International Journal of Differential Equations

International Journal of Differential Equations / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 279818 | https://doi.org/10.1155/2009/279818

H. Holden, K. H. Karlsen, D. Mitrovic, "Zero Diffusion-Dispersion-Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function", International Journal of Differential Equations, vol. 2009, Article ID 279818, 33 pages, 2009. https://doi.org/10.1155/2009/279818

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

Academic Editor: Philippe G. LeFloch
Received02 Apr 2009
Revised24 Aug 2009
Accepted24 Sep 2009
Published03 Dec 2009

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

1. Introduction

We consider the convergence of smooth solutions 𝑢=𝑢𝜀(𝑡,𝑥) with (𝑡,𝑥)𝐑+×𝐑𝑑 of the nonlinear partial differential equation

𝜕𝑡𝑢+div𝑥𝑓𝜚(𝑡,𝑥,𝑢)=𝜀div𝑥𝑏(𝑢)+𝛿𝑑𝑗=1𝜕3𝑥𝑗𝑥𝑗𝑥𝑗𝑢(1.1) as 𝜀0 and 𝛿=𝛿(𝜀),𝜚=𝜚(𝜀)0. Here 𝑓𝐶(𝐑;𝐵𝑉(𝐑+𝑡×𝐑𝑑𝑥)) is the Caratheodory flux vector such that

max|𝑢|𝑙||𝑓𝜚||(𝑡,𝑥,𝑢)𝑓(𝑡,𝑥,𝑢)0,𝜚0,in𝐿𝑝loc𝐑+×𝐑𝑑,(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:

𝜕𝑡𝑢+div𝑥𝑓(𝑡,𝑥,𝑢)=0,𝑢=𝑢(𝑡,𝑥),𝑥𝐑𝑑,𝑡0.(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 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., [1417]). 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 Section 2 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 flux 𝑓 in (1.1) is smooth in all variables. Consider a sequence (𝑢𝜀,𝛿)𝜀,𝛿 of solutions of

𝜕𝑡𝑢+div𝑥𝑓(𝑡,𝑥,𝑢)=𝜀div𝑥𝑏(𝑢)+𝛿𝑑𝑗=1𝜕3𝑥𝑗𝑥𝑗𝑥𝑗𝑢,𝑢(𝑥,0)=𝑢0(𝑥),𝑥𝐑𝑑.(2.1)

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,𝛿𝐿([0,𝑇];𝐻4(𝐑𝑑)).

Later on, we will assume that the initial data 𝑢0 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 write 𝑢𝜀 instead of 𝑢𝜀,𝛿.

We will need the following assumptions on the diffusion term 𝑏(𝜆)=(𝑏1(𝜆),,𝑏𝑛(𝜆)).

(H1) For some positive constants 𝐶1,𝐶2 we have

𝐶1||𝜆||2𝜆𝑏(𝜆)𝐶2||𝜆||2𝜆𝐑𝑑.(2.2)

(H2) The gradient matrix 𝐷𝑏(𝜆) is a positive definite matrix, uniformly in 𝜆𝐑𝑑, that is, for every 𝜆,𝜚𝐑𝑑, there exists a positive constant 𝐶3 such that we have

𝜚𝑇𝐷𝑏(𝜆)𝜚𝐶3||𝜚||2.(2.3)

We use the following notation:

||𝐷2𝑢||2=𝑑𝑖,𝑘=1||𝜕2𝑥𝑖𝑥𝑘𝑢||2.(2.4) In the sequel, for a vector valued function 𝑔=(𝑔1,,𝑔𝑑) defined on 𝐑+×𝐑𝑑×𝐑, we denote

||𝑔||2=𝑑𝑖=1||𝑔𝑖||2.(2.5) The partial derivative 𝜕𝑥𝑖 in the point (𝑡,𝑥,𝑢), where 𝑢 possibly depends on (𝑡,𝑥), is defined by the formula

𝜕𝑥𝑖𝑔𝐷(𝑡,𝑥,𝑢(𝑡,𝑥))=𝑥𝑖𝑔(𝑡,𝑥,𝜆)𝜆=𝑢(𝑡,𝑥).(2.6) In particular, the total derivative 𝐷𝑥𝑖 and the partial derivative 𝜕𝑥𝑖 are connected by the identity

𝐷𝑥𝑖𝑔(𝑡,𝑥,𝑢)=𝜕𝑥𝑖𝑔(𝑡,𝑥,𝑢)+𝜕𝑢𝑔(𝑡,𝑥,𝑢)𝜕𝑥𝑖𝑢.(2.7) Finally we use

div𝑥𝑔(𝑡,𝑥,𝑢)=𝑑𝑖=1𝐷𝑥𝑖𝑔𝑖𝑔(𝑡,𝑥,𝑢),𝑔=1,,𝑔𝑑,Δ𝑥𝑞(𝑡,𝑥,𝑢)=𝑑𝑖=1𝐷2𝑥𝑖𝑥𝑖𝑞(𝑡,𝑥,𝑢),𝑞𝐶2𝐑+×𝐑𝑑.×𝐑(2.8)

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

(H3) The growth of the velocity variable 𝑢 and the spatial derivative of the flux 𝑓 are such that for some 𝐶,𝛼>0, 𝑝1, and every 𝑙>0, we have

max||𝜆||<𝑙||𝑓𝑖||(𝑡,𝑥,𝜆)𝐿𝑝𝐑+×𝐑𝑑,𝑖=1,,𝑑,𝑑𝑖=1||𝜕𝑢𝑓𝑖||(𝑡,𝑥,𝑢)𝐶,𝑑𝑖,𝑗=1||𝜕𝑥𝑖𝑓𝑗||(𝑡,𝑥,𝑢)𝜇(𝑡,𝑥)1+|𝑢|1+𝛼,(2.9) where 𝜇(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 function 𝑓=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: 𝐑𝑑||𝑢𝜀(||𝑡,𝑥)2𝑑𝑥+𝜀𝑡0𝐑𝑑||𝑢𝜀𝑡||,𝑥2𝑑𝑥𝑑𝑡𝐶4𝐑𝑑||𝑢0||(𝑥)2𝑑𝑥𝑡0𝐑𝑑𝑢𝜀𝑡0,𝑥div𝑥𝑓𝑡,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡,𝜀(2.10)2𝐑𝑑||||𝑢𝜀(𝑡,𝑥)|2𝑑𝑥+𝜀3𝑡0𝐑𝑑||||𝐷2𝑢𝜀𝑡|,𝑥2𝑑𝑥𝑑𝑡𝐶5𝜀2𝐑𝑑||𝑢0||(𝑥)2𝑑𝑥+𝜀𝑡0𝐑𝑑𝑑𝑘=1||𝜕𝑥𝑘𝑓𝑡,𝑥,𝑢𝜀𝑡||,𝑥2𝑑𝑥𝑑𝑡+𝜕𝑢𝑓2𝐿(𝐑+×𝐑𝑑×𝐑),(2.11) for some constants 𝐶4 and 𝐶5.

