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

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=𝑢_𝜀(𝑡,𝑥) with (𝑡,𝑥)∈𝐑⁺×𝐑^𝑑 of the nonlinear partial differential equation

𝜕_𝑡𝑢+div_𝑥𝑓_𝜚(𝑡,𝑥,𝑢)=𝜀div_𝑥𝑏(∇𝑢)+𝛿^𝑑_𝑗=1𝜕^3𝑥_{𝑗𝑥𝑗𝑥𝑗}𝑢(1.1) as 𝜀→0 and 𝛿=𝛿(𝜀),𝜚=𝜚(𝜀)→0. Here 𝑓∈𝐶(𝐑;𝐵𝑉(𝐑^+𝑡×𝐑^𝑑𝑥)) is the Caratheodory flux vector such that

max_{|𝑢|≤𝑙}||𝑓_𝜚(𝑡,𝑥,𝑢)−𝑓(𝑡,𝑥,𝑢)||⟶0,𝜚⟶0,in𝐿^𝑝loc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:

𝜕_𝑡𝑢+div_𝑥𝑓(𝑡,𝑥,𝑢)=0,𝑢=𝑢(𝑡,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 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 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 (𝑢_𝜀,𝛿)_𝜀,𝛿 of solutions of

𝜕_𝑡𝑢+div_𝑥𝑓(𝑡,𝑥,𝑢)=𝜀div_𝑥𝑏(∇𝑢)+𝛿^𝑑_𝑗=1𝑢(𝑥,0)=𝑢₀𝜕^3𝑥_{𝑗𝑥𝑗𝑥𝑗}𝑢,(𝑥),𝑥∈𝐑^𝑑.(2.1)

We assume that (𝑢_𝜀,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 𝑢_𝜀,𝛿∈𝐿^∞([0,𝑇];𝐻⁴(𝐑^𝑑)).

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 𝑢_𝜀 instead of 𝑢_𝜀,𝛿.

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

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

𝐶₁||𝜆||²≤𝜆⋅𝑏(𝜆)≤𝐶₂||𝜆||²∀𝜆∈𝐑^𝑑.(2.2)

(H2) The gradient matrix 𝐷𝑏(𝜆) is a positive definite matrix, uniformly in 𝜆∈𝐑^𝑑, that is, for every 𝜆,𝜚∈𝐑^𝑑, there exists a positive constant 𝐶₃ such that we have

𝜚^𝑇𝐷𝑏(𝜆)𝜚≥𝐶₃||𝜚||².(2.3)

We use the following notation:

||𝐷²𝑢||²=^𝑑_𝑖_,𝑘=1||𝜕^2𝑥_𝑖𝑥𝑘𝑢||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_𝑖=1||𝑔_𝑖||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

𝜕_𝑥𝑖𝑔(𝑡,𝑥,𝑢(𝑡,𝑥))=𝐷_𝑥𝑖𝑔(𝑡,𝑥,𝜆)∣_{𝜆=𝑢(𝑡,𝑥)}.(2.6) In particular, the total derivative 𝐷_𝑥𝑖 and the partial derivative 𝜕_𝑥𝑖 are connected by the identity

𝐷_𝑥𝑖𝑔(𝑡,𝑥,𝑢)=𝜕_𝑥𝑖𝑔(𝑡,𝑥,𝑢)+𝜕_𝑢𝑔(𝑡,𝑥,𝑢)𝜕_𝑥𝑖𝑢.(2.7) Finally we use

div_𝑥𝑔(𝑡,𝑥,𝑢)=^𝑑_𝑖=1Δ_𝑥𝐷_𝑥𝑖𝑔_𝑖(𝑡,𝑥,𝑢),𝑔=𝑔₁,…,𝑔_𝑑,𝑞(𝑡,𝑥,𝑢)=^𝑑_𝑖=1𝐷^2𝑥_𝑖𝑥𝑖𝑞(𝑡,𝑥,𝑢),𝑞∈𝐶²𝐑⁺×𝐑^𝑑×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, 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_𝑖=1||𝜕_𝑢𝑓_𝑖(𝑡,𝑥,𝑢)||≤𝐶,^𝑑_𝑖_,𝑗=1||𝜕_𝑥𝑖𝑓_𝑗(𝑡,𝑥,𝑢)||≤𝜇(𝑡,𝑥)1+|𝑢|^1+𝛼,(2.9) where 𝜇∈ℳ(𝐑⁺×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=𝑓(𝑡,𝑥,𝑢) satisfies (H3) and that it is Lipschitz continuous on 𝐑⁺×𝐑^𝑑×𝐑. Assume also that initial data 𝑢₀ belongs to 𝐿²(𝐑^𝑑). Under conditions (H1)-(H2) the sequence of solutions (𝑢_𝜀)_𝜀>0 of (2.1) for every 𝑡∈[0,𝑇] satisfies the following inequalities: _𝐑𝑑||𝑢_𝜀(𝑡,𝑥)||²𝑑𝑥+𝜀^𝑡0_𝐑𝑑||∇𝑢_𝜀𝑡^,𝑥||²𝑑𝑥𝑑𝑡^≤𝐶₄_𝐑^𝑑||𝑢₀(𝑥)||²𝑑𝑥−^𝑡0_𝐑𝑑^{𝑢𝜀𝑡′0,𝑥}div_𝑥𝑓𝑡^,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡^𝜀,(2.10)₂_𝐑𝑑||||∇𝑢_𝜀(𝑡,𝑥)|²𝑑𝑥+𝜀³^𝑡0_𝐑𝑑||||𝐷₂𝑢_𝜀𝑡^,𝑥|²𝑑𝑥𝑑𝑡^≤𝐶₅𝜀²_𝐑𝑑||∇𝑢₀(𝑥)||²𝑑𝑥+𝜀^𝑡0_𝐑𝑑^𝑑_𝑘=1||𝜕_𝑥𝑘𝑓𝑡^,𝑥,𝑢_𝜀𝑡^,𝑥||²𝑑𝑥𝑑𝑡^+‖‖𝜕_𝑢𝑓‖‖^2𝐿_{∞(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), 𝑢∈𝐑, we define 𝑞_𝑖(𝑡,𝑥,𝑢)=^𝑢0𝜂^(𝑣)𝜕_𝑣𝑓_𝑖(𝑡,𝑥,𝑣)𝑑𝑣,𝑖=1,…,𝑑.(2.12) If we multiply (2.1) by 𝜂^(𝑢), it becomes 𝜕_𝑡𝜂𝑢_𝜀+^𝑑_𝑖=1𝜕_𝑥𝑖𝑞_𝑖𝑡,𝑥,𝑢_𝜀−^𝑑_𝑖=1^𝑢𝜀₀𝜕^2𝑥_𝑖𝑣𝑓_𝑖(𝑡,𝑥,𝑣)𝜂^(𝑣)𝑑𝑣+^𝑑_𝑖=1=𝜀_𝑑𝜂^𝑢_𝜀𝜕_𝑥𝑖𝑓_𝑖𝑡,𝑥,𝑢_𝜀_𝑖=1𝜕_𝑥𝑖𝜂^𝑢_𝜀𝑏_𝑖∇𝑢_𝜀−𝜀𝜂^𝑢_𝜀^𝑑_𝑖=1𝑏_𝑖∇𝑢_𝜀𝜕_𝑥𝑖𝑢_𝜀+𝛿^𝑑_𝑖=1−𝛿𝜕_𝑥𝑖𝜂^𝑢_𝜀𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀2𝜂^𝑢_𝜀^𝑑_𝑖=1𝜕_𝑥𝑖𝜕_𝑥𝑖𝑢_𝜀².(2.13) Choosing here 𝜂(𝑢)=𝑢²/2 and integrating over [0,𝑡)×𝐑^𝑑, we get _𝐑𝑑||𝑢_𝜀(𝑡,𝑥)||²𝑑𝑥+𝜀^𝑡0_𝐑𝑑∇𝑢_𝜀𝑡^,𝑥⋅𝑏∇𝑢_𝜀𝑡^,𝑥𝑑𝑥𝑑𝑡^=_𝐑𝑑||𝑢₀(𝑥)||²𝑑𝑥+^𝑑_𝑗=1^𝑡0_𝐑𝑑^{𝑢𝜀𝑡′0,𝑥}𝑣𝐷^2𝑥_𝑗𝑣𝑓_𝑗𝑡^,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡^−^𝑑_𝑖=1^𝑡0_𝐑𝑑𝑢_𝜀𝑡^,𝑥𝜕_𝑥𝑖𝑓_𝑖𝑡^,𝑥,𝑢_𝜀𝑡^,𝑥𝑑𝑥𝑑𝑡^=_𝐑𝑑||𝑢₀(𝑥)||²𝑑𝑥−^𝑑_𝑖=1^𝑡0_𝐑𝑑^{𝑢𝜀𝑡′0,𝑥}𝜕_𝑥𝑖𝑓_𝑖𝑡^,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡^,(2.14) where the second equality sign is justified by the following partial integration: ^𝑡0_𝐑𝑑^𝑢𝜀₀𝑣𝐷^2𝑥_𝑗𝑣𝑓_𝑗𝑡^,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡^=^𝑡0_𝐑𝑑𝑢_𝜀𝜕_𝑥𝑖𝑓_𝑖𝑡^,𝑥,𝑢_𝜀𝑑𝑥𝑑𝑡^−^𝑡0_𝐑𝑑^𝑢𝜀₀𝜕_𝑥𝑖𝑓_𝑖𝑡^,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡^.(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||𝑢_𝜀(𝑡,𝑥)||²𝑑𝑥+𝜀^𝑡0_𝐑𝑑∇𝑢_𝜀𝑡^,𝑥⋅𝑏∇𝑢_𝜀𝑡^,𝑥𝑑𝑥𝑑𝑡^=_𝐑𝑑||𝑢₀(𝑥)||²𝑑𝑥−^𝑑_𝑖=1^𝑡0_𝐑𝑑^{𝑢𝜀𝑡′0,𝑥}𝜕_𝑥𝑖𝑓_𝑖𝑡^,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡^≤_𝐑𝑑||𝑢₀(𝑥)||²𝑑𝑥+^𝑑_𝑖=1^𝑡0_𝐑𝑑^{𝑢𝜀𝑡′0,𝑥}||𝜕_𝑥𝑖𝑓_𝑖𝑡^,𝑥,𝑣||𝑑𝑣𝑑𝑥𝑑𝑡^≤_𝐑𝑑||𝑢₀(𝑥)||²𝑑𝑥+^𝑡0_𝐑𝑑_𝐑𝜇𝑡^,𝑥1+|𝑣|^1+𝛼𝑑𝑣𝑑𝑥𝑑𝑡^≤_𝐑𝑑||𝑢₀(𝑥)||²𝑑𝑥+𝐶^𝑡0_𝐑𝑑𝜇𝑡^,𝑥𝑑𝑥𝑑𝑡^,(2.16) where 𝐶=∫_𝐑(𝑑𝑣/(1+|𝑣|^1+𝛼)).
From here, using (H3), we conclude in particular that 𝜀^𝑡0_𝐑𝑑||∇𝑢_𝜀𝑡^,𝑥||²𝑑𝑥𝑑𝑡^≤𝐶₁₁,(2.17) for some constant 𝐶₁₁ independent of 𝜀.
Next, we differentiate (2.1) with respect to 𝑥_𝑘 and multiply the expression by 𝜕_𝑥𝑘𝑢. Integrating over 𝐑^𝑑, using integration by parts and then summing over 𝑘=1,…,𝑑, we get: 12_𝐑^𝑑𝜕_𝑡||∇𝑢_𝜀||²𝑑𝑥−^𝑑_𝑘=1_𝐑𝑑∇𝜕_𝑥𝑘=−𝜀_𝑑𝑢_𝜀⋅𝜕_𝑥𝑘𝑓_𝑘𝑡,𝑥,𝑢_𝜀+𝜕_𝑢𝑓_𝑘𝜕_𝑥𝑘𝑢_𝜀𝑑𝑥_𝑘=1_𝐑𝑑∇𝜕_𝑥𝑘𝑢_𝜀^𝑇𝐷𝑏∇𝑢_𝜀∇𝜕_𝑥𝑘𝑢_𝜀𝑑𝑥.(2.18) Integrating this over [0,𝑡] and using the Cauchy-Schwarz inequality and condition (H2), we find 12_𝐑^𝑑||∇𝑢_𝜀(𝑡,⋅)||²𝑑𝑥+𝜀𝐶_3𝑑_𝑘=1^𝑡0_𝐑𝑑||∇𝜕_𝑥𝑘𝑢_𝜀||²𝑑𝑥𝑑𝑡^≤12_𝐑^𝑑||∇𝑢₀||²𝑑𝑥+^𝑑_𝑘=1‖‖∇𝜕_𝑥𝑘𝑢_𝜀‖‖_𝐿²_𝐑⁺_×𝐑^𝑑_‖‖𝜕_𝑥𝑘𝑓_𝑘⋅,⋅,𝑢_𝜀+𝜕_𝑢𝑓_𝑘𝜕_𝑥𝑘𝑢_𝜀‖‖_{𝐿2𝐑+×𝐑𝑑},(2.19) where 𝐶₃ is independent of 𝜀. Then, using Young's inequality (the constant 𝐶₃ is the same as previously mentioned) 𝑎𝑏≤𝐶³𝜀2𝑎²+12𝐶₃𝜀𝑏²,𝑎,𝑏∈𝐑,(2.20) we obtain 12_𝐑^𝑑||∇𝑢_𝜀(𝑡,⋅)||²𝑑𝑥+𝜀𝐶_3𝑑_𝑘=1^𝑡0_𝐑𝑑||∇𝜕_𝑥𝑘𝑢_𝜀||²𝑑𝑥𝑑𝑡^≤12_𝐑^𝑑||∇𝑢₀||²𝑑𝑥+𝐶₃𝜀2^𝑑_𝑘=1^𝑡0_𝐑𝑑||∇𝜕_𝑥𝑘𝑢_𝜀||²𝑑𝑥𝑑𝑡^+12𝐶₃𝜀^𝑡0_𝐑𝑑^𝑑_𝑘=1||𝜕_𝑥𝑘𝑓_𝑘𝑡^,𝑥,𝑢_𝜀+𝜕_𝑢𝑓_𝑘𝜕_𝑥𝑘𝑢_𝜀||²𝑑𝑥𝑑𝑡^.(2.21) Multiplying this by 𝜀², using (𝑎+𝑏)²≤2𝑎²+2𝑏², and applying (2.17), we conclude 𝜀²2_𝐑^𝑑||∇𝑢_𝜀(𝑡,⋅)||²𝑑𝑥+𝐶₃𝜀³2_𝐑^𝑑^𝑡0||𝐷²𝑢_𝜀||²𝑑𝑥𝑑𝑡^≤𝜀²2_𝐑^𝑑||∇𝑢₀||²𝑑𝑥𝑑𝑡^+𝜀𝐶₃^𝑡0_𝐑𝑑^𝑑_𝑘=1||𝜕_𝑥𝑘𝑓_𝑘𝑡^,𝑥,𝑢_𝜀𝑡^,𝑥||²𝑑𝑥𝑑𝑡^+𝐶¹¹𝐶₃‖‖𝜕_𝑢𝑓_𝑘‖‖^2𝐿_{∞𝐑+×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

