} ^{d}

*i,k1*

∂^{2}_{x}_{i}_{x}_{k}*u*^{2}*.* 2.4

In the sequel, for a vector valued function*g* g1*, . . . , g** _{d}*defined on

**R**

^{}×

**R**

*×*

^{d}**R, we denote**

*g*

^{2}

^{d}*i1*

*g*_{i}^{2}*.* 2.5

The partial derivative*∂*_{x}* _{i}* in the pointt, x, u, where

*u*possibly depends ont, x, is defined by the formula

*∂**x**i**gt, x, ut, x *

*D**x**i**gt, x, λ*

|_{λut,x}*.* 2.6

In particular, the total derivative *D**x**i* and the partial derivative *∂**x**i* are connected by the
identity

*D**x**i**gt, x, u ∂**x**i**gt, x, u ∂**u**gt, x, u∂**x**i**u.* 2.7
Finally we use

div_{x}*g*t, x, u ^{d}

*i1*

*D*_{x}_{i}*g** _{i}*t, x, u,

*g*

*g*_{1}*, . . . , g*_{d}*,*

Δ*x**qt, x, u *^{d}

*i1*

*D*^{2}_{x}_{i}_{x}_{i}*qt, x, u,* *q*∈*C*^{2}

**R**^{}×**R*** ^{d}*×

**R**

*.*

2.8

With the previous conventions, we introduce the following assumption on the flux
vector*f.*

H3The growth of the velocity variable*u*and the spatial derivative of the flux*f*are
such that for some*C, α >*0,*p*≥1, and every*l >*0, we have

max|λ|<l*f** _{i}*t, x, λ∈

*L*

^{p}**R**^{}×**R**^{d}

*,* *i*1, . . . , d,
*d*

*i1*

*∂*_{u}*f** _{i}*t, x, u≤

*C,*

*d*
*i,j1*

*∂*_{x}_{i}*f** _{j}*t, x, u≤

*μt, x*1|u|

^{1α}

*,*

2.9

where *μ* ∈ MR^{} ×**R*** ^{d}* 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 function***f* *ft, x, usatisfies (H3) and that it is Lipschitz*
*continuous on***R**^{}×R* ^{d}*×R. Assume also that initial data

*u*

_{0}

*belongs toL*

^{2}R

*. Under conditions (H1)-*

^{d}*(H2) the sequence of solutions*u

*ε*

_{ε>0}*of*2.1

*for everyt*∈0, T

*satisfies the following inequalities:*

**R*** ^{d}*|u

*ε*t, x|

^{2}

*dxε*

_{t}0

**R**^{d}

∇u*ε*

*t, x*^{2}*dxdt*

≤*C*4

**R*** ^{d}*|u0x|

^{2}

*dx*−

_{t}0

**R**^{d}

_{u}_{ε}_{t}_{,x}

0

div*x**f*
*t, x, v*

*dvdxdt*

*,*

2.10

*ε*^{2}

**R**^{d}

∇u*ε*t, x|^{2}*dxε*^{3}
_{t}

0

**R**^{d}

*D*^{2}*u**ε*

*t, x*

|^{2}*dxdt*

≤*C*_{5} *ε*^{2}

**R*** ^{d}*|∇u0x|

^{2}

*dxε*

_{t}0

**R**^{d}

*d*
*k1*

*∂*_{x}_{k}*f*

*t, x, u*_{ε}

*t, x*^{2}*dxdt* *∂*_{u}*f*^{2}

*L*^{∞}R^{}×R* ^{d}*×R

*,*
2.11

*for some constantsC*_{4}*andC*_{5}*.*

*Proof. We follow the procedure from*19. Given a smooth function*ηηu,u*∈**R, we define**