Proof. We follow the procedure from [19]. Given a smooth function 𝜂=𝜂(𝑢), 𝑢𝐑, we define 𝑞𝑖(𝑡,𝑥,𝑢)=𝑢0𝜂(𝑣)𝜕𝑣𝑓𝑖(𝑡,𝑥,𝑣)𝑑𝑣,𝑖=1,,𝑑.(2.12) If we multiply (2.1) by 𝜂(𝑢), it becomes 𝜕𝑡𝜂𝑢𝜀+𝑑𝑖=1𝜕𝑥𝑖𝑞𝑖𝑡,𝑥,𝑢𝜀𝑑𝑖=1𝑢𝜀0𝜕2𝑥𝑖𝑣𝑓𝑖(𝑡,𝑥,𝑣)𝜂(𝑣)𝑑𝑣+𝑑𝑖=1𝜂𝑢𝜀𝜕𝑥𝑖𝑓𝑖𝑡,𝑥,𝑢𝜀=𝜀𝑑𝑖=1𝜕𝑥𝑖𝜂𝑢𝜀𝑏𝑖𝑢𝜀𝜀𝜂𝑢𝜀𝑑𝑖=1𝑏𝑖𝑢𝜀𝜕𝑥𝑖𝑢𝜀+𝛿𝑑𝑖=1𝜕𝑥𝑖𝜂𝑢𝜀𝜕2𝑥𝑖𝑥𝑖𝑢𝜀𝛿2𝜂𝑢𝜀𝑑𝑖=1𝜕𝑥𝑖𝜕𝑥𝑖𝑢𝜀2.(2.13) Choosing here 𝜂(𝑢)=𝑢2/2 and integrating over [0,𝑡)×𝐑𝑑, we get 𝐑𝑑||𝑢𝜀(||𝑡,𝑥)2𝑑𝑥+𝜀𝑡0𝐑𝑑𝑢𝜀𝑡,𝑥𝑏𝑢𝜀𝑡,𝑥𝑑𝑥𝑑𝑡=𝐑𝑑||𝑢0||(𝑥)2𝑑𝑥+𝑑𝑗=1𝑡0𝐑𝑑𝑢𝜀𝑡0,𝑥𝑣𝐷2𝑥𝑗𝑣𝑓𝑗𝑡,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡𝑑𝑖=1𝑡0𝐑𝑑𝑢𝜀𝑡𝜕,𝑥𝑥𝑖𝑓𝑖𝑡,𝑥,𝑢𝜀𝑡,𝑥𝑑𝑥𝑑𝑡=𝐑𝑑||𝑢0||(𝑥)2𝑑𝑥𝑑𝑖=1𝑡0𝐑𝑑𝑢𝜀𝑡0,𝑥𝜕𝑥𝑖𝑓𝑖𝑡,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡,(2.14) where the second equality sign is justified by the following partial integration: 𝑡0𝐑𝑑𝑢𝜀0𝑣𝐷2𝑥𝑗𝑣𝑓𝑗𝑡,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡=𝑡0𝐑𝑑𝑢𝜀𝜕𝑥𝑖𝑓𝑖𝑡,𝑥,𝑢𝜀𝑑𝑥𝑑𝑡𝑡0𝐑𝑑𝑢𝜀0𝜕𝑥𝑖𝑓𝑖𝑡,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡.(2.15)
Now inequality (2.10) follows from (2.14), using (H1).
As for inequality (2.11), we start by using (2.14), namely, 𝐑𝑑||𝑢𝜀(||𝑡,𝑥)2𝑑𝑥+𝜀𝑡0𝐑𝑑𝑢𝜀𝑡,𝑥𝑏𝑢𝜀𝑡,𝑥𝑑𝑥𝑑𝑡=𝐑𝑑||𝑢0||(𝑥)2𝑑𝑥𝑑𝑖=1𝑡0𝐑𝑑𝑢𝜀𝑡0,𝑥𝜕𝑥𝑖𝑓𝑖𝑡,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡𝐑𝑑||𝑢0||(𝑥)2𝑑𝑥+𝑑𝑖=1𝑡0𝐑𝑑𝑢𝜀𝑡0,𝑥||𝜕𝑥𝑖𝑓𝑖𝑡||,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡𝐑𝑑||𝑢0||(𝑥)2𝑑𝑥+𝑡0𝐑𝑑𝐑𝜇𝑡,𝑥1+|𝑣|1+𝛼𝑑𝑣𝑑𝑥𝑑𝑡𝐑𝑑||𝑢0||(𝑥)2𝑑𝑥+𝐶𝑡0𝐑𝑑𝜇𝑡,𝑥𝑑𝑥𝑑𝑡,(2.16) where 𝐶=𝐑(𝑑𝑣/(1+|𝑣|1+𝛼)).
From here, using (H3), we conclude in particular that 𝜀𝑡0𝐑𝑑||𝑢𝜀𝑡||,𝑥2𝑑𝑥𝑑𝑡𝐶11,(2.17) 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: 12𝐑𝑑𝜕𝑡||𝑢𝜀||2𝑑𝑥𝑑𝑘=1𝐑𝑑𝜕𝑥𝑘𝑢𝜀𝜕𝑥𝑘𝑓𝑘𝑡,𝑥,𝑢𝜀+𝜕𝑢𝑓𝑘𝜕𝑥𝑘𝑢𝜀𝑑𝑥=𝜀𝑑𝑘=1𝐑𝑑𝜕𝑥𝑘𝑢𝜀𝑇𝐷𝑏𝑢𝜀𝜕𝑥𝑘𝑢𝜀𝑑𝑥.(2.18) Integrating this over [0,𝑡] and using the Cauchy-Schwarz inequality and condition (H2), we find 12𝐑𝑑||𝑢𝜀||(𝑡,)2𝑑𝑥+𝜀𝐶3𝑑𝑘=1𝑡0𝐑𝑑||𝜕𝑥𝑘𝑢𝜀||2𝑑𝑥𝑑𝑡12𝐑𝑑||𝑢0||2𝑑𝑥+𝑑𝑘=1𝜕𝑥𝑘𝑢𝜀𝐿2𝐑+×𝐑𝑑𝜕𝑥𝑘𝑓𝑘,,𝑢𝜀+𝜕𝑢𝑓𝑘𝜕𝑥𝑘𝑢𝜀𝐿2𝐑+×𝐑𝑑,(2.19) where 𝐶3 is independent of 𝜀. Then, using Young's inequality (the constant 𝐶3 is the same as previously mentioned) 𝐶𝑎𝑏3𝜀2𝑎2+12𝐶3𝜀𝑏2,𝑎,𝑏𝐑,(2.20) we obtain 12𝐑𝑑||𝑢𝜀||(𝑡,)2𝑑𝑥+𝜀𝐶3𝑑𝑘=1𝑡0𝐑𝑑||𝜕𝑥𝑘𝑢𝜀||2𝑑𝑥𝑑𝑡12𝐑𝑑||𝑢0||2𝑑𝑥+𝐶3𝜀2𝑑𝑘=1𝑡0𝐑𝑑||𝜕𝑥𝑘𝑢𝜀||2𝑑𝑥𝑑𝑡+12𝐶3𝜀𝑡0𝐑𝑑𝑑𝑘=1||𝜕𝑥𝑘𝑓𝑘𝑡,𝑥,𝑢𝜀+𝜕𝑢𝑓𝑘𝜕𝑥𝑘𝑢𝜀||2𝑑𝑥𝑑𝑡.(2.21) Multiplying this by 𝜀2, using (𝑎+𝑏)22𝑎2+2𝑏2, and applying (2.17), we conclude 𝜀22𝐑𝑑||𝑢𝜀(||𝑡,)2𝑑𝑥+𝐶3𝜀32𝐑𝑑𝑡0||𝐷2𝑢𝜀||2𝑑𝑥𝑑𝑡𝜀22𝐑𝑑||𝑢0||2𝑑𝑥𝑑𝑡+𝜀𝐶3𝑡0𝐑𝑑𝑑𝑘=1||𝜕𝑥𝑘𝑓𝑘𝑡,𝑥,𝑢𝜀𝑡||,𝑥2𝑑𝑥𝑑𝑡+𝐶11𝐶3𝜕𝑢𝑓𝑘2𝐿𝐑+×𝐑𝑑×𝐑.(2.22) 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. Find 𝑢=𝑢(𝑡,𝑥) such that