𝑢𝜕(𝑥_𝑡𝑢,0+)d=iv𝑢_𝑥0𝑓(𝑡,𝑥,𝑢)=0,(𝑥),𝑥∈𝐑^𝑑,(3.1) where 𝑢₀∈𝐿²(𝐑^𝑑) is a given initial data.

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

(H4a) For the flux 𝑓=𝑓(𝑡,𝑥,𝑢), (𝑡,𝑥,𝑢)∈𝐑⁺×𝐑^𝑑×𝐑, we assume that 𝑓∈𝐶(𝐑;𝐵𝑉(𝐑⁺×𝐑^𝑑)) and that for every 𝑙∈𝐑⁺ we have max_{𝑢∈[−𝑙,𝑙]}|𝑓(𝑡,𝑥,𝑢)|∈𝐿^𝑝(𝐑⁺×𝐑^𝑑), 𝑝>2 .

(H4b) There exists a sequence 𝑓_𝜚=(𝑓_1𝜚,…,𝑓_𝑑𝜚), 𝜚∈(0,1), such that 𝑓_𝜚=𝑓_𝜚(𝑡,𝑥,𝑢)∈𝐶¹(𝐑⁺×𝐑^𝑑×𝐑), satisfying for some 𝑝>2 and every 𝑙∈𝐑⁺:

max_{𝑧∈[−𝑙,𝑙]}||𝑓_𝜚(⋅,⋅,𝑧)−𝑓(⋅,⋅,𝑧)||⟶_𝜚→00in𝐿^𝑝𝐑⁺×𝐑^𝑑=0,(3.2a)_𝑑_𝑖=1_{𝐑+×𝐑𝑑}||𝜕_𝑥𝑖𝑓_𝑖𝜚(𝑡,𝑥,𝑢)||𝑑𝑥𝑑𝑡≤𝐶¹1+|𝑢|^1+𝛼𝜚,(3.2b)_𝑑𝑖_=1,𝑘_{𝐑+×𝐑𝑑}||𝜕_𝑥𝑘𝑓_𝑖𝜚(𝑡,𝑥,𝑢)||²𝑑𝑥𝑑𝑡≤𝐶₂,(3.2c)_𝑑_𝑖=1||𝜕_𝑢𝑓_𝑖𝜚(𝑡,𝑥,𝑢)||≤𝐶𝛽(𝜚),(3.2d)_𝑑_𝑖=1_{𝐑+×𝐑𝑑}||𝜕^2𝑥_𝑖𝑢𝑓_𝑖𝜚(𝑡,𝑥,𝑢)||𝑑𝑥𝑑𝑡≤𝐶³1+|𝑢|^1+𝛼,(3.2e) where 𝐶_𝑖, 𝑖=1,2,3, and 𝐶 are constants, while the function 𝛽∶𝐑→𝐑 is such that lim_𝜌→0𝛽(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 𝜑₁∈𝐶^∞0(𝐑⁺×𝐑^𝑑) and 𝜑₂∈𝐶^∞0(𝐑) with total mass one denote 𝜑_𝜚(𝑧,𝑢)=1𝜚^𝑑+1𝜑₁𝜚𝑧𝛽1(𝜚)𝜑²𝛽𝑢(𝜚),(3.3)𝑧∈𝐑⁺×𝐑^𝑑 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⁺×𝐑^𝑑))∩𝐵𝑉(𝐑×𝐑⁺×𝐑^𝑑)) is bounded, straightforward computation shows that the sequence 𝑓_𝜚=𝑓⋆𝜑_𝜚=(𝑓_1𝜚,…,𝑓_𝑑𝜚) 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⁺×𝐑^𝑑 and every 𝜉∈𝐑^𝑑⧵{0}, the mapping