*q** _{i}*t, x, u

_{u}0

*η*v∂*v**f** _{i}*t, x, vdv,

*i*1, . . . , d. 2.12

If we multiply2.1by*η*u, it becomes

*∂*_{t}*ηu**ε*

*d*
*i1*

*∂*_{x}_{i}*q** _{i}*t, x, u

*ε*−

^{d}*i1*

_{u}_{ε}

0

*∂*^{2}_{x}_{i}_{v}*f** _{i}*t, x, vηvdv

^{d}*i1*

*η*u*ε*∂*x**i**f** _{i}*t, x, u

*ε*

*ε*
*d*

*i1*

*∂*_{x}_{i}

*η*u*ε*b*i*∇u*ε*

−*εη*^{ }u*ε*^{d}

*i1*

*b** _{i}*∇u

*ε*∂

*x*

*i*

*u*

_{ε}*δ*

*d*

*i1*

*∂*_{x}_{i}

*η*u*ε*∂^{2}_{x}_{i}_{x}_{i}*u*_{ε}

−*δ*

2*η*^{ }u*ε*^{d}

*i1*

*∂*_{x}* _{i}*∂

*x*

*i*

*u*

_{ε}^{2}

*.*

2.13

Choosing here*ηu u*^{2}*/2 and integrating over*0, t×**R*** ^{d}*, we get

**R*** ^{d}*|u

*ε*t, x|

^{2}

*dxε*

_{t}0

**R*** ^{d}*∇u

*ε*

*t, x*

·*b*

∇u*ε*

*t, x*
*dxdt*

**R*** ^{d}*|u0x|

^{2}

*dx*

^{d}*j1*

_{t}

0

**R**^{d}

_{u}_{ε}_{t}_{,x}

0

*vD*^{2}_{x}_{j}_{v}*f**j*

*t, x, v*
*dvdxdt*

−^{d}

*i1*

_{t}

0

**R**^{d}*u*_{ε}*t, x*

*∂*_{x}_{i}*f*_{i}

*t, x, u*_{ε}*t, x*

*dxdt*

**R*** ^{d}*|u0x|

^{2}

*dx*−

^{d}*i1*

_{t}

0

**R**^{d}

_{u}_{ε}_{t}_{,x}

0

*∂*_{x}_{i}*f*_{i}*t, x, v*

*dvdxdt,*

2.14

where the second equality sign is justified by the following partial integration:

_{t}

0

**R**^{d}

_{u}_{ε}

0

*vD*^{2}_{x}_{j}_{v}*f*_{j}*t, x, v*

*dvdxdt*

_{t}

0

**R**^{d}*u*_{ε}*∂*_{x}_{i}*f*_{i}

*t, x, u*_{ε}*dxdt* −

_{t}

0

**R**^{d}

_{u}_{ε}

0

*∂*_{x}_{i}*f*_{i}*t, x, v*

*dvdxdt.*

2.15

Now inequality2.10follows from2.14, usingH1.

As for inequality2.11, we start by using2.14, namely,

**R*** ^{d}*|u

*ε*t, x|

^{2}

*dxε*

_{t}0

**R*** ^{d}*∇u

*ε*

*t, x*

·*b*

∇u*ε*

*t, x*
*dxdt*

**R*** ^{d}*|u0x|

^{2}

*dx*−

^{d}*i1*

_{t}

0

**R**^{d}

_{u}_{ε}_{t}_{,x}

0

*∂*_{x}_{i}*f*_{i}*t, x, v*

*dvdxdt*

≤

**R*** ^{d}*|u0x|

^{2}

*dx*

^{d}*i1*

_{t}

0

**R**^{d}

_{u}_{ε}_{t}_{,x}

0

*∂*_{x}_{i}*f*_{i}

*t, x, v* *dvdxdt*

≤

**R*** ^{d}*|u0x|

^{2}

*dx*

_{t}0

**R**^{d}

**R**

*μt, x*

1|v|^{1α}*dvdxdt*

≤

**R*** ^{d}*|u0x|

^{2}

*dxC*

_{t}0

**R**^{d}*μ*
*t, x*

*dxdt,*

2.16

where*C*

**R**dv/1|v|^{1α}.

From here, usingH3, we conclude in particular that

*ε*
_{t}

0

**R**^{d}

∇u*ε*

*t, x*^{2}*dxdt* ≤*C*_{11}*,* 2.17

for some constant*C*_{11}independent of*ε.*

Next, we diﬀerentiate 2.1with respect to*x**k* and multiply the expression by*∂**x**k**u.*

Integrating over**R*** ^{d}*, using integration by parts and then summing over

*k*1, . . . , d,we get:

1 2

**R**^{d}*∂**t*|∇u*ε*|^{2}*dx*−^{d}

*k1*

**R*** ^{d}*∇∂

*x*

*k*

*u*

*ε*·

*∂**x**k**f**k*t, x, u*ε* *∂**u**f**k**∂**x**k**u**ε*

*dx*

−ε^{d}

*k1*

**R*** ^{d}*∇∂

*x*

*k*

*u*

_{ε}

^{T}*Db∇u*

*ε*∇∂

*x*

*k*

*u*

*dx.*

_{ε}2.18

Integrating this over0, tand using the Cauchy-Schwarz inequality and conditionH2, we find