𝜕𝑡𝑢+div𝑥𝑢𝑓(𝑡,𝑥,𝑢)=0,(𝑥,0)=𝑢0(𝑥),𝑥𝐑𝑑,(3.1) where 𝑢0𝐿2(𝐑𝑑) is a given initial data.

For the flux 𝑓=(𝑓1,,𝑓𝑑) we need the following assumption, denoted (H4).

(H4a) For the flux 𝑓=𝑓(𝑡,𝑥,𝑢), (𝑡,𝑥,𝑢)𝐑+×𝐑𝑑×𝐑, we assume that 𝑓𝐶(𝐑;𝐵𝑉(𝐑+×𝐑𝑑)) and that for every 𝑙𝐑+ we have max𝑢[𝑙,𝑙]|𝑓(𝑡,𝑥,𝑢)|𝐿𝑝(𝐑+×𝐑𝑑), 𝑝>2 .

(H4b) There exists a sequence 𝑓𝜚=(𝑓1𝜚,,𝑓𝑑𝜚), 𝜚(0,1), such that 𝑓𝜚=𝑓𝜚(𝑡,𝑥,𝑢)𝐶1(𝐑+×𝐑𝑑×𝐑), satisfying for some 𝑝>2 and every 𝑙𝐑+:

max𝑧[𝑙,𝑙]||𝑓𝜚||(,,𝑧)𝑓(,,𝑧)𝜚00in𝐿𝑝𝐑+×𝐑𝑑=0,(3.2a)𝑑𝑖=1𝐑+×𝐑𝑑||𝜕𝑥𝑖𝑓𝑖𝜚(||𝐶𝑡,𝑥,𝑢)𝑑𝑥𝑑𝑡11+|𝑢|1+𝛼𝜚,(3.2b)𝑑𝑖=1,𝑘𝐑+×𝐑𝑑||𝜕𝑥𝑘𝑓𝑖𝜚||(𝑡,𝑥,𝑢)2𝐶𝑑𝑥𝑑𝑡2,(3.2c)𝑑𝑖=1||𝜕𝑢𝑓𝑖𝜚||𝐶(𝑡,𝑥,𝑢)𝛽(𝜚),(3.2d)𝑑𝑖=1𝐑+×𝐑𝑑||𝜕2𝑥𝑖𝑢𝑓𝑖𝜚||𝐶(𝑡,𝑥,𝑢)𝑑𝑥𝑑𝑡31+|𝑢|1+𝛼,(3.2e) where 𝐶𝑖, 𝑖=1,2,3, and 𝐶 are constants, while the function 𝛽𝐑𝐑 is such that lim𝜌0𝛽(𝜌)=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 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, nonnegative 𝜑1𝐶0(𝐑+×𝐑𝑑) and 𝜑2𝐶0(𝐑) with total mass one denote 𝜑𝜚1(𝑧,𝑢)=𝜚𝑑+1𝜑1𝑧𝜚1𝜑𝛽(𝜚)2𝑢𝛽(𝜚),(3.3)𝑧𝐑+×𝐑𝑑 and 𝑢𝐑, where 𝛽 is a positive function tending to zero as 𝜚0. In the case when the flux 𝑓𝐶(𝐑;𝐵𝑉(𝐑+×𝐑𝑑))𝐵𝑉(𝐑×𝐑+×𝐑𝑑)) is bounded, straightforward computation shows that the sequence 𝑓𝜚=𝑓𝜑𝜚=(𝑓1𝜚,,𝑓𝑑𝜚) satisfies (H4b) with 𝛽(𝜚)=𝜚.