𝐑∋𝜆⟼^𝑑_𝑖=1𝑓_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

𝜕_𝑡𝑢+div_𝑥𝑓_𝜚(𝑡,𝑥,𝑢)=𝜀div_𝑥𝑏(∇𝑢)+𝛿^𝑑_𝑗=1𝑢(𝑥,0)=𝑢_0,𝜀𝜕^3𝑥_{𝑗𝑥𝑗𝑥𝑗}𝑢,(3.5)(𝑥),𝑥∈𝐑^𝑑,(3.6) where the flux 𝑓_𝜚 satisfies the conditions (H4b). We denote the solution of (3.5)-(3.6) by 𝑢_𝜀=𝑢_𝜀(𝑡,𝑥). We assume that

‖‖𝑢_0,𝜀−𝑢₀‖‖_{𝐿2𝐑𝑑}⟶0,‖‖𝑢_0,𝜀‖‖_{𝐿2𝐑𝑑}+𝜀‖‖𝑢_0,𝜀‖‖_{𝐻1𝐑𝑑}≤𝐶.(3.7) 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 (𝑢_𝜀)_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 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 (𝑢_𝜀)_𝜀>0 of (3.5)-(3.6) satisfies the following inequalities for every 𝑡∈[0,𝑇]: _𝐑𝑑||𝑢_𝜀(𝑡,𝑥)||²𝑑𝑥+𝜀^𝑡0_𝐑𝑑||∇𝑢_𝜀(𝑥,𝑠)||²𝑑𝑥𝑑𝑠≤𝐶₄_𝐑𝑑||𝑢_0,𝜀(𝑥)||²𝑑𝑥+𝐶₁₀𝜀,(3.8)₂_𝐑𝑑||∇𝑢_𝜀(𝑡,𝑥)||²𝑑𝑥+𝜀³^𝑡0_𝐑𝑑||𝐷²𝑢_𝜀𝑡^,𝑥||²𝑑𝑥𝑑𝑡^≤𝐶₅𝜀²_𝐑𝑑||∇𝑢_0,𝜀(𝑥)||²𝑑𝑥+𝜀𝜚𝐶¹¹+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 𝑓_𝜚=(𝑓_1𝜚,…,𝑓_𝑑𝜚) is smooth, and, due to (H4), we see that 𝑓_𝜚 satisfies (H3). This means that we can apply Theorem 2.1.
Replacing the flux 𝑓 by 𝑓_𝜚 from (3.5) and 𝑢₀ by 𝑢_0,𝜀 from (3.6) in (2.10) and (2.11), we get _𝐑𝑑||𝑢_𝜀(𝑡,𝑥)||²𝑑𝑥+𝜀^𝑡0_𝐑𝑑≤𝐶₃||∇𝑢_𝜀(𝑥,𝑠)||²𝑑𝑥𝑑𝑠_𝐑^𝑑||𝑢_0,𝜀(𝑥)||²𝑑𝑥−^𝑡0_𝐑𝑑^{𝑢𝜀𝑡′0,𝑥}div_𝑥𝑓_𝜚𝑡^,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡^𝜀,(3.10)₂_𝐑𝑑||∇𝑢_𝜀(𝑡,𝑥)||²𝑑𝑥+𝜀³^𝑡0_𝐑𝑑||𝐷²𝑢_𝜀𝑡^,𝑥||²𝑑𝑥𝑑𝑡^≤𝐶₄𝜀²_𝐑𝑑||∇𝑢_0,𝜀(𝑥)||²𝑑𝑥+‖‖𝜕_𝑢𝑓_𝜚‖‖^2𝐿_{∞𝐑+×𝐑𝑑×𝐑}+𝜀^𝑡0_𝐑𝑑^𝑑_𝑑𝑘=1_𝑖=1𝜕_𝑥𝑘𝑓_𝑖𝜚𝑡^,𝑥,𝑢_𝜀𝑡^,𝑥²𝑑𝑥𝑑𝑡^.(3.11) To proceed, we use assumption (H4). We have ^𝑡0_𝐑𝑑^{𝑢𝜀𝑡′0,𝑥}div𝑓_𝑖𝜚𝑡^,𝑥,𝑣𝑑𝑣𝑑𝑥𝑑𝑡^≤^𝑡0_𝐑𝑑_{𝐑𝑑𝑖=1}≤||𝜕_𝑥𝑖𝑓_𝑖𝜚𝑡^,𝑥,𝑣||𝑑𝑣𝑑𝑥𝑑𝑡_𝐑𝐶₁1+|𝑣|^1+𝛼𝑑𝑣≤𝐶₁₀,(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 (𝑢_𝜀)_𝜀>0 of solutions to (3.5)-(3.6) in the case when