1 2

**R*** ^{d}*|∇u

*ε*t,·|

^{2}

*dxεC*

_{3}

*d*

*k1*

_{t}

0

**R*** ^{d}*|∇∂

*x*

*k*

*u*

*|*

_{ε}^{2}

*dxdt*

≤ 1 2

**R*** ^{d}*|∇u0|

^{2}

*dx*

^{d}*k1*

∇∂*x**k**u*_{ε}* _{L}*2

^{R}^{}

^{×R}

^{d}*∂*

_{x}

_{k}*f*

*·,·, u*

_{k}*ε*

*∂*

_{u}*f*

_{k}*∂*

_{x}

_{k}*u*

_{ε}*L*^{2}^{R}^{}^{×R}^{d}*,*

2.19

where*C*3is independent of*ε. Then, using Young’s inequality*the constant*C*3is the same as
previously mentioned

*ab*≤ *C*_{3}*ε*

2 *a*^{2} 1

2C_{3}*εb*^{2}*,* *a, b*∈**R,** 2.20

we obtain

1 2

**R*** ^{d}*|∇u

*ε*t,·|

^{2}

*dxεC*

_{3}

*d*

*k1*

_{t}

0

**R*** ^{d}*|∇∂

*x*

*k*

*u*

*|*

_{ε}^{2}

*dxdt*

≤ 1 2

**R*** ^{d}*|∇u0|

^{2}

*dxC*

_{3}

*ε*2

*d*
*k1*

_{t}

0

**R*** ^{d}*|∇∂

*x*

*k*

*u*

*|*

_{ε}^{2}

*dxdt*1

2C3*ε*
_{t}

0

**R**^{d}

*d*
*k1*

*∂*_{x}_{k}*f*_{k}

*t, x, u*_{ε}

*∂*_{u}*f*_{k}*∂*_{x}_{k}*u*_{ε}^{2}*dxdt.*

2.21

Multiplying this by*ε*^{2}, usinga*b*^{2}≤2a^{2}2b^{2}, and applying2.17, we conclude

*ε*^{2}
2

**R*** ^{d}*|∇u

*ε*t, · |

^{2}

*dxC*

_{3}

*ε*

^{3}2

**R**^{d}

_{t}

0

*D*^{2}*u*_{ε}^{2}*dxdt*

≤ *ε*^{2}
2

**R*** ^{d}*|∇u0|

^{2}

*dxdt*

*ε*

*C*3

_{t}

0

**R**^{d}

*d*
*k1*

*∂**x**k**f**k*

*t, x, u**ε*

*t, x*^{2}*dxdt* *C*11

*C*3

*∂**u**f**k*^{2}

*L*^{∞}^{R}^{}^{×R}^{d}^{×R}*.*
2.22

This inequality is actually inequality 2.11 when we take *C*_{5} 2 max{1,1/C_{3}, *C*_{11}*/*
*C*3}/min{1, C3}.

**3. The Multidimensional Case**

Consider the following initial-value problem. Find*uut, x*such that

*∂**t**u*div*x**ft, x, u *0,
*ux,*0 *u*0x, *x*∈**R**^{d}*,*

3.1

where*u*_{0}∈*L*^{2}R* ^{d}*is a given initial data.

For the flux*f* f1*, . . . , f** _{d}*we need the following assumption, denotedH4.

H4aFor the flux*fft, x, u,*t, x, u∈**R**^{}×R* ^{d}*×R, we assume that

*f*∈

*CR;BV*R

^{}×

**R**

*and that for every*

^{d}*l*∈

**R**

^{}we have max

*|ft, x, u| ∈*

_{u∈−l,l}*L*

*R*

^{p}^{}×

**R**

*,*

^{d}*p >*2 .