We also need to assume that the flux 𝑓 is genuinely nonlinear, that is, for every (𝑡,𝑥)𝐑+×𝐑𝑑 and every 𝜉𝐑𝑑{0}, the mapping

𝐑𝜆𝑑𝑖=1𝑓𝑖𝜉(𝑡,𝑥,𝜆)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

𝜕𝑡𝑢+div𝑥𝑓𝜚(𝑡,𝑥,𝑢)=𝜀div𝑥𝑏(𝑢)+𝛿𝑑𝑗=1𝜕3𝑥𝑗𝑥𝑗𝑥𝑗𝑢,(3.5)𝑢(𝑥,0)=𝑢0,𝜀(𝑥),𝑥𝐑𝑑,(3.6) 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

𝑢0,𝜀𝑢0𝐿2𝐑𝑑𝑢0,0,𝜀𝐿2𝐑𝑑𝑢+𝜀0,𝜀𝐻1𝐑𝑑𝐶.(3.7) We also assume that 𝜚=𝜚(𝜀)0 and 𝛿=𝛿(𝜀)0 as 𝜀0. 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 𝜀0. 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 flux 𝑓(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]: 𝐑𝑑||𝑢𝜀(||𝑡,𝑥)2𝑑𝑥+𝜀𝑡0𝐑𝑑||𝑢𝜀(||𝑥,𝑠)2𝑑𝑥𝑑𝑠𝐶4𝐑𝑑||𝑢0,𝜀(||𝑥)2𝑑𝑥+𝐶10,𝜀(3.8)2𝐑𝑑||𝑢𝜀(||𝑡,𝑥)2𝑑𝑥+𝜀3𝑡0𝐑𝑑||𝐷2𝑢𝜀𝑡||,𝑥2𝑑𝑥𝑑𝑡𝐶5𝜀2𝐑𝑑||𝑢0,𝜀(||𝑥)2𝜀𝑑𝑥+𝜚𝐶11+𝐶12𝛽(𝜚)2,(3.9) for some constants 𝐶10,𝐶11,𝐶12 (the constants 𝐶4,𝐶5 are introduced in Theorem 2.1).

Proof. For every fixed 𝜚, 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 flux 𝑓 by 𝑓𝜚 from (3.5) and 𝑢0 by 𝑢0,𝜀 from (3.6) in (2.10) and (2.11), we get 𝐑𝑑||𝑢𝜀(||𝑡,𝑥)2𝑑𝑥+𝜀𝑡0𝐑𝑑||𝑢𝜀(||𝑥,𝑠)2𝑑𝑥𝑑𝑠𝐶3𝐑𝑑||𝑢0,𝜀||(𝑥)2𝑑𝑥𝑡0𝐑𝑑𝑢𝜀𝑡0,𝑥div𝑥𝑓𝜚𝑡,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡,𝜀(3.10)2𝐑𝑑||𝑢𝜀(||𝑡,𝑥)2𝑑𝑥+𝜀3𝑡0𝐑𝑑||𝐷2𝑢𝜀𝑡||,𝑥2𝑑𝑥𝑑𝑡𝐶4𝜀2𝐑𝑑||𝑢0,𝜀||(𝑥)2𝜕𝑑𝑥+𝑢𝑓𝜚2𝐿𝐑+×𝐑𝑑×𝐑+𝜀𝑡0𝐑𝑑𝑑𝑑𝑘=1𝑖=1𝜕𝑥𝑘𝑓𝑖𝜚𝑡,𝑥,𝑢𝜀𝑡,𝑥2𝑑𝑥𝑑𝑡.(3.11) To proceed, we use assumption (H4). We have 𝑡0𝐑𝑑𝑢𝜀𝑡0,𝑥div𝑓𝑖𝜚𝑡,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡𝑡0𝐑𝑑𝐑𝑑𝑖=1||𝜕𝑥𝑖𝑓𝑖𝜚𝑡||,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡𝐑𝐶11+|𝑣|1+𝛼𝑑𝑣𝐶10,(3.12) 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

𝑏𝜆1,,𝜆𝑑=𝜆1,,𝜆𝑑(3.13) 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 the 𝐻-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 vector 𝑓(𝑡,𝑥,𝑢) is genuinely nonlinear in the sense of (3.4). Then each family (𝑣𝜀(𝑡,𝑥))𝜀>0𝐿(𝐑+×𝐑𝑑) such that for every 𝑐𝐑 the distribution 𝜕𝑡𝜃𝑣𝜀𝑣𝑐𝜀𝑐+div𝑥𝜃𝑣𝜀𝑓𝑐𝑡,𝑥,𝑣𝜀𝑓(𝑡,𝑥,𝑐)(3.14) is precompact in 𝐻1loc contains a subsequence convergent in 𝐿1loc(𝐑+×𝐑𝑑).