𝑏𝜆₁,…,𝜆_𝑑=𝜆₁,…,𝜆_𝑑(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 (𝑣_𝜀(𝑡,𝑥))_𝜀>0⊂𝐿^∞(𝐑⁺×𝐑^𝑑) such that for every 𝑐∈𝐑 the distribution 𝜕_𝑡𝜃𝑣_𝜀−𝑐𝑣_𝜀−𝑐+div_𝑥𝜃𝑣_𝜀−𝑐𝑓𝑡,𝑥,𝑣_𝜀−𝑓(𝑡,𝑥,𝑐)(3.14) is precompact in 𝐻^−1loc contains a subsequence convergent in 𝐿^1loc(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 𝜚=𝜀,𝛿=𝜀²𝜌²(𝜀)with𝜌(𝜀)=𝒪(𝛽(𝜀)),(3.15) and that 𝑢_0,𝜀 satisfies (3.7). Then, there exists a subsequence of the family (𝑢_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 family (𝑢_𝜀)_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), 𝑢∈𝐑, and multiply the regularized equation (3.5) by 𝑆^(𝑢_𝜀). As usual, put 𝑞(𝑡,𝑥,𝑢)=^𝑢0𝑆^(𝑣)𝜕_𝑢𝑓_𝜚𝑑𝑣,𝑞=𝑞₁,…,𝑞_𝑑.(3.16)
We easily find that 𝜕_𝑡𝑆𝑢_𝜀+div_𝑥𝑞𝑡,𝑥,𝑢_𝜀−div_𝑥𝑞(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}+𝑆^𝑢_𝜀div_𝑥𝑓_𝜚(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}=𝜀div_𝑥𝑆^𝑢_𝜀∇𝑢_𝜀−𝜀𝑆^𝑢_𝜀||∇𝑢_𝜀||²+𝛿^𝑑_𝑗=1𝐷_𝑥𝑗𝑆^𝑢_𝜀𝜕^2𝑥_𝑗𝑥𝑗𝑢_𝜀−𝛿^𝑑_𝑗=1𝑆^𝑢_𝜀𝜕_𝑥𝑗𝑢_𝜀𝜕^2𝑥_𝑗𝑥𝑗𝑢_𝜀.(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 (𝑇_𝑙(𝑢_𝜀))_𝜀>0, where the truncation function 𝑇_𝑙 is defined for every fixed 𝑙∈𝐍 as 𝑇_𝑙(𝑢)=⎧⎪⎨⎪⎩−𝑢𝑙,,𝑙,𝑢−𝑢𝑙≤≥≤𝑙−.𝑢𝑙≤,𝑙,(3.18) We will prove that the sequence (𝑇_𝑙(𝑢_𝜀))_𝜀>0 is precompact for every fixed 𝑙. Denote by 𝑢_𝑙 a subsequential limit (in 𝐿^1loc) of the family (𝑇_𝑙(𝑢_𝜀))_𝜀>0, which gives raise to a new sequence (𝑢_𝑙)_𝑙>1 that we prove converges to a weak solution of (3.1).
To carry out this plan, we must replace 𝑇_𝑙 by a 𝐶² regularization 𝑇_𝑙,𝜎∶𝐑→𝐑. We define 𝑇_𝑙,𝜎∶𝐑→𝐑 by 𝑇_𝑙,𝜎(0)=0 and 𝑇^𝑙,𝜎(𝑢)=⎧⎪⎨⎪⎩1𝑙,−|𝑢|+𝜎|𝑢|<𝑙,0,𝜎,𝑙|𝑢<|>|𝑢𝑙|+<𝜎𝑙.+𝜎,(3.19)
Next, we want to estimate ‖𝑇^{𝑙,𝜎}(𝑢_𝜀)∇𝑢_𝜀‖_{𝐿2(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)=⎧⎪⎨⎪⎩1𝑙,+𝜎−𝑢𝑢<𝑙,𝑇0,𝜎,𝑙𝑢<>𝑢𝑙+<𝜎𝑙,+𝜎,(3.20)_−𝑙,𝜎^(𝑢)=⎧⎪⎨⎪⎩1𝑙,+𝜎+𝑢𝑢>−𝑙,0,𝜎,−𝑢𝑙<−−𝜎𝑙−<𝜎𝑢.<−𝑙,(3.21) Notice that 𝑇𝑇^±𝑙,𝜎_+𝑙,𝜎(𝑢)^≤1,||𝑇^±𝑙,𝜎(𝑢)||≤|𝑢|+𝜎2,(𝑢)=𝑇^−𝑙,𝜎(𝑢)for−𝑙≤𝑢≤𝑙.(3.22) By inserting 𝑆(𝑢)=−𝑇^+𝑙,𝜎(𝑢), 𝑞=𝑞₊(𝑡,𝑥,𝑢)=−∫^𝑢0(𝑇^+𝑙,𝜎)^(𝑣)𝜕_𝑢𝑓_𝜚𝑑𝑣 in (3.17) and integrating over Π_𝑡=[0,𝑡]×𝐑^𝑑, we get −_𝐑𝑑𝑇^+𝑙,𝜎𝑢_𝜀𝑑𝑥+_𝐑𝑑𝑇^+𝑙,𝜎𝑢₀𝑑𝑥+𝜀𝜎_{Π𝑡∩𝑙<𝑢𝜀<𝑙+𝜎}=||∇𝑢_𝜀||²𝑑𝑥𝑑𝑡_Π𝑡−𝛿div_𝑥𝑞₊(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}𝑑𝑥𝑑𝑡+_Π𝑡𝑇^+𝑙,𝜎^𝑢_𝜀div_𝑥𝑓_𝜚(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}𝑑𝑥𝑑𝑡𝜎_{Π𝑡∩𝑙<𝑢𝜀𝑑<𝑙+𝜎}_𝑗=1𝜕_𝑥𝑗𝑢_𝜀𝜕^2𝑥_𝑗𝑥𝑗𝑢_𝜀𝑑𝑥𝑑𝑡.(3.23) Similarly, for 𝑆(𝑢)=𝑇^−𝑙,𝜎(𝑢), 𝑞=𝑞₋(𝑡,𝑥,𝑢)=∫^𝑢0(𝑇^−𝑙,𝜎)^(𝑣)𝜕_𝑢𝑓_𝜚𝑑𝑣, we have from (3.17) _𝐑𝑑𝑇^−𝑙,𝜎𝑢_𝜀𝑑𝑥−_𝐑𝑑𝑇^−𝑙,𝜎𝑢₀𝑑𝑥+𝜀𝜎_{Π𝑡∩−𝑙−𝜎<𝑢𝜀<−𝑙}=||∇𝑢_𝜀||²𝑑𝑥𝑑𝑡_Π𝑡+𝛿div_𝑥𝑞₋(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}𝑑𝑥𝑑𝑡−_Π𝑡𝑇^−𝑙,𝜎^𝑢_𝜀div_𝑥𝑓_𝜚(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}𝑑𝑥𝑑𝑡𝜎_{Π𝑡∩−𝑙−𝜎<𝑢𝜀𝑑<−𝑙}_𝑗=1𝜕_𝑥𝑗𝑢_𝜀𝜕^2𝑥_𝑗𝑥𝑗𝑢_𝜀𝑑𝑥𝑑𝑡.(3.24) Adding (3.23) to (3.24), we get 𝜀𝜎_{Π𝑡∩||𝑢𝑙<𝜀||<𝑙+𝜎}=−_𝐑𝑑||∇𝑢_𝜀||²𝑑𝑥𝑑𝑡+𝑇^−𝑙,𝜎𝑢_𝜀−𝑇^+𝑙,𝜎𝑢_𝜀𝑑𝑥+_𝐑𝑑𝑇^−𝑙,𝜎𝑢₀−𝑇^+𝑙,𝜎𝑢₀𝑑𝑥_Π𝑡−div_𝑥𝑞₋(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}𝑑𝑥𝑑𝑡+_Π𝑡div_𝑥𝑞₊(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}𝑑𝑥𝑑𝑡_Π𝑡+𝛿𝑇^−𝑙,𝜎^𝑢_𝜀div_𝑥𝑓_𝜚(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}𝑑𝑥𝑑𝑡+_Π𝑡𝑇^+𝑙,𝜎^𝑢_𝜀div_𝑥𝑓_𝜚(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}𝑑𝑥𝑑𝑡𝜎_{Π𝑡∩−𝑙−𝜎<𝑢𝜀𝑑<−𝑙}_𝑗=1𝜕_𝑥𝑗𝑢_𝜀𝜕^2𝑥_𝑗𝑥𝑗𝑢_𝜀𝑑𝑥𝑑𝑡−𝛿𝜎_{Π𝑡∩𝑙<𝑢𝜀𝑑<𝑙+𝜎}_𝑗=1𝜕_𝑥𝑗𝑢_𝜀𝜕^2𝑥_𝑗𝑥𝑗𝑢_𝜀𝑑𝑥𝑑𝑡.(3.25) From (3.22) and the definition of 𝑞₋ and 𝑞₊, it follows 𝜀𝜎_{Π𝑡∩||𝑢𝑙<𝜀||<𝑙+𝜎}||∇𝑢_𝜀||²𝑑𝑥𝑑𝑡≤+2_{||𝑢𝜀||>𝑙}2_Π𝑡||𝑢_𝜀||𝑑𝑥+_{||𝑢0||>𝑙}2||𝑢₀||𝑑𝑥_{𝐑𝑑𝑖=1}+2_Π𝑡𝑑||𝐷^2𝑥_𝑖𝑣𝑓_𝑖𝜚(𝑡,𝑥,𝑣)||𝑑𝑣𝑑𝑥𝑑𝑡_𝑖=1+2𝛿||𝜕_𝑥𝑖𝑓_𝑖𝜚𝑡,𝑥,𝑢_𝜀||𝑑𝑥𝑑𝑡𝜎_{Π𝑡∩||𝑢𝑙−𝜎<𝜀||𝑑<𝑙}_𝑗=1|||𝜕_𝑥𝑗𝑢_𝜀𝜕^2𝑥_𝑗𝑥𝑗𝑢_𝜀|||𝑑𝑥𝑑𝑡.(3.26) Without loss of generality, we can assume that 𝑙>1. Having this in mind, we get from (H4) and (3.26) 𝜀𝜎_{Π𝑡∩||𝑢𝑙<𝜀||<𝑙+𝜎}≤||∇𝑢_𝜀||²𝑑𝑥𝑑𝑡_{||𝑢𝜀||>𝑙}2||𝑢_𝜀||²𝑑𝑥+_{||𝑢0||>𝑙}2||𝑢₀||²𝑑𝑥+2_𝐑𝑑_𝑖=1𝐶₃1+|𝑣|^1+𝛼+2_Π𝑡𝑑𝑑𝑣_𝑖=1||𝜕_𝑥𝑖𝑓_𝑖𝜚𝑡,𝑥,𝑢_𝜀||𝑑𝑥𝑑𝑡+2𝛿𝜎_{Π𝑡∩||𝑢𝑙<𝜀||𝑑<𝑙+𝜎}_𝑗=1|||𝜕_𝑥𝑗𝑢_𝜀𝜕^2𝑥_𝑗𝑥𝑗𝑢_𝜀≤|||𝑑𝑥𝑑𝑡_𝐑𝑑2||𝑢_𝜀(𝑥,𝑡)||²+||𝑢₀(𝑥,𝑡)||²𝑑𝑥+𝐾¹+𝐾₂+2𝛿𝜎𝜀^2𝑑_𝑖=1‖‖𝜀^1/2𝜕_𝑥𝑖𝑢_𝜀‖‖_{𝐿2𝐑+×𝐑𝑑}×‖‖𝜀^3/2𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀‖‖_{𝐿2(𝐑+×𝐑𝑑)}≤𝐾₅+𝛿²𝜎²𝜀⁴(𝛽(𝜚))²+𝛿²𝜎²𝜀⁴^1/2𝐾₃𝐾₄,(3.27) where 𝐾_𝑖, 𝑖=1,…,5, are constants such that (cf. (3.8) and (3.9)) 2_𝐑𝑑_𝑖=1𝐶₃1+|𝑣|^1+𝛼2_Π𝑡𝑑𝑑𝑣≤𝐾₁,_𝑖=1||𝜕_𝑥𝑖𝑓_𝑖𝜚𝑡,𝑥,𝑢_𝜀||𝑑𝑥𝑑𝑡≤𝐾₂,_𝑑_𝑖=1‖‖𝜀^1/2𝜕_𝑥𝑖𝑢_𝜀‖‖_{𝐿2𝐑+×𝐑𝑑}≤𝐾₃,_𝑑_𝑖=1‖‖𝜀^3/2𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀‖‖_{𝐿2𝐑+×𝐑𝑑}≤1(𝛽(𝜚))²+𝜀𝜚^1/2_𝐑𝑑𝐾₄,2||𝑢_𝜀(𝑥,𝑡)||²+||𝑢₀(𝑥,𝑡)||²𝑑𝑥+𝐾¹+𝐾₂≤𝐾₅.(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 𝛿𝜎𝜀²‖𝜀^1/2∇𝑢_𝜀‖_{𝐿2𝐑+×𝐑𝑑𝑑}_𝑖=1‖𝜀^3/2𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀‖_{𝐿2𝐑+×𝐑𝑑}≤𝛿²𝜎²𝜀⁴𝛽²(𝜀)+𝜎𝛿²𝜀²⁴^1/2𝐾₃𝐾₄.(3.29) Thus, in view of (3.27), 𝜀𝜎_{Π𝑡∩||𝑢𝑙<𝜀||<𝑙+𝜎}||∇𝑢_𝜀||²𝑑𝑥𝑑𝑡≤𝐾₅+𝛿²𝜎²𝜀⁴𝛽²(𝜀)+𝜎𝛿²𝜀²⁴^1/2𝐾₃𝐾₄,(3.30) which is the sought for estimate for ‖𝑇^{𝑙,𝜎}(𝑢_𝜀)∇𝑢_𝜀‖_{𝐿2(𝐑+×𝐑𝑑)}.
Next, take a function 𝑈_𝜌(𝑧) satisfying 𝑈_𝜌(0)=0 and 𝑈^𝜌(𝑧)=⎧⎪⎨⎪⎩0𝑧,𝑧<0,1𝜌,,𝑧0><𝜌𝑧.<𝜌,(3.31) Clearly, 𝑈_𝜌 is convex, and 𝑈^𝜌(𝑧)→𝜃(𝑧) in 𝐿^𝑝loc(𝐑) as 𝜌→0, for any 𝑝<∞; as before, 𝜃 denotes the Heaviside function.
Inserting 𝑆(𝑢_𝜀)=𝑈_𝜌(𝑇_𝑙,𝜎(𝑢_𝜀)−𝑐) in (3.17), we get 𝜕_𝑡𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐+div_𝑥^𝑢𝜀=𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣_𝑢𝜀−𝑈𝑈^𝜌_𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)div_𝑥𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑇^𝑙,𝜎𝑢_𝜀div_𝑥𝑓_𝜚(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}+𝜀Δ_𝑥𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐−𝜀𝐷^2𝑢𝑢𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐||∇𝑢_𝜀||²+𝛿^𝑑_𝑖=1−𝛿_𝑑𝐷_𝑥𝑖𝐷_𝑢𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀_𝑖=1𝐷^2𝑢𝑢𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝜕_𝑥𝑖𝑢_𝜀𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀.(3.32) We rewrite the previous expression in the following manner: 𝜕_𝑡=𝜃Γ𝑇_𝑙1,𝜀𝑢_𝜀−𝑐𝑇_𝑙𝑢_𝜀−𝑐+div_𝑥𝜃𝑇_𝑙𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)+Γ_2,𝜀+Γ_3,𝜀+Γ_4,𝜀+Γ_5,𝜀+Γ_6,𝜀+Γ_7,𝜀,(3.33) where ΓΓ_1,𝜀=𝜕_𝑡𝜃𝑇_𝑙𝑢_𝜀−𝑐𝑇_𝑙𝑢_𝜀−𝑐−𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐_2,𝜀,=div_𝑥−𝜃𝑇^𝑙𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)_𝑢𝜀Γ𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣_3,𝜀,=−𝑈^𝑢𝜀𝑈^𝜌_𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)div_𝑥𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣Γ_4,𝜀𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑇^𝑙,𝜎𝑢_𝜀div_𝑥𝑓_𝜚(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀},=𝜀Δ_𝑥𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐+𝛿^𝑑_𝑖=1Γ_5,𝜀𝐷_𝑥𝑖𝐷_𝑢𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀,Γ_6,𝜀=−𝜀𝑈^𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑇^{𝑙,𝜎}(𝑢𝜀)||∇𝑢𝜀||²,=−𝛿^𝑑_𝑖=1Γ_7,𝜀𝐷^2𝑢𝑢𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝜕_𝑥𝑖𝑢_𝜀𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀,=−𝜀𝑈^𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑇^𝑙,𝜎𝑢_𝜀²||∇𝑢_𝜀||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 (𝑇_𝑙(𝑢_𝜀))_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 𝐻^−1loc(𝐑⁺×𝐑^𝑑).
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 div𝑄_𝜀, we have 𝑄_𝜀∈𝐿^𝑝loc(𝐑⁺×𝐑^𝑑) for 𝑝>2.
(ii) The right-hand side of (3.33) is of the form ℳ_loc,𝐵+𝐻^−1loc,𝑐, where ℳ_loc,𝐵 denotes a set of families which are locally bounded in the space of measures, and 𝐻^−1loc,𝑐 is a set of families precompact in 𝐻^−1loc.
First, since 𝑇_𝑙(𝑢_𝜀) 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 Γ_1,𝜀=𝜕_𝑡𝜃𝑇_𝑙𝑢_𝜀−𝑐𝑇_𝑙𝑢_𝜀−𝑐−𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐∈𝐻^−1loc,𝑐.(3.36) We have 𝜃=𝑇_𝑙𝜃𝑢_𝜀𝑇_𝑙−𝑐𝑇_𝑙𝑢_𝜀−𝑐−𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐+𝜃𝑇𝑢_{𝜀𝑙,𝜎}−𝑐𝑇_𝑙𝑢_𝜀−𝑐−𝜃𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑢_𝜀−𝑐𝑇_𝑙,𝜎𝑢_𝜀−𝑐−𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐.(3.37) Since the function 𝜃(𝑧−𝑐)(𝑧−𝑐) is Lipschitz continuous in 𝑧 with the Lipschitz constant one, and, according to definition of 𝑈_𝜌, it holds |𝑈_𝜌(𝑧)−𝜃(𝑧)𝑧|≤1/2𝜌, we conclude from the last expression ||𝜃𝑇_𝑙𝑢_𝜀−𝑐𝑇_𝑙𝑢_𝜀−𝑐−𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐||≤||𝑇_𝑙𝑢_𝜀−𝑇_𝑙,𝜎𝑢_𝜀||+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 𝜀→0𝜃𝑇_𝑙𝑢_𝜀−𝑐𝑇_𝑙𝑢_𝜀−𝑐−𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐⟶0(3.39) in 𝐿^𝑝loc for all 𝑝<∞. Thus, (since we can take 𝑝=2 as well) we see that Γ_1,𝜀∈𝐻^−1loc,𝑐.
Next, we will prove that Γ_2,𝜀=div_𝑥−𝜃𝑇^𝑙𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)_𝑢𝜀𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣∈𝐻^−1loc,𝑐+ℳ_loc,𝐵.(3.40) Indeed, 𝜃=𝑇_𝑙𝜃𝑢_𝜀𝑇_𝑙−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)−^𝑢𝜀𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣+𝜃𝑇𝑢_{𝜀𝑙,𝜎}−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)−𝜃𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑓_𝜚𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓_𝜚(𝑡,𝑥,𝑐)−𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)−^𝑢𝜀𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣_𝑢𝜀𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)−𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣.(3.41) Since 𝑇_𝑙(𝑢)=𝑢 if |𝑢|≤𝑙 and 𝑇^𝑙(𝑢)=0 if |𝑢|≥𝑙, ^𝑢𝜀𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣=^𝑢𝜀𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚𝑡,𝑥,𝑇_𝑙(𝑣)𝑑𝑣,(3.42) from which we conclude 𝜃𝑇_𝑙−𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)_𝑢𝜀=𝜃𝑇_𝑙𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣−𝜃𝑇𝑢_{𝜀𝑙,𝜎}−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)+𝜃𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)−𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)_𝑢𝜀−𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚𝑡,𝑥,𝑇_𝑙(𝑣)𝑑𝑣_𝑢𝜀=𝜃𝑇_𝑙𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)−𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣−𝜃𝑇𝑢_{𝜀𝑙,𝜎}−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)+𝜃𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)−𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)_𝑢𝜀−𝜃𝑇_𝑙,𝜎(𝑣)−𝑐𝐷_𝑣𝑓_𝜚𝑡,𝑥,𝑇_𝑙(𝑣)𝑑𝑣_𝑢𝜀−𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)−𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣_𝑢𝜀=Γ_12,𝜀𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐−𝜃𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚𝑡,𝑥,𝑇_𝑙(𝑣)𝑑𝑣+Γ^22,𝜀+Γ^32,𝜀,(3.43) with ΓΓ^12,𝜀=𝜃𝑇_𝑙𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)−𝜃𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)_22,𝜀,Γ=𝜃𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)−^𝑢𝜀𝜃𝑇_𝑙,𝜎(𝑣)−𝑐𝐷_𝑣𝑓_𝜚𝑡,𝑥,𝑇_𝑙(𝑣)𝑑𝑣,_32,𝜀=−−^𝑢𝜀𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)−𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣_𝑢𝜀𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐−𝜃𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚𝑡,𝑥,𝑇_𝑙(𝑣)𝑑𝑣.(3.44)
Consider now each term on the right-hand side of (3.43). Since 𝑇_𝑙 is a continuous function and 𝑇_𝑙(𝑢)∈[−𝑙,𝑙], the function 𝑓(𝑡,𝑥,𝑇_𝑙(𝑢)) is uniformly continuous in 𝑢∈𝐑. Therefore, we have pointwise on 𝐑⁺×𝐑^𝑑: ||Γ^12,𝜀||=||−𝜃𝜃𝑇_𝑙𝑇𝑢_{𝜀𝑙,𝜎}−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)||⟶0a𝑠𝜎⟶0.(3.45) Since max_{𝑢∈[−𝑙,𝑙]}𝑓(𝑡,𝑥,𝑢)∈𝐿^𝑝(𝐑⁺×𝐑^𝑑), 𝑝>2, Lebesgue's dominated convergence theorem yields |Γ^12,𝜀|=𝑜_𝜎,𝐿^𝑝loc(1), where ∫_{𝐑+×𝐑𝑑}|𝑜_𝜎,𝐿^𝑝(1)|^𝑝𝑑𝑥𝑑𝑡→0 as 𝜎→0. Thus, we conclude div_𝑥Γ^12,𝜀∈𝐻^−1loc𝐑⁺×𝐑^𝑑.(3.46)