H4bThere exists a sequence*f* f1*, . . . , f**d*,∈0,1, such that*f**f*t, x, u∈
*C*^{1}R^{}×**R*** ^{d}*×

**R, satisfying for some**

*p >*2 and every

*l*∈

**R**

^{}:

*z∈−l,l*max*f** _{}*·,·, z−

*f·,*·, z −→

*→*00 in *L*^{p}

**R**^{}×**R**^{d}

0, 3.2a
*d*

*i1*

**R**^{}×R^{d}

*∂*_{x}_{i}*f** _{i}*t, x, u

*dxdt*≤

*C*

_{1}

1|u|^{1α}*,* 3.2b

*d*
*i1,k*

**R**^{}×R^{d}

*∂*_{x}_{k}*f** _{i}*t, x, u

^{2}

*dxdt*≤

*C*

_{2}

*,*3.2c

*d*
*i1*

*∂**u**f**i*t, x, u≤ *C*
*β*

*,* 3.2d

*d*
*i1*

**R**^{}×R^{d}

∂^{2}_{x}_{i}_{u}*f**i*t, x, udxdt≤ *C*3

1|u|^{1α}*,* 3.2e

where *C**i*, *i* 1,2,3, and *C* are constants, while the function *β* : **R** → **R is such that**
lim_{ρ→}_{0}*βρ *0.

In the case when we have only vanishing diﬀusion, it is usually possible to obtain
uniform *L*^{∞} bound for the corresponding sequence of solutions under relatively mild
assumptions on the flux and initial datasee, 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 boundsseeTheorem 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 dispersionsee, e.g.,16,17,19.

*Remark 3.1. For an arbitrary compactly supported, nonnegativeϕ*1 ∈*C*^{∞}_{0} R^{}×**R*** ^{d}*and

*ϕ*2 ∈

*C*

^{∞}

_{0}Rwith total mass one denote

*ϕ*z, u 1
^{d1}*ϕ*1

*z*

1
*β*

*ϕ*2 *u*
*β*

*,* 3.3

*z* ∈ **R**^{} ×**R*** ^{d}* and

*u*∈

**R, where**

*β*is a positive function tending to zero as → 0. In the case when the flux

*f*∈

*CR;BV*R

^{}×

**R**

*∩*

^{d}*BV*R×

**R**

^{}×

**R**

*is bounded, straightforward computation shows that the sequence*

^{d}*f*

*f ϕ*f1

*, . . . , f*

*d*satisfiesH4bwith

*β .*We also need to assume that the flux

*f*is genuinely nonlinear, that is, for everyt, x∈

**R**

^{}×

**R**

*and every*

^{d}*ξ*∈

**R**

*\ {0}, the mapping*

^{d}**R***λ*−→^{d}

*i1*

*f** _{i}*t, x, λ

*ξ*

_{i}|ξ| 3.4

is nonconstant on every nondegenerate interval of the real line.

We will analyze the vanishing diﬀusion-dispersion-smoothing limit of the problem

*∂*_{t}*u*div_{x}*f** _{}*t, x, u

*εdiv*

_{x}*b∇u δ*

*d*

*j1*

*∂*^{3}_{x}_{j}_{x}_{j}_{x}_{j}*u,* 3.5

*ux,*0 *u*0,εx, *x*∈**R**^{d}*,* 3.6

where the flux*f** _{}*satisfies the conditionsH4b. We denote the solution of3.5-3.6by

*u*

_{ε}*u*

*ε*t, x. We assume that

u0,ε−*u*0* _{L}*2

^{R}*−→0, u0,ε*

^{d}*2*

_{L}

^{R}

^{d}*εu*0,ε

*1*

_{H}

^{R}*≤*

^{d}*C.*3.7

We also assume that*ε* → 0 and*δδε* → 0 as*ε* → 0. We want to prove that under
certain conditions, a sequence of solutionsu*ε** _{ε>0}*of3.5-3.6converges to a weak solution
of problem3.1as

*ε*→ 0. To do this in the multidimensional case we use the approach of

*H-measures, introduced in*11and further developed in10,21. In the one-dimensional case, we use the compensated compactness method, following13.

In order to accomplish the plan we need the following a priori estimates.

**Theorem 3.2**a priori inequalities. Suppose that the flux *ft, x, usatisfies (H4). Also assume*
*that the initial datau*0*satisfies*3.7. Under these conditions the sequence of smooth solutionsu*ε*_{ε>0}*of* 3.5-3.6*satisfies the following inequalities for everyt*∈0, T:

**R*** ^{d}*|u

*ε*t, x|

^{2}

*dxε*

_{t}0

**R*** ^{d}*|∇u

*ε*x, s|

^{2}

*dxds*≤

*C*

_{4}

**R*** ^{d}*|u0,εx|

^{2}

*dxC*

_{10}

*,* 3.8

*ε*^{2}

**R*** ^{d}*|∇u

*ε*t, x|

^{2}

*dxε*

^{3}

_{t}0

**R**^{d}

D^{2}*u*_{ε}

*t, x*^{2}*dxdt* ≤C5 *ε*^{2}

**R*** ^{d}*|∇u0,εx|

^{2}

*dx*

*ε*

*C*_{11} *C*12

*β*
_{2}

*,*
3.9

*for some constantsC*_{10}*, C*_{11}*, C*_{12}*(the constantsC*_{4}*, C*_{5}*are introduced inTheorem 2.1).*

*Proof. For every fixed, the functionf* f1*, . . . , f**d*is smooth, and, due toH4, we see
that*f** _{}*satisfiesH3. This means that we can applyTheorem 2.1.