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 𝜚=𝜀,𝛿=𝜀2𝜌2(𝜀)with𝜌(𝜀)=𝒪(𝛽(𝜀)),(3.15) 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 Theorem 3.3. 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 arbitrary 𝐶2 function 𝑆=𝑆(𝑢), 𝑢𝐑, and multiply the regularized equation (3.5) by 𝑆(𝑢𝜀). As usual, put 𝑞(𝑡,𝑥,𝑢)=𝑢0𝑆(𝑣)𝜕𝑢𝑓𝜚𝑞𝑑𝑣,𝑞=1,,𝑞𝑑.(3.16)
We easily find that 𝜕𝑡𝑆𝑢𝜀+div𝑥𝑞𝑡,𝑥,𝑢𝜀div𝑥𝑞(𝑡,𝑥,𝑣)𝑣=𝑢𝜀+𝑆𝑢𝜀div𝑥𝑓𝜚(𝑡,𝑥,𝑣)𝑣=𝑢𝜀=𝜀div𝑥𝑆𝑢𝜀𝑢𝜀𝜀𝑆𝑢𝜀||𝑢𝜀||2+𝛿𝑑𝑗=1𝐷𝑥𝑗𝑆𝑢𝜀𝜕2𝑥𝑗𝑥𝑗𝑢𝜀𝛿𝑑𝑗=1𝑆𝑢𝜀𝜕𝑥𝑗𝑢𝜀𝜕2𝑥𝑗𝑥𝑗𝑢𝜀.(3.17) We will apply this formula repeatedly with different choices for 𝑆(𝑢).
In order to apply Theorem 3.3, we will consider a truncated sequence (𝑇𝑙(𝑢𝜀))𝜀>0, where the truncation function 𝑇𝑙 is defined for every fixed 𝑙𝐍 as 𝑇𝑙(𝑢)=𝑙,𝑢𝑙,𝑢,𝑙𝑢𝑙,𝑙,𝑢𝑙.(3.18) We will prove that the sequence (𝑇𝑙(𝑢𝜀))𝜀>0 is precompact for every fixed 𝑙. Denote by 𝑢𝑙 a subsequential limit (in 𝐿1loc) 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 replace 𝑇𝑙 by a 𝐶2 regularization 𝑇𝑙,𝜎𝐑𝐑. We define 𝑇𝑙,𝜎𝐑𝐑 by 𝑇𝑙,𝜎(0)=0 and 𝑇𝑙,𝜎(𝑢)=1,|𝑢|<𝑙,𝑙|𝑢|+𝜎𝜎,𝑙<|𝑢|<𝑙+𝜎,0,|𝑢|>𝑙+𝜎.(3.19)
Next, we want to estimate 𝑇𝑙,𝜎(𝑢𝜀)𝑢𝜀𝐿2(𝐑+×𝐑𝑑). To accomplish this, we insert the functions 𝑇±𝑙,𝜎 for 𝑆 in (3.17) where 𝑇±𝑙,𝜎 are defined by 𝑇±𝑙,𝜎(0)=0 and 𝑇+𝑙,𝜎(𝑢)=1,𝑢<𝑙,𝑙+𝜎𝑢𝜎𝑇,𝑙<𝑢<𝑙+𝜎,0,𝑢>𝑙+𝜎,(3.20)𝑙,𝜎(𝑢)=1,𝑢>𝑙,𝑙+𝜎+𝑢𝜎,𝑙𝜎<𝑢<𝑙,0,𝑢<𝑙𝜎.(3.21) Notice that 𝑇±𝑙,𝜎(𝑢)||𝑇1,±𝑙,𝜎||𝜎(𝑢)|𝑢|+2,𝑇+𝑙,𝜎(𝑢)=𝑇𝑙,𝜎(𝑢)for𝑙𝑢𝑙.(3.22) By inserting 𝑆(𝑢)=𝑇+𝑙,𝜎(𝑢), 𝑞=𝑞+(𝑡,𝑥,𝑢)=𝑢0(𝑇+𝑙,𝜎)(𝑣)𝜕𝑢𝑓𝜚𝑑𝑣 in (3.17) and integrating over Π𝑡=[0,𝑡]×𝐑𝑑, we get 𝐑𝑑𝑇+𝑙,𝜎𝑢𝜀𝑑𝑥+𝐑𝑑𝑇+𝑙,𝜎𝑢0𝜀𝑑𝑥+𝜎Π𝑡𝑙<𝑢𝜀<𝑙+𝜎||𝑢𝜀||2=𝑑𝑥𝑑𝑡Π𝑡div𝑥𝑞+(𝑡,𝑥,𝑣)𝑣=𝑢𝜀𝑑𝑥𝑑𝑡+Π𝑡𝑇+𝑙,𝜎𝑢𝜀div𝑥𝑓𝜚(𝑡,𝑥,𝑣)𝑣=𝑢𝜀𝛿𝑑𝑥𝑑𝑡𝜎Π𝑡𝑙<𝑢𝜀𝑑<𝑙+𝜎𝑗=1𝜕𝑥𝑗𝑢𝜀𝜕2𝑥𝑗𝑥𝑗𝑢𝜀𝑑𝑥𝑑𝑡.(3.23) Similarly, for 𝑆(𝑢)=𝑇𝑙,𝜎(𝑢), 𝑞=𝑞(𝑡,𝑥,𝑢)=𝑢0(𝑇𝑙,𝜎)(𝑣)𝜕𝑢𝑓𝜚𝑑𝑣, we have from (3.17) 𝐑𝑑𝑇𝑙,𝜎𝑢𝜀𝑑𝑥𝐑𝑑𝑇𝑙,𝜎𝑢0𝜀𝑑𝑥+𝜎Π𝑡𝑙𝜎<𝑢𝜀<𝑙||𝑢𝜀||2=𝑑𝑥𝑑𝑡Π𝑡div𝑥𝑞(𝑡,𝑥,𝑣)𝑣=𝑢𝜀𝑑𝑥𝑑𝑡Π𝑡𝑇𝑙,𝜎𝑢𝜀div𝑥𝑓𝜚(𝑡,𝑥,𝑣)𝑣=𝑢𝜀+𝛿𝑑𝑥𝑑𝑡𝜎Π𝑡𝑙𝜎<𝑢𝜀𝑑<𝑙𝑗=1𝜕𝑥𝑗𝑢𝜀𝜕2𝑥𝑗𝑥𝑗𝑢𝜀𝑑𝑥𝑑𝑡.(3.24) Adding (3.23) to (3.24), we get 𝜀𝜎Π𝑡||𝑢𝑙<𝜀||<𝑙+𝜎||𝑢𝜀||2𝑑𝑥𝑑𝑡=𝐑𝑑𝑇𝑙,𝜎𝑢𝜀𝑇+𝑙,𝜎𝑢𝜀𝑑𝑥+𝐑𝑑𝑇𝑙,𝜎𝑢0𝑇+𝑙,𝜎𝑢0+𝑑𝑥Π𝑡div𝑥𝑞(𝑡,𝑥,𝑣)𝑣=𝑢𝜀𝑑𝑥𝑑𝑡+Π𝑡div𝑥𝑞+(𝑡,𝑥,𝑣)𝑣=𝑢𝜀𝑑𝑥𝑑𝑡Π𝑡𝑇𝑙,𝜎𝑢𝜀div𝑥𝑓𝜚(𝑡,𝑥,𝑣)𝑣=𝑢𝜀𝑑𝑥𝑑𝑡+Π𝑡𝑇+𝑙,𝜎𝑢𝜀div𝑥𝑓𝜚(𝑡,𝑥,𝑣)𝑣=𝑢𝜀+𝛿𝑑𝑥𝑑𝑡𝜎Π𝑡𝑙𝜎<𝑢𝜀𝑑<𝑙𝑗=1𝜕𝑥𝑗𝑢𝜀𝜕2𝑥𝑗𝑥𝑗𝑢𝜀𝛿𝑑𝑥𝑑𝑡𝜎Π𝑡𝑙<𝑢𝜀𝑑<𝑙+𝜎𝑗=1𝜕𝑥𝑗𝑢𝜀𝜕2𝑥𝑗𝑥𝑗𝑢𝜀𝑑𝑥𝑑𝑡.