We pass to Γ^22,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 𝑇_𝑙(𝑐)=𝑐 and 𝑇_𝑙,𝜎(𝑐)=𝑐. Thus, ||Γ^22,𝜀||=||||𝜃−𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)_𝑢𝜀=||||𝜃𝑇𝜃𝑇_𝑙,𝜎(𝑣)−𝑐𝐷_𝑣𝑓_𝜚𝑡,𝑥,𝑇_𝑙(𝑣)𝑑𝑣_𝑙,𝜎||||−𝜃𝑇𝑢_{𝜀𝑙,𝜎}−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)=||𝜃𝑇𝑢_𝜀−𝑐^𝑢𝜀_𝑐𝐷_𝑣𝑓_𝜚𝑡,𝑥,𝑇_𝑙(𝑣)𝑑𝑣_𝑙,𝜎||||−𝜃𝑇𝑢_{𝜀𝑙,𝜎}−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)≤||𝜃𝑇𝑢_𝜀−𝑐𝑓_𝜚𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓_𝜚(𝑡,𝑥,𝑐)_𝑙,𝜎||+||𝜃𝑇𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓_𝜚𝑡,𝑥,𝑇_𝑙𝑢_𝜀_𝑙,𝜎||≤||𝜃𝑇𝑢_𝜀−𝑐𝑓(𝑡,𝑥,𝑐)−𝑓_𝜚(𝑡,𝑥,𝑐)_𝑙,𝜎||+||𝜃𝑇𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓_𝜚𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀_𝑙,𝜎||+||𝜃𝑇𝑢_𝜀−𝑐𝑓_𝜚𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓_𝜚𝑡,𝑥,𝑇_𝑙𝑢_𝜀_𝑙,𝜎||=𝑜_{𝜚,𝐿𝑝loc}𝑢_𝜀−𝑐𝑓(𝑡,𝑥,𝑐)−𝑓_𝜚(𝑡,𝑥,𝑐)||(1)+𝒪𝜎𝛽(𝜚)+𝑜^{𝜚,𝐿𝑝loc}(1)=𝒪(1)+𝑜_𝜚,𝐿^𝑝loc(1),(3.47) where 𝑜_𝜚,𝐿^𝑝loc(1) appears due to (3.2a), and 𝒪(1) comes from (3.35).
For 𝑐>𝑙 we have 𝑐≥𝑙+𝜎 for a 𝜎 small enough, and therefore 𝜃(𝑇_𝑙,𝜎(𝑢_𝜀)−𝑐)≡0. On the other hand, for 𝑐<−𝑙 we have 𝑐≤−𝑙−𝜎, and so 𝜃(𝑇_𝑙,𝜎(𝑢_𝜀)−𝑐)≡1. Thus, the problematic case is when 𝑐<−𝑙. In this case, we have instead of (3.47) Γ^22,𝜀==𝑓𝜃𝑇𝑡,𝑥_𝑙,𝜎,𝑇𝑢_{𝜀𝑙,𝜎}−𝑐𝑓𝑡,𝑥,𝑇_𝑙,𝜎𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)−^𝑢𝜀𝜃𝑇_𝑙,𝜎(𝑣)−𝑐𝐷_𝑣𝑓_𝜚𝑡,𝑥,𝑇_𝑙(𝑣)𝑑𝑣𝑢_𝜀−𝑓_𝜚𝑡,𝑥,𝑇_𝑙𝑢_𝜀+𝑓_𝜚(𝑡,𝑥,−𝑙)−𝑓(𝑡,𝑥,𝑐)(3.48) implying div_𝑥Γ^22,𝜀∈𝐻^−1loc,𝑐+ℳ_loc,𝐵,(3.49) since 𝑓(𝑡,𝑥,𝑇_𝑙,𝜎(𝑢_𝜀))−𝑓_𝜚(𝑡,𝑥,𝑇_𝑙(𝑢_𝜀))→0 in 𝐿^𝑝loc(𝐑⁺×𝐑^𝑑) for 𝑝≥2, and 𝑓_𝜚(𝑡,𝑥,−𝑙)−𝑓(𝑡,𝑥,𝑐)∈𝐵𝑉(𝐑⁺×𝐑^𝑑).
It remains to estimate Γ^32,𝜀. Noticing that |𝑈^𝜌|,|𝑇^𝑙,𝜎|≤1, we get ||||^𝑢_𝜀𝑈≤^𝜌𝑇𝐶_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)−𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣||||𝛽(𝜌)_𝐑||𝑇^𝑙,𝜎(𝑣)−𝑇^𝑙(𝑣)||𝑑𝑣=𝒪𝜎𝛽(𝜚)^{(3.15),(3.35)}=𝒪(1),(3.50) where 𝐶 is the constant given by (3.2d).
Similarly, from (3.2d) and since |𝑇^𝑙(𝑣)|≤1, we have ||||^𝑢_𝜀≤𝑈𝐶^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐−𝜃𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙(𝑣)𝜕_𝑣𝑓_𝜚𝑡,𝑥,𝑇_𝑙(𝑣)𝑑𝑣||||𝛽(𝜌)^𝑙−𝑙||𝑈^𝜌𝑇_𝑙,𝜎(𝑣)−𝑐−𝜃𝑇_𝑙,𝜎(𝑣)−𝑐||𝑑𝑣=𝒪𝜚𝛽(𝜚)^(3.15)=𝒪(1),(3.51) from which we conclude that Γ^32,𝜀 is bounded in 𝐿^2loc. 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 𝐿^2loc from which it follows that Γ_2,𝜀∈𝐻^−1loc,𝑐.
The next term is Γ_3,𝜀=−𝑈^𝑢𝜀𝑈^𝜌_𝜌𝑇_𝑙,𝜎(𝑣)−𝑐𝑇^𝑙,𝜎(𝑣)div_𝑥𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑇^𝑙,𝜎𝑢_𝜀div_𝑥𝑓_𝜚(𝑡,𝑥,𝑣)∣_{𝑣=𝑢𝜀}.(3.52) According to (H4), it is clear that Γ_3,𝜀∈ℳ_{l𝑜𝑐,𝐵}. Indeed, since |𝑈^𝜌|,|𝑇^𝑙,𝜎|≤1 we have from (3.2b) and (3.2e) _{𝐑+×𝐑𝑑}||Γ_3,𝜀||𝑑𝑥𝑑𝑡≤_𝐑𝐶₃1+|𝑣|^1+𝛼𝑑𝑣+𝐶₁≤𝐾₆(3.53) for a constant 𝐾₆, implying the claim.
Next, we claim that Γ_4,𝜀=^𝑑_𝑖=1𝐷_𝑥𝑖𝜀𝐷_𝑥𝑖𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐+𝛿𝐷_𝑢𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀∈𝐻^−1loc,𝑐.(3.54) Due to a priori estimates (3.8) and (3.9) and, again, the fact that |𝑇^𝑙,𝜎|,|𝑈^𝜌|≤1, we see that for every 𝑖=1,…,𝑑𝜀𝐷_𝑥𝑖𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐+𝛿𝐷_𝑢𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀⟶0(3.55) in 𝐿²(𝐑⁺×𝐑^𝑑). Therefore, Γ_4,𝜀∈𝐻^−1loc,𝑐.
Further, we claim that Γ_5,𝜀=𝜀𝑈^𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑇^{𝑙,𝜎}𝑢_𝜀||∇𝑢_𝜀||²∈ℳ_loc,𝐵.(3.56) Since |𝑈^𝜌|≤1 and |𝑇^{𝑙,𝜎}|≤1𝜎 we have from (3.30) (recall (3.28) for the definition of the constants 𝐾_𝑗 for 𝑗=3,4,5) 𝜀≤𝜀_{𝐑+×𝐑𝑑}||𝑈^𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑇^{𝑙,𝜎}𝑢_𝜀||||∇𝑢_𝜀||²𝑑𝑥𝑑𝑡𝜎_{||𝑢𝑙<𝜀||<𝑙+𝜎}||∇𝑢_𝜀||²𝑑𝑥𝑑𝑡≤^(3.30)𝐾₅+𝛿²𝜎²𝜀⁴+𝛿²𝜎²(𝛽(𝜚))²𝜀⁴^1/2𝐾₃𝐾₄≤𝐾₆,(3.57) 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 Γ_5,𝜀∈ℳ_loc,𝐵.
Next, we need to show Γ_6,𝜀=𝛿^𝑑_𝑖=1𝐷^2𝑢𝑢𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝜕_𝑥𝑖𝑢_𝜀𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀∈ℳ_{l𝑜𝑐,𝐵}.(3.58) In view of a priori estimates (3.8) and (3.9), and assumptions (3.15) and (3.35), it holds _{𝐑+×𝐑𝑑}𝛿|||||^𝑑_𝑖=1𝐷^2𝑢𝑢𝑈_𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝜕_𝑥𝑖𝑢_𝜀𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀≤|||||𝑑𝑥𝑑𝑡_𝑑_𝑖=1𝜎𝛿+𝛿𝜌_𝐑⁺_×𝐑^𝑑||𝜕_𝑥𝑖𝑢_𝜀||²𝑑𝑥𝑑𝑡^1/2_{𝐑+×𝐑𝑑}||𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀||²𝑑𝑥𝑑𝑡^1/2≤𝛿𝜀²𝜎+𝜀𝛿²𝜌^𝑑_𝑖=1𝜀_{𝐑+×𝐑𝑑}||𝜕_𝑥𝑖𝑢_𝜀||²𝑑𝑥𝑑𝑡^1/2𝜀³_{𝐑+×𝐑𝑑}||𝜕^2𝑥_𝑖𝑥𝑖𝑢_𝜀||²𝑑𝑥𝑑𝑡^1/2≤𝐾₇𝜀𝛿²𝜎+𝜀𝛿²𝜌𝜚𝜀+(𝛽(1𝜚))²^1/2≤𝐾₈,(3.59) for some constants 𝐾₇ and 𝐾₈. The second estimate holds since 𝑈^𝜌≤1/𝜌 and 𝑇^{𝑙,𝜎}≤1/𝜎 implying |𝐷_𝑢𝑢[𝑈_𝜌(𝑇_𝑙,𝜎(𝑢_𝜀)−𝑐)]|≤(1/𝜎+1/𝜌). Therefore, Γ_6,𝜀∈ℳ_loc,𝐵.
Finally, we will prove that Γ_7𝜀=−𝜀𝑈^𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑇^𝑙,𝜎𝑢_𝜀²||∇𝑢_𝜀||²∈ℳ_loc,𝐵.(3.60) First, notice that supp𝑈^𝜌=(0,𝜚), and therefore 𝑈^𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐≠0for𝑐≤𝑇_𝑙,𝜎𝑢_𝜀≤𝑐+𝜌.(3.61) 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 𝑈^𝜌(𝑇_𝑙,𝜎(𝑢_𝜀)−𝑐)≠0 only if 𝑙+𝜎≤𝑇_𝑙,𝜎(𝑢_𝜀)≤𝑙+𝜎+𝜌 which is never fulfilled according to the definition of 𝑇_𝑙,𝜎 (see (3.22)). So, in this case, Γ_7𝜀≡0∈ℳ_loc,𝐵.(3.62)
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 𝑈^𝜌(𝑇_𝑙,𝜎(𝑢_𝜀)−𝑐)≠0 if −𝑙≤𝑐≤𝑇_𝑙,𝜎(𝑢_𝜀)≤𝑐+𝜌≤𝑙 implying that 𝑈^𝜌(𝑇_𝑙,𝜎(𝑢_𝜀)−𝑐)=𝑈^𝜌(𝑢_𝜀−𝑐). Thus, Γ_7𝜀=−𝜀𝑈^𝜌𝑢_𝜀−𝑐||∇𝑢_𝜀||²∈ℳ_loc,𝐵,(3.63) according to (3.30) (we put there 𝑙=𝑐).
Finally, assume that |𝑐|=𝑙. From (3.61) and (3.19), we conclude 𝑇^𝑙,𝜎𝑢_𝜀²𝑈^𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑙≠0iiffff𝑙𝑙≤≤𝑢𝑇_{𝑙,𝜎𝜀}𝑢_𝜀≤𝑙+𝜌and𝑢_𝜀≤𝑙+𝜎≤𝑙+𝜎.(3.64) From here and since |𝑇^𝑙,𝜎|≤1, it follows (recall that we assume 𝜌=𝜎) _Π𝑡𝜀𝑈^𝜌𝑇_𝑙,𝜎𝑢_𝜀−𝑐𝑇^𝑙,𝜎𝑢_𝜀²||∇𝑢_𝜀||²𝑑𝑥𝑑𝑡≤𝜀𝜎_{Π𝑡∩||𝑢𝑙<𝜀||<𝑙+𝜎}||∇𝑢_𝜀||²𝑑𝑥𝑑𝑡<∞,(3.65) according to (3.30). From here, (3.62) and (3.63), we conclude (3.60).
Collecting the previous items, due to the properties of Γ_𝑖,𝜀, 𝑖=1,…,7, it follows from (3.33) that 𝜕_𝑡𝜃+𝑇_𝑙di𝑢v_𝜀𝑥−𝑐𝑇_𝑙𝑢_𝜀−𝑐𝜃𝑇_𝑙𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)∈ℳ_loc,𝐵+𝐻^−1loc,𝑐.(3.66) Therefore, we see that (ii) is satisfied and we can use Murat's lemma to conclude that 𝜕_𝑡𝜃𝑇_𝑙𝑢_𝜀−𝑐𝑇_𝑙𝑢_𝜀−𝑐+div_𝑥𝜃𝑇_𝑙𝑢_𝜀−𝑐𝑓𝑡,𝑥,𝑇_𝑙𝑢_𝜀−𝑓(𝑡,𝑥,𝑐)∈𝐻^−1loc,𝑐.(3.67) Thus we conclude that the conditions of Theorem 3.3 are satisfied, and we find that for every 𝑙>0 the sequence (𝑇_𝑙(𝑢_𝜀))_𝜀>0 is precompact in 𝐿¹_l𝑜𝑐(𝐑⁺×𝐑).
Since the sequence (𝑢_𝜀)_𝜀>0 is uniformly bounded in 𝐿²(𝐑⁺×𝐑^𝑑), from [23, Lemma 7], we conclude that (𝑢_𝜀)_𝜀>0 is precompact in 𝐿¹_l𝑜𝑐(𝐑⁺×𝐑^𝑑).