Replacing the flux*f*by*f*from3.5and*u*0by*u*0,ε from3.6in2.10and2.11, we
get

**R*** ^{d}*|u

*ε*t, x|

^{2}

*dxε*

_{t}0

**R*** ^{d}*|∇u

*ε*x, s|

^{2}

*dxds*

≤*C*_{3}

**R*** ^{d}*|u0,εx|

^{2}

*dx*−

_{t}0

**R**^{d}

_{u}_{ε}_{t}_{,x}

0

div_{x}*f*_{}*t, x, v*

*dvdxdt*

*,*

3.10

*ε*^{2}

**R*** ^{d}*|∇u

*ε*t, x|

^{2}

*dxε*

^{3}

_{t}0

**R**^{d}

D^{2}*u**ε*

*t, x*^{2}*dxdt*

≤*C*4 *ε*^{2}

**R*** ^{d}*|∇u0,εx|

^{2}

*dx∂*

*u*

*f*

^{2}

*L*^{∞}^{R}^{}^{×R}^{d}^{×R}
ε

_{t}

0

**R**^{d}

*d*
*k1*

*d*
*i1*

*∂*_{x}_{k}*f*_{i}

*t, x, u*_{ε}*t, x*2

*dxdt*

*.*

3.11

To proceed, we use assumptionH4. We have

_{t}

0

**R**^{d}

_{u}_{ε}_{t}_{,x}

0

div*f**i*

*t, x, v*

*dvdxdt* ≤
_{t}

0

**R**^{d}

**R**

*d*
*i1*

*∂**x**i**f**i*

*t, x, vdvdxdt*

≤

**R**

*C*1

1|v|^{1α}*dv*≤*C*10*,*

3.12

which together with3.10immediately gives3.8.

Similarly, combiningH4and3.11, and arguing as in3.12, we get3.9.

In this section, we will inspect the convergence of a familyu*ε** _{ε>0}*of solutions to3.5-
3.6in the case when

*bλ*1*, . . . , λ**d* λ1*, . . . , λ**d* 3.13

for the function*b*appearing in the right-hand side of3.5. 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 the *H-measures*
approachsee, e.g.,10, Corollary 2 and Remark 3. We let*θ*denote the Heaviside function.

**Theorem 3.3** see10. Assume that the vector*ft, x, u* *is genuinely nonlinear in the sense of*
3.4. Then each familyv*ε*t, x* _{ε>0}*⊂

*L*

^{∞}R

^{}×

**R**

^{d}*such that for everyc*∈

**R the distribution***∂**t*θv*ε*−*cv**ε*−*c *div*x*

*θv**ε*−*c*

*ft, x, v**ε*−*ft, x, c*

3.14
*is precompact inH*_{loc}^{−1}*contains a subsequence convergent inL*^{1}_{loc}R^{}×**R*** ^{d}*.

We can now prove the following theorem.

**Theorem 3.4. Assume that the flux vector**f*is genuinely nonlinear in the sense of* 3.4*and that it*
*satisfies (H4). Furthermore, assume that*

*ε,* *δε*^{2}*ρ*^{2}ε *with* *ρε *O

*βε*

*,* 3.15

*and that* *u*_{0,ε} *satisfies* 3.7. Then, there exists a subsequence of the familyu*ε*_{ε>0}*of solutions to*
3.5–3.6*that converges to a weak solution of problem*3.1.

*Proof. We will use*Theorem 3.3. Since it is well known that the familyu*ε** _{ε>0}*of solutions of
problem3.5–3.6 is not uniformly bounded, we cannot directly apply the conditions of
Theorem 3.3.