(3.25) From (3.22) and the definition of 𝑞 and 𝑞+, it follows 𝜀𝜎Π𝑡||𝑢𝑙<𝜀||<𝑙+𝜎||𝑢𝜀||2𝑑𝑥𝑑𝑡||𝑢𝜀||>𝑙2||𝑢𝜀||𝑑𝑥+||𝑢0||>𝑙2||𝑢0||𝑑𝑥+2Π𝑡𝐑𝑑𝑖=1||𝐷2𝑥𝑖𝑣𝑓𝑖𝜚(||𝑡,𝑥,𝑣)𝑑𝑣𝑑𝑥𝑑𝑡+2Π𝑡𝑑𝑖=1||𝜕𝑥𝑖𝑓𝑖𝜚𝑡,𝑥,𝑢𝜀||𝛿𝑑𝑥𝑑𝑡+2𝜎Π𝑡||𝑢𝑙𝜎<𝜀||𝑑<𝑙𝑗=1|||𝜕𝑥𝑗𝑢𝜀𝜕2𝑥𝑗𝑥𝑗𝑢𝜀|||𝑑𝑥𝑑𝑡.(3.26) Without loss of generality, we can assume that 𝑙>1. Having this in mind, we get from (H4) and (3.26) 𝜀𝜎Π𝑡||𝑢𝑙<𝜀||<𝑙+𝜎||𝑢𝜀||2𝑑𝑥𝑑𝑡||𝑢𝜀||>𝑙2||𝑢𝜀||2𝑑𝑥+||𝑢0||>𝑙2||𝑢0||2𝑑𝑥+2𝐑𝑑𝑖=1𝐶31+|𝑣|1+𝛼𝑑𝑣+2Π𝑡𝑑𝑖=1||𝜕𝑥𝑖𝑓𝑖𝜚𝑡,𝑥,𝑢𝜀||𝛿𝑑𝑥𝑑𝑡+2𝜎Π𝑡||𝑢𝑙<𝜀||𝑑<𝑙+𝜎𝑗=1|||𝜕𝑥𝑗𝑢𝜀𝜕2𝑥𝑗𝑥𝑗𝑢𝜀|||𝑑𝑥𝑑𝑡𝐑𝑑2||𝑢𝜀||(𝑥,𝑡)2+||𝑢0||(𝑥,𝑡)2𝑑𝑥+𝐾1+𝐾2𝛿+2𝜎𝜀2𝑑𝑖=1𝜀1/2𝜕𝑥𝑖𝑢𝜀𝐿2𝐑+×𝐑𝑑×𝜀3/2𝜕2𝑥𝑖𝑥𝑖𝑢𝜀𝐿2(𝐑+×𝐑𝑑)𝐾5+𝛿2𝜎2𝜀4(𝛽(𝜚))2+𝛿2𝜎2𝜀41/2𝐾3𝐾4,(3.27) where 𝐾𝑖, 𝑖=1,,5, are constants such that (cf. (3.8) and (3.9)) 2𝐑𝑑𝑖=1𝐶31+|𝑣|1+𝛼𝑑𝑣𝐾1,2Π𝑡𝑑𝑖=1||𝜕𝑥𝑖𝑓𝑖𝜚𝑡,𝑥,𝑢𝜀||𝑑𝑥𝑑𝑡𝐾2,𝑑𝑖=1𝜀1/2𝜕𝑥𝑖𝑢𝜀𝐿2𝐑+×𝐑𝑑𝐾3,𝑑𝑖=1𝜀3/2𝜕2𝑥𝑖𝑥𝑖𝑢𝜀𝐿2𝐑+×𝐑𝑑1(𝛽(𝜚))2+𝜀𝜚1/2𝐾4,𝐑𝑑2||𝑢𝜀(||𝑥,𝑡)2+||𝑢0(||𝑥,𝑡)2𝑑𝑥+𝐾1+𝐾2𝐾5.(3.28) 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 𝛿𝜎𝜀2𝜀1/2𝑢𝜀𝐿2𝐑+×𝐑𝑑𝑑𝑖=1𝜀3/2𝜕2𝑥𝑖𝑥𝑖𝑢𝜀𝐿2𝐑+×𝐑𝑑𝛿2𝜎2𝜀4𝛽2+𝛿(𝜀)2𝜎2𝜀41/2𝐾3𝐾4.(3.29) Thus, in view of (3.27), 𝜀𝜎Π𝑡||𝑢𝑙<𝜀||<𝑙+𝜎||𝑢𝜀||2𝑑𝑥𝑑𝑡𝐾5+𝛿2𝜎2𝜀4𝛽2+𝛿(𝜀)2𝜎2𝜀41/2𝐾3𝐾4,(3.30) which is the sought for estimate for 𝑇𝑙,𝜎(𝑢𝜀)𝑢𝜀𝐿2(𝐑+×𝐑𝑑).
Next, take a function 𝑈𝜌(𝑧) satisfying 𝑈𝜌(0)=0 and 𝑈𝜌𝑧(𝑧)=0,𝑧<0,𝜌,0<𝑧<𝜌,1,𝑧>𝜌.(3.31) Clearly, 𝑈𝜌 is convex, and 𝑈𝜌(𝑧)𝜃(𝑧) in 𝐿𝑝loc(𝐑) as 𝜌0, for any 𝑝<; as before, 𝜃 denotes the Heaviside function.
Inserting 𝑆(𝑢𝜀)=𝑈𝜌(𝑇𝑙,𝜎(𝑢𝜀)𝑐) in (3.17), we get 𝜕𝑡𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝑐+div𝑥𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙,𝜎(𝑣)𝜕𝑣𝑓𝜚=(𝑡,𝑥,𝑣)𝑑𝑣𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙,𝜎(𝑣)div𝑥𝜕𝑣𝑓𝜚(𝑡,𝑥,𝑣)𝑑𝑣𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝑇𝑐𝑙,𝜎𝑢𝜀div𝑥𝑓𝜚(𝑡,𝑥,𝑣)𝑣=𝑢𝜀+𝜀Δ𝑥𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝑐𝜀𝐷2𝑢𝑢𝑈𝜌𝑇𝑙,𝜎𝑢𝜀||𝑐𝑢𝜀||2+𝛿𝑑𝑖=1𝐷𝑥𝑖𝐷𝑢𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝜕𝑐2𝑥𝑖𝑥𝑖𝑢𝜀𝛿𝑑𝑖=1𝐷2𝑢𝑢𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝜕𝑐𝑥𝑖𝑢𝜀𝜕2𝑥𝑖𝑥𝑖𝑢𝜀.