4. The One-Dimensional Case

We will analyze the convergence of the sequence (𝑢_𝜀)_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⁺×𝐑_𝑥))∩𝐿^∞(𝐑×𝐑⁺×𝐑_𝑥) and 𝜕_𝑢𝑓∈𝐿^∞(𝐑×𝐑⁺×𝐑_𝑥).

(H4b) There exists a sequence (𝑓_𝜚)_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:

max_{|𝑧|≤𝑙}||𝑓_𝜚(𝑡,𝑥,𝑧)−𝑓(𝑡,𝑥,𝑧)||⟶0,𝜚⟶0in𝐿^𝑝loc𝐑⁺×𝐑_𝐑+×𝐑,||𝜕_𝑥𝑓_𝜚(𝑡,𝑥,𝑢)||𝑑𝑥𝑑𝑡≤𝐶¹1+|𝑢|^1+𝛼,_𝐑+×𝐑𝜚||𝜕_𝑥𝑓_𝜚(𝑡,𝑥,𝑢)||²𝑑𝑥𝑑𝑡≤𝐶₂(𝑡,𝑥),_𝐑+×𝐑||𝜕^2𝑥𝑢𝑓_𝜚(𝑡,𝑥,𝑢)||𝑑𝑥𝑑𝑡≤𝐶³1+|𝑢|^1+𝛼||𝜕_𝑢,𝑓_𝜚(𝑡,𝑥,𝑢)||≤𝐶,(4.1) where 𝐶_𝑖, 𝑖=1,2,3, and 𝐶 are constants independent on (𝑡,𝑥,𝑢)∈𝐑⁺×𝐑^𝑑×𝐑.

Under these assumptions we will prove the following.

(i)Without assuming nondegeneracy of the flux, the sequence (𝑢_𝜀)_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 assume 𝑓∈𝐶²(𝐑;𝐵𝑉(𝐑⁺×𝐑_𝑥))∩𝐿^∞(𝐑×𝐑⁺×𝐑_𝑥), and that 𝑓 is genuinely nonlinear in the sense of (4.12), the sequence (𝑢_𝜀)_𝜀>0 of solutions of problem (3.5)–(3.6) is strongly precompact in 𝐿^1loc(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 (𝑢_𝜀𝑘) is uniformly bounded in 𝐿^𝑝loc(𝐑⁺×𝐑^𝑑)), 𝑝≥1. Then, there exists a subsequence (not relabeled) (𝑢_𝜀𝑘) and a sequence of probability measures 𝜈_(𝑡,𝑥)∈ℳ(𝐑),(𝑡,𝑥)∈𝐑⁺×𝐑^𝑑(4.2) such that the limit 𝑔(𝑡,𝑥)∶=lim_𝑘→∞𝑔𝑡,𝑥,𝑢_𝜀𝑘(𝑡,𝑥)(4.3) exists in the distributional sense for all 𝑔 measurable with respect to (𝑡,𝑥)∈𝐑⁺×𝐑^𝑑, continuous in 𝑢∈𝐑 and satisfying uniformly in (𝑡,𝑥): ||𝑔(𝑡,𝑥,𝑢)||≤𝐶(1+|𝑢|^𝑞),(4.4) for a constant 𝐶 independent of 𝑢, and 𝑞 such that 0≤𝑞<𝑝. The limit is represented by the expectation value 𝑔(𝑡,𝑥)=_{𝐑+×𝐑𝑑}𝑔(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥)(𝜆),(4.5) for almost all points (𝑡,𝑥)∈𝐑⁺×𝐑^𝑑.
One refers to such a sequence of measures 𝜈=(𝜈_(𝑡,𝑥)) as the Young measures associated to the sequence (𝑢_𝜀𝑘)_𝑘∈𝐍.
Furthermore, 𝑢_𝜀𝑘⟶𝑢in𝐿^𝑟loc𝐑⁺×𝐑^𝑑,1≤𝑟<𝑝,(4.6) if and only if 𝜈_𝑦=𝛿_𝑢(𝑦)𝑎.𝑒.(4.7)

Before we continue, we need to recall the celebrated Div-Curl lemma.

Lemma 4.3 (Div-Curl). Let 𝑄⊂𝐑² be a bounded domain, and suppose that 𝑤𝑣^1𝜀_1𝜀⇀𝑣¹,𝑣^2𝜀⇀𝑣²,⇀𝑤¹,𝑤^2𝜀⇀𝑤²,(4.8) in 𝐿²(𝑄) as 𝜀↓0. Assume also that the two sequences {div(𝑣^1𝜀,𝑣^2𝜀)}_𝜀>0 and {curl(𝑤^1𝜀,𝑤^2𝜀)}_𝜀>0 lie in a (common) compact subset of 𝐻^−1loc(𝑄), where div(𝑣^1𝜀,𝑣^2𝜀)=𝜕_𝑥1𝑣^1𝜀+𝜕_𝑥2𝑣^2𝜀 and curl(𝑤^1𝜀,𝑤^2𝜀)=𝜕_𝑥1𝑤^2𝜀−𝜕_𝑥2𝑤^1𝜀. Then along a subsequence 𝑣^1𝜀,𝑣^2𝜀⋅𝑤^1𝜀,𝑤^2𝜀⟶𝑣¹,𝑣²⋅𝑤¹,𝑤²in𝒟^(Q)as𝜀↓0.(4.9)

Lemma 4.4. Assume that (𝑢_𝜀)_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(𝐑_𝜆;𝐿^∞∩𝐵𝑉(𝐑^+𝑡×𝐑_𝑥)).
By 𝜂_𝑛 one denotes the truncation of the function 𝜂: 𝜂_𝑛(𝑡,𝑥,𝜆)=𝜂0,(𝑡,𝑥,𝜆),||||𝜆𝜆||||<>𝑛2,𝑛,(𝑡,𝑥)∈𝐑⁺×𝐑,(4.10) and 𝑞_𝑛(𝑡,𝑥,𝜆) the corresponding entropy flux.
If for every 𝑛∈𝐍 one has div_(𝑡,𝑥)𝜂_𝑛𝑡,𝑥,𝑢_𝜀,𝑞_𝑛𝑡,𝑥,𝑢_𝜀∈𝐻^−1loc,𝑐𝐑⁺×𝐑,(4.11) then the limit function 𝑢 is a weak solution of (1.3).
Furthermore, if the flux function 𝑓=𝑓(𝑡,𝑥,𝜆) is twice differentiable with respect to 𝜆, and is genuinely nonlinear, that is, for every (𝑡,𝑥)∈𝐑⁺×𝐑^𝑑 the mapping 𝐑∋𝜆⟼𝜕_𝜆f(t,x,𝜆)isnonconstant(4.12) on nondegenerate intervals, then (𝑢_𝜀)_𝜀>0 converges strongly along a subsequence to 𝑢 in 𝐿^1loc(𝐑⁺×𝐑).