Take an arbitrary*C*^{2}function*S* *Su,u*∈**R, and multiply the regularized equation**
3.5by*S*u*ε*. As usual, put

*qt, x, u *
_{u}

0

*S*v∂*u**f* *dv,* *q*

*q*1*, . . . , q**d*

*.* 3.16

We easily find that

*∂**t**Su**ε* div*x**qt, x, u**ε*−div*x**qt, x, v|*_{vu}_{ε}*S*u*ε*div*x**f*t, x, v|_{vu}_{ε}*εdiv**x*

*S*u*ε*∇u*ε*

−*εS*^{ }u*ε*|∇u*ε*|^{2}*δ*
*d*

*j1*

*D**x**j*

*S*u*ε*∂^{2}_{x}_{j}_{x}_{j}*u**ε*

−*δ*
*d*

*j1*

*S*^{ }u*ε*∂*x**j**u**ε**∂*^{2}_{x}_{j}_{x}_{j}*u**ε**.*
3.17
We will apply this formula repeatedly with diﬀerent choices for*Su.*

In order to apply Theorem 3.3, we will consider a truncated sequence T*l*u*ε** _{ε>0}*,
where the truncation function

*T*

*is defined for every fixed*

_{l}*l*∈

**N as**

*T** _{l}*u

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−l, u≤ −l,
*u,* −l≤*u*≤*l,*
*l,* *u*≥*l.*

3.18

We will prove that the sequenceT*l*u*ε** _{ε>0}* is precompact for every fixed

*l. Denote byu*

*a subsequential limitin*

_{l}*L*

^{1}

_{loc}of the family T

*l*u

*ε*

*, which gives raise to a new sequence u*

_{ε>0}*l*

*that we prove converges to a weak solution of3.1.*

_{l>1}To carry out this plan, we must replace*T**l* by a*C*^{2} regularization *T**l,σ* : **R** → **R. We**
define*T** _{l,σ}*:

**R**→

**R by**

*T*

*0 0 and*

_{l,σ}*T** _{l,σ}*u

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

1, |u|*< l,*
*l*− |*u|σ*

*σ* *, l <*|u|*< lσ,*
0, |u|*> lσ.*

3.19

Next, we want to estimate T_{l,σ}^{ } u*ε*∇u*ε*_{L}_{2}_{R}_{}_{×R}_{d}_{}. To accomplish this, we insert the
functions*T*_{l,σ}^{±} for*S*in3.17where*T*_{l,σ}^{±} are defined by*T*_{l,σ}^{±} 0 0 and