(3.32) We rewrite the previous expression in the following manner: 𝜕𝑡𝜃𝑇𝑙𝑢𝜀𝑇𝑐𝑙𝑢𝜀𝑐+div𝑥𝜃𝑇𝑙𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙𝑢𝜀𝑓(𝑡,𝑥,𝑐)=Γ1,𝜀+Γ2,𝜀+Γ3,𝜀+Γ4,𝜀+Γ5,𝜀+Γ6,𝜀+Γ7,𝜀,(3.33) where Γ1,𝜀=𝜕𝑡𝜃𝑇𝑙𝑢𝜀𝑇𝑐𝑙𝑢𝜀𝑐𝑈𝜌𝑇𝑙,𝜎𝑢𝜀,Γ𝑐2,𝜀=div𝑥𝜃𝑇𝑙𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙𝑢𝜀𝑓(𝑡,𝑥,𝑐)𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙,𝜎(𝑣)𝜕𝑣𝑓𝜚,Γ(𝑡,𝑥,𝑣)𝑑𝑣3,𝜀=𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙,𝜎(𝑣)div𝑥𝜕𝑣𝑓𝜚(𝑡,𝑥,𝑣)𝑑𝑣𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝑇𝑐𝑙,𝜎𝑢𝜀div𝑥𝑓𝜚(𝑡,𝑥,𝑣)𝑣=𝑢𝜀,Γ4,𝜀=𝜀Δ𝑥𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝑐+𝛿𝑑𝑖=1𝐷𝑥𝑖𝐷𝑢𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝜕𝑐2𝑥𝑖𝑥𝑖𝑢𝜀,Γ5,𝜀=𝜀𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝑇𝑐𝑙,𝜎||||(𝑢𝜀)𝑢𝜀2,Γ6,𝜀=𝛿𝑑𝑖=1𝐷2𝑢𝑢𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝜕𝑐𝑥𝑖𝑢𝜀𝜕2𝑥𝑖𝑥𝑖𝑢𝜀,Γ7,𝜀=𝜀𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝑇𝑐𝑙,𝜎𝑢𝜀2||𝑢𝜀||2.(3.34)
To continue, we assume that 𝜎 depends on 𝜀 in the following way: 𝜎=𝜌=𝒪(𝛽(𝜀)).(3.35)
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 𝐻1loc(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 𝑄𝜀𝐿𝑝loc(𝐑+×𝐑𝑑) for 𝑝>2.
(ii) The right-hand side of (3.33) is of the form loc,𝐵+𝐻1loc,𝑐, where loc,𝐵 denotes a set of families which are locally bounded in the space of measures, and 𝐻1loc,𝑐 is a set of families precompact in 𝐻1loc.
First, since 𝑇𝑙(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 Γ1,𝜀=𝜕𝑡𝜃𝑇𝑙𝑢𝜀𝑇𝑐𝑙𝑢𝜀𝑐𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝑐𝐻1loc,𝑐.(3.36) We have 𝜃𝑇𝑙𝑢𝜀𝑇𝑐𝑙𝑢𝜀𝑐𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝑇𝑐=𝜃𝑙𝑢𝜀𝑇𝑐𝑙𝑢𝜀𝑇𝑐𝜃𝑙,𝜎𝑢𝜀𝑇𝑐𝑙,𝜎𝑢𝜀𝑇𝑐+𝜃𝑙,𝜎𝑢𝜀𝑇𝑐𝑙,𝜎𝑢𝜀𝑐𝑈𝜌𝑇𝑙,𝜎𝑢𝜀.𝑐(3.37) Since the function 𝜃(𝑧𝑐)(𝑧𝑐) 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 ||𝜃𝑇𝑙𝑢𝜀𝑇𝑐𝑙𝑢𝜀𝑐𝑈𝜌𝑇𝑙,𝜎𝑢𝜀||||𝑇𝑐𝑙𝑢𝜀𝑇𝑙,𝜎𝑢𝜀||+𝒪(𝜌)𝒪(𝜎)+𝒪(𝜌).(3.38)
From this and assumptions (3.15) and (3.35) on 𝜎=𝜎(𝜀) and 𝜌=𝜌(𝜀), it follows that as 𝜀0𝜃𝑇𝑙𝑢𝜀𝑇𝑐𝑙𝑢𝜀𝑐𝑈𝜌𝑇𝑙,𝜎𝑢𝜀𝑐0(3.39) in 𝐿𝑝loc for all 𝑝<. Thus, (since we can take 𝑝=2 as well) we see that Γ1,𝜀𝐻1loc,𝑐.
Next, we will prove that Γ2,𝜀=div𝑥𝜃𝑇𝑙𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙𝑢𝜀𝑓(𝑡,𝑥,𝑐)𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙,𝜎(𝑣)𝜕𝑣𝑓𝜚(𝑡,𝑥,𝑣)𝑑𝑣𝐻1loc,𝑐+loc,𝐵.(3.40) Indeed, 𝜃𝑇𝑙𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙𝑢𝜀𝑓(𝑡,𝑥,𝑐)𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙,𝜎(𝑣)𝜕𝑣𝑓𝜚𝑇(𝑡,𝑥,𝑣)𝑑𝑣=𝜃𝑙𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙𝑢𝜀𝑇𝑓(𝑡,𝑥,𝑐)𝜃𝑙,𝜎𝑢𝜀𝑓𝑐𝜚𝑡,𝑥,𝑇𝑙,𝜎𝑢𝜀𝑓𝜚𝑇(𝑡,𝑥,𝑐)+𝜃𝑙,𝜎𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙,𝜎𝑢𝜀𝑓(𝑡,𝑥,𝑐)𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙(𝑣)𝜕𝑣𝑓𝜚(𝑡,𝑥,𝑣)𝑑𝑣𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙,𝜎(𝑣)𝑇𝑙𝜕(𝑣)𝑣𝑓𝜚(𝑡,𝑥,𝑣)𝑑𝑣.