*T*_{l,σ}^{}

u

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

1, *u < l,*
*lσ*−*u*

*σ* *, l < u < lσ,*
0, *u > lσ,*

3.20

*T*_{l,σ}^{−}

u

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

1, *u >*−l,
*lσu*

*σ* *,* −l−*σ < u <*−l,
0, *u <*−l−*σ.*

3.21

Notice that

*T*_{l,σ}^{±}u

≤1, T_{l,σ}^{±}u≤ |*u|σ*
2*,*
*T*_{l,σ}^{}u *T*_{l,σ}^{−} u for −*l*≤*u*≤*l.*

3.22

By inserting*Su * −T_{l,σ}^{}u,*q* *q*_{}t, x, u −_{u}

0T_{l,σ}^{} v∂*u**f**dv*in3.17and integrating
overΠ*t* 0, t×**R*** ^{d}*, we get

−

**R**^{d}*T*_{l,σ}^{}u*ε*dx

**R**^{d}*T*_{l,σ}^{}u0dx *ε*
*σ*

Π*t*∩{l<u*ε**<lσ}*|∇u*ε*|^{2}*dxdt*

Π*t*

div_{x}*q*_{}t, x, v|_{vu}_{ε}*dxdt*

Π*t*

*T*_{l,σ}^{}

u*ε*div*x**f** _{}*t, x, v|

_{vu}

_{ε}*dxdt*

−*δ*
*σ*

Π*t*∩{l<u*ε**<lσ}*

*d*
*j1*

*∂*_{x}_{j}*u*_{ε}*∂*^{2}_{x}_{j}_{x}_{j}*u*_{ε}*dxdt.*

3.23

Similarly, for*Su T*_{l,σ}^{−}u,*qq*_{−}t, x, u _{u}

0T_{l,σ}^{−} v∂*u**f**dv, we have from*3.17

**R**^{d}*T*_{l,σ}^{−}u*ε*dx−

**R**^{d}*T*_{l,σ}^{−} u0dx *ε*
*σ*

Π*t*∩{−l−σ<u*ε**<−l}*|∇u*ε*|^{2}*dxdt*

Π*t*

div*x**q*_{−}t, x, v|_{vu}_{ε}*dxdt*−

Π*t*

*T*_{l,σ}^{−}

u*ε*div*x**f*t, x, v|_{vu}_{ε}*dxdt*

*δ*
*σ*

Π*t*∩{−l−σ<u*ε**<−l}*

*d*
*j1*

*∂**x**j**u**ε**∂*^{2}_{x}_{j}_{x}_{j}*u**ε**dxdt.*

3.24

Adding3.23to3.24, we get

*ε*
*σ*

Π*t*∩{l<|u*ε*|<lσ}|∇u*ε*|^{2}*dxdt*
−

**R**^{d}

*T*_{l,σ}^{−} u*ε*−*T*_{l,σ}^{}u*ε*
*dx*

**R**^{d}

*T*_{l,σ}^{−}u0−*T*_{l,σ}^{} u0
*dx*

Π*t*

div*x**q*_{−}t, x, v|_{vu}_{ε}*dxdt*

Π*t*

div*x**q*_{}t, x, v|_{vu}_{ε}*dxdt*

−

Π*t*

*T*_{l,σ}^{−}

u*ε*div*x**f** _{}*t, x, v|

_{vu}

_{ε}*dxdt*

Π*t*

*T*_{l,σ}^{}

u*ε*div*x**f** _{}*t, x, v|

_{vu}

_{ε}*dxdt*

*δ*
*σ*

Π*t*∩{−l−σ<u*ε**<−l}*

*d*
*j1*

*∂*_{x}_{j}*u*_{ε}*∂*^{2}_{x}_{j}_{x}_{j}*u*_{ε}*dxdt*− *δ*
*σ*

Π*t*∩{l<u*ε**<lσ}*

*d*
*j1*

*∂*_{x}_{j}*u*_{ε}*∂*^{2}_{x}_{j}_{x}_{j}*u*_{ε}*dxdt.*

3.25

From3.22and the definition of*q*_{−}and*q*_{}, it follows

*ε*
*σ*

Π*t*∩{l<|u*ε*|<lσ}|∇u*ε*|^{2}*dxdt*≤

|u*ε*|>l2|u*ε*|dx

|u0|>l2|u0|dx 2

Π*t*

**R**

*d*
*i1*

*D*^{2}_{x}_{i}_{v}*f** _{i}*t, x, v

*dvdxdt*2

Π*t*

*d*
*i1*

*∂*_{x}_{i}*f** _{i}*t, x, u

*ε*

*dxdt*2

*δ*

*σ*

Π*t*∩{l−σ<|u*ε*|<l}

*d*
*j1*

∂*x**j**u*_{ε}*∂*^{2}_{x}_{j}_{x}_{j}*u** _{ε}*dxdt.

3.26

Without loss of generality, we can assume that*l >*1. Having this in mind, we get fromH4
and3.26

*ε*
*σ*

Π*t*∩{l<|u*ε*|<lσ}|∇u*ε*|^{2}*dxdt*

≤

|u*ε*|>l2|u*ε*|^{2}*dx*

|u0|>l2|u0|^{2}*dx*2

**R**

*d*
*i1*

*C*_{3}
1|v|^{1α}*dv*
2

Π*t*

*d*
*i1*

*∂*_{x}_{i}*f** _{i}*t, x, u

*ε*

*dxdt*2

*δ*

*σ*

Π*t*∩{l<|u*ε*|<lσ}

*d*
*j1*

∂*x**j**u*_{ε}*∂*^{2}_{x}_{j}_{x}_{j}*u** _{ε}*dxdt

≤

**R*** ^{d}*2

|u*ε*x, t|^{2}|u0x, t|^{2}

*dxK*1*K*22 *δ*
*σε*^{2}

*d*
*i1*

ε^{1/2}*∂**x**i**u**ε*

*L*^{2}^{R}^{}^{×R}^{d}

×*ε*^{3/2}*∂*^{2}_{x}_{i}_{x}_{i}*u*_{ε}

*L*^{2}R^{}×R^{d}

≤*K*_{5} *δ*^{2}
*σ*^{2}*ε*^{4}

*β*

_{2} *δ*^{2}
*σ*^{2}*ε*^{4}

_{1/2}
*K*_{3}*K*_{4}*,*

3.27

where*K** _{i}*,

*i*1, . . . ,5, are constants such thatcf.3.8and3.9

2

**R**

*d*
*i1*

*C*_{3}

1|v|^{1α}*dv*≤*K*_{1}*,*
2

Π*t*

*d*
*i1*

*∂*_{x}_{i}*f** _{i}*t, x, u

*ε*

*dxdt*≤

*K*

_{2}

*,*

*d*
*i1*

*ε*^{1/2}*∂*_{x}_{i}*u*_{ε}

*L*^{2}^{R}^{}^{×R}* ^{d}*≤

*K*

_{3}

*,*

*d*

*i1*

ε^{3/2}*∂*^{2}_{x}_{i}_{x}_{i}*u*_{ε}

*L*^{2}^{R}^{}^{×R}* ^{d}*≤ 1

*β*

_{2} *ε*

1/2

*K*_{4}*,*

**R*** ^{d}*2

|u*ε*x, t|^{2}|u0x, t|^{2}

*dxK*1*K*2≤*K*5*.*

3.28

These estimates follow fromH4and the a priori estimates3.8,3.9. If in addition we use
the assumption*ε*from3.15, we conclude

*δ*

*σε*^{2}ε^{1/2}∇u*ε*_{L}^{2}^{R}^{}^{×R}^{d}*d*

*i1*

ε^{3/2}*∂*^{2}_{x}_{i}_{x}_{i}*u**ε** _{L}*2

^{R}^{}

^{×R}

*≤*

^{d}*δ*

^{2}

*σ*^{2}*ε*^{4}*β*^{2}ε *δ*^{2}
*σ*^{2}*ε*^{4}

_{1/2}

*K*3*K*4*.* 3.29

Thus, in view of3.27,

*ε*
*σ*

Π*t*∩{l<|u*ε*|<lσ}|∇u*ε*|^{2}*dxdt*≤*K*5 *δ*^{2}

*σ*^{2}*ε*^{4}*β*^{2}ε *δ*^{2}
*σ*^{2}*ε*^{4}

_{1/2}

*K*3*K*4*,* 3.30

which is the sought for estimate forT_{l,σ}^{ } u*ε*∇u*ε*_{L}_{2}_{R}_{}_{×R}_{d}_{}.
Next, take a function*U** _{ρ}*zsatisfying

*U*

*0 0 and*

_{ρ}*U** _{ρ}*z

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

0, *z <*0,
*z*

*ρ,* 0*< z < ρ,*
1, *z > ρ.*

3.31

Clearly,*U**ρ*is convex, and*U** _{ρ}*z →

*θz*in

*L*

^{p}_{loc}Ras

*ρ*→ 0, for any

*p <*∞; as before,

*θ*denotes the Heaviside function.

Inserting*Su**ε* *U** _{ρ}*T

*l,σ*u

*ε*−

*c*in3.17, we get

*∂*_{t}*U** _{ρ}*T

*l,σ*u

*ε*−

*c*div

_{x}

_{u}

_{ε}*U** _{ρ}*T

*l,σ*v−

*cT*

*v∂*

_{l,σ}*v*

*f*

*t, x, vdv*

_{}
_{u}_{ε}

*U** _{ρ}*T

*l,σ*v−

*cT*

*vdiv*

_{l,σ}*x*

*∂*

_{v}*f*

*t, x, vdv*

_{}−*U** _{ρ}*T

*l,σ*u

*ε*−

*cT*

*u*

_{l,σ}*ε*div

*x*

*f*

*t, x, v|*

_{}

_{vu}

_{ε}*εΔ*

*x*

*U*

*T*

_{ρ}*l,σ*u

*ε*−

*c*−

*εD*

_{uu}^{2}

*U** _{ρ}*T

*l,σ*u

*ε*−

*c*

|∇u*ε*|^{2}
*δ*

*d*
*i1*

*D*_{x}_{i}*D*_{u}

*U** _{ρ}*T

*l,σ*u

*ε*−

*c*

*∂*^{2}_{x}_{i}_{x}_{i}*u*_{ε}

−*δ*
*d*
*i1*

*D*_{uu}^{2}

*U** _{ρ}*T

*l,σ*u

*ε*−

*c*

*∂*_{x}_{i}*u*_{ε}*∂*^{2}_{x}_{i}_{x}_{i}*u*_{ε}*.*

3.32

We rewrite the previous expression in the following manner:

*∂** _{t}*θT

*l*u

*ε*−

*cT*

*l*u

*ε*−

*c*div

_{x}*θT**l*u*ε*−*c*

*f*t, x, T*l*u*ε*−*ft, x, c*

Γ1,ε Γ2,ε Γ3,ε Γ4,ε Γ5,ε Γ6,ε Γ7,ε*,* 3.33