(3.41) Since 𝑇𝑙(𝑢)=𝑢 if |𝑢|𝑙 and 𝑇𝑙(𝑢)=0 if |𝑢|𝑙, 𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙(𝑣)𝜕𝑣𝑓𝜚(𝑡,𝑥,𝑣)𝑑𝑣=𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙(𝑣)𝜕𝑣𝑓𝜚𝑡,𝑥,𝑇𝑙(𝑣)𝑑𝑣,(3.42) from which we conclude 𝜃𝑇𝑙𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙𝑢𝜀𝑓(𝑡,𝑥,𝑐)𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙,𝜎(𝑣)𝜕𝑣𝑓𝜚𝑇(𝑡,𝑥,𝑣)𝑑𝑣=𝜃𝑙𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙𝑢𝜀𝑇𝑓(𝑡,𝑥,𝑐)𝜃𝑙,𝜎𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙,𝜎𝑢𝜀𝑇𝑓(𝑡,𝑥,𝑐)+𝜃𝑙,𝜎𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙,𝜎𝑢𝜀𝑓(𝑡,𝑥,𝑐)𝑢𝜀𝑈𝜌𝑇𝑙,𝜎(𝑇𝑣)𝑐𝑙(𝑣)𝜕𝑣𝑓𝜚𝑡,𝑥,𝑇𝑙(𝑣)𝑑𝑣𝑢𝜀𝑈𝜌𝑇𝑙,𝜎(𝑇𝑣)𝑐𝑙,𝜎(𝑣)𝑇𝑙(𝜕𝑣)𝑣𝑓𝜚(𝑇𝑡,𝑥,𝑣)𝑑𝑣=𝜃𝑙𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙𝑢𝜀𝑇𝑓(𝑡,𝑥,𝑐)𝜃𝑙,𝜎𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙,𝜎𝑢𝜀𝑇𝑓(𝑡,𝑥,𝑐)+𝜃𝑙,𝜎𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙,𝜎𝑢𝜀𝑓(𝑡,𝑥,𝑐)𝑢𝜀𝜃𝑇𝑙,𝜎𝐷(𝑣)𝑐𝑣𝑓𝜚𝑡,𝑥,𝑇𝑙(𝑣)𝑑𝑣𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙,𝜎(𝑣)𝑇𝑙𝜕(𝑣)𝑣𝑓𝜚(𝑡,𝑥,𝑣)𝑑𝑣𝑢𝜀𝑈𝜌𝑇𝑙,𝜎(𝑇𝑣)𝑐𝜃𝑙,𝜎(𝑇𝑣)𝑐𝑙(𝑣)𝜕𝑣𝑓𝜚𝑡,𝑥,𝑇𝑙(𝑣)𝑑𝑣=Γ12,𝜀+Γ22,𝜀+Γ32,𝜀,(3.43) with Γ12,𝜀𝑇=𝜃𝑙𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙𝑢𝜀𝑇𝑓(𝑡,𝑥,𝑐)𝜃𝑙,𝜎𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙,𝜎𝑢𝜀,Γ𝑓(𝑡,𝑥,𝑐)22,𝜀𝑇=𝜃𝑙,𝜎𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙,𝜎𝑢𝜀𝑓(𝑡,𝑥,𝑐)𝑢𝜀𝜃𝑇𝑙,𝜎𝐷(𝑣)𝑐𝑣𝑓𝜚𝑡,𝑥,𝑇𝑙Γ(𝑣)𝑑𝑣,32,𝜀=𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝑙,𝜎(𝑣)𝑇𝑙𝜕(𝑣)𝑣𝑓𝜚(𝑡,𝑥,𝑣)𝑑𝑣𝑢𝜀𝑈𝜌𝑇𝑙,𝜎𝑇(𝑣)𝑐𝜃𝑙,𝜎𝑇(𝑣)𝑐𝑙(𝑣)𝜕𝑣𝑓𝜚𝑡,𝑥,𝑇𝑙(𝑣)𝑑𝑣.(3.44)
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 𝑢𝐑. Therefore, we have pointwise on 𝐑+×𝐑𝑑: ||Γ12,𝜀||=||𝜃𝑇𝑙𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙𝑢𝜀𝑇𝑓(𝑡,𝑥,𝑐)𝜃𝑙,𝜎𝑢𝜀𝑓𝑐𝑡,𝑥,𝑇𝑙,𝜎𝑢𝜀||𝑓(𝑡,𝑥,𝑐)0a𝑠𝜎0.(3.45) Since max𝑢[𝑙,𝑙]𝑓(𝑡,𝑥,𝑢)𝐿𝑝(𝐑+×𝐑𝑑), 𝑝>2, Lebesgue's dominated convergence theorem yields |Γ12,𝜀|=𝑜𝜎,𝐿𝑝loc(1), where 𝐑+×𝐑𝑑|𝑜𝜎,𝐿𝑝(1)|𝑝𝑑𝑥𝑑𝑡0 as 𝜎0. Thus, we conclude div𝑥Γ12,𝜀𝐻1loc𝐑+×𝐑𝑑.(3.46)
We pass to Γ22,𝜀. 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)=