Proof. We will apply the method of compensated compactness as in [13].
First, notice that according to Theorem 4.2 there exist a subsequence (𝑢_𝜀𝑘)⊂(𝑢_𝜀) and a sequence of probability measures 𝜈_(𝑡,𝑥)∈ℳ(𝐑),(𝑡,𝑥)∈𝐑⁺×𝐑(4.13) such that the limit 𝑔(𝑡,𝑥)∶=lim_𝑘→∞𝑔𝑡,𝑥,𝑢_𝜀𝑘(𝑡,𝑥)(4.14) 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 𝑔(𝑡,𝑥)=_𝐑+×𝐑𝑔(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥)(𝜆),(4.15) 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|≤𝐶(1+|𝑢|), for a constant 𝐶 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 lim_𝑘→∞𝑓𝑡,𝑥,𝑢_𝜀𝑘(𝑡,𝑥)=_𝐑+×𝐑𝑓(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥)(𝜆).(4.16)
To continue, notice that 𝑢(𝑡,𝑥)=𝜆𝑑𝜈_(𝑡,𝑥)(𝜆).(4.17)
Take 𝜂(𝑢)=𝐼(𝑢)=𝑢 in (4.10), and consider the vector fields (𝐼_𝑛(𝑢_𝜀),𝑓_𝑛(𝑡,𝑥,𝑢_𝜀)) where 𝜕_𝜆𝑓_𝑛(𝑡,𝑥,𝑢_𝜀)=𝐼^𝑛(𝑣)𝜕_𝜆𝑓(𝑡,𝑥,𝑢_𝜀), and (−𝜓_𝑛(𝑡,𝑥,𝑢_𝜀),𝜙_𝑛(𝑢_𝜀)), where 𝜙∈𝐶¹(𝐑) is an arbitrary entropy, and 𝜓_𝑛 is the entropy flux corresponding to 𝜙_𝑛. Here 𝐼_𝑛 and 𝜙_𝑛 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: 𝐼^𝑛=(𝜆)𝜓_𝑛(𝑡,𝑥,𝜆)−𝜙_𝑛(𝜆)𝑓_𝑛(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥)(𝜆)𝑢_𝑛(𝑡,𝑥)𝜓_𝑛(𝑡,𝑥,𝜆)−𝑓_𝑛(𝑡,𝑥)𝜙_𝑛(𝜆)𝑑𝜈_(𝑡,𝑥)(𝜆),(4.18) where 𝑓_𝑛(𝑡,𝑥)=𝑓_𝑛(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥)(𝜆),𝑢_𝑛(𝑡,𝑥)=𝐼_𝑛(𝜆)𝑑𝜈_(𝑡,𝑥)(𝜆).(4.19)
Then, put 𝜙(𝜆)=|𝜆−𝑢(𝑡,𝑥)|. Notice that for |𝜆|<𝑛 it holds 𝜓_𝑛(𝑡,𝑥,𝜆)=sgn(𝜆−𝑢(𝑡,𝑥))(𝑓(𝑡,𝑥,𝜆)−𝑓(𝑡,𝑥,𝑢(𝑡,𝑥))). Therefore, we have from (4.18) ^𝑛−𝑛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)_𝑛−𝑛=−_{−𝑛−∞}𝑢(𝑡,𝑥)sgn(𝜆−𝑢(𝑡,𝑥))(𝑓(𝑡,𝑥,𝜆)−𝑓(𝑡,𝑥,𝑢(𝑡,𝑥)))−||𝑢(𝑡,𝑥)−𝜆||𝑓_𝑛𝑑𝜈^(𝑡,𝑥)(𝜆)+^∞𝑛𝐼^𝑛+(𝜆)𝜓_𝑛(𝑡,𝑥,𝜆)−𝜙_𝑛(𝜆)𝑓_𝑛(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥)(𝜆)_{−𝑛−∞}+^∞𝑛𝑢(𝑡,𝑥)𝜓^𝑛+(𝑡,𝑥,𝜆)−𝑓_𝑛𝜙_𝑛(𝜆)𝑑𝜈_(𝑡,𝑥)(𝜆)_{−𝑛−∞}+^∞𝑛𝐼^𝑛(𝜆)−𝜆𝑑𝜈_(𝑡,𝑥)(𝜆)𝜓_𝑛(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥)(𝜆).(4.20) It is clear that for every fixed (𝑡,𝑥)∈𝐑⁺×𝐑^𝑑 the right-hand side of (4.20) tends to zero as 𝑛→∞ implying (due to the Lebesgue dominated convergence theorem) 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)−−||(𝑢𝑢((𝑡𝑡,,𝑥𝑥))−sg𝜆n||(𝜆−𝑢(𝑡,𝑥))(𝑓(𝑡,𝑥,𝜆)−𝑓(𝑡,𝑥,𝑢(𝑡,𝑥)))(𝜆)𝑓(𝑡,𝑥)𝑑𝜈_(𝑡,𝑥)(𝜆)=0in𝒟^𝐑⁺×𝐑^𝑑.(4.21) Now, a standard procedure gives (see, e.g., [6, Remark 2.3]) 𝑓(𝑡,𝑥,𝑢(𝑡,𝑥))−𝑓(𝑡,𝑥)||𝜆−𝑢(𝑡,𝑥)||𝑑𝜈_(𝑡,𝑥)(𝜆)=0,(4.22) where 𝑓(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 that 𝑓∈𝐶²(𝐑;𝐵𝑉(𝐑⁺×𝐑_𝑥))∩𝐿^∞(𝐑×𝐑⁺×𝐑_𝑥), and that it is genuinely nonlinear in the sense of (4.12).
Then, take arbitrary entropies 𝜂₁(𝑡,𝑥,𝑢)∈𝐶¹((𝐑;𝐿^∞∩𝐵𝑉(𝐑^+𝑡×𝐑_𝑥))) and 𝜂₂∈𝐶¹(𝐑), and denote by 𝑞₁(𝑡,𝑥,𝑢) and 𝑞₂(𝑢), respectively, their corresponding entropy fluxes. Assume that 𝜂_𝑖, 𝑞_𝑖, 𝜂_𝑖𝑞_𝑖, 𝑖=1,2, satisfy (4.4) for 𝑞<2. Notice that 𝜕_𝑢𝜂₁ depends explicitly on (𝑡,𝑥), while 𝐷_𝑢𝜂₂ does not. Denote by 𝜂_1,𝑛 and 𝜂_2,𝑛 the appropriate smooth truncations (cf. (4.10)) and by 𝑞_1,𝑛 and 𝑞_2,𝑛 the corresponding entropy fluxes, that is, 𝑞𝑞_1,𝑛(𝑡,𝑥,𝜆)=^𝜆𝜕_𝑧𝜂_1,𝑛(𝑡,𝑥,𝑧)𝜕_𝑧𝑓(𝑡,𝑥,𝑧)𝑑𝑧,_1,𝑛(𝑡,𝑥,𝜆)=^𝜆𝜕_𝑧𝜂_2,𝑛(𝑧)𝜕_𝑧𝑓(𝑡,𝑥,𝑧)𝑑𝑧.(4.23) Due to (4.11) and the Div-Curl lemma the following commutation relation holds: _𝐑𝜂_1,𝑛(𝑡,𝑥,𝜆)𝑞_2,𝑛(𝑡,𝑥,𝜆)−𝜂₂(𝜆)𝑞_1,𝑛(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥)=_𝐑𝜂_1,𝑛(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥)_𝐑𝑞_2,𝑛(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥)−_𝐑𝜂_2,𝑛(𝜆)𝑑𝜈_(𝑡,𝑥)_𝐑𝑞_1,𝑛(𝑡,𝑥,𝜆)𝑑𝜈_(𝑡,𝑥).(4.24) Letting 𝑛→∞ as in (4.20), we get _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).(4.25)
Next, recall that the function 𝑓 satisfies (4.4) for 𝑞=1. Therefore, the following entropy-entropy fluxes are admissible: 𝜂₁(𝑡,𝑥,𝜆𝜂)=𝑓(𝑡,𝑥,𝜆)−𝑓(𝑡,𝑥,𝑢(𝑡,𝑥)),𝑞₁(𝑡,𝑥,𝜆)=^{𝜆𝑢(𝑡,𝑥)}𝜕_𝑣𝑓(𝑡,𝑥,𝑣)²𝑑𝑣,₂(𝜆)=𝜆−𝑢(𝑡,𝑥),𝑞₂(𝑡,𝑥,𝜆)=𝑓(𝑡,𝑥,𝜆)−𝑓(𝑡,𝑥,𝑢(𝑡,𝑥)).(4.26) Then, following [6], we insert the last quantities in (4.25) which yields the following relation: _𝐑(𝑓(𝑡,𝑥,𝜆)−𝑓(𝑡,𝑥,𝑢(𝑡,𝑥)))𝑑𝜈_(𝑡,𝑥)²+_𝐑(𝜆−𝑢(𝑡,𝑥))^{𝜆𝑢(𝑡,𝑥)}𝜕_𝜚𝑓(𝑡,𝑥,𝜚)²𝑑𝜚−(𝑓(𝑡,𝑥,𝜆)−𝑓(𝑡,𝑥,𝑢))²𝑑𝜈^(𝑡,𝑥)(𝜆)=0.(4.27) By the Cauchy–Schwarz inequality (𝑓(𝑡,𝑥,𝜆)−𝑓(𝑡,𝑥,𝑢))²=^{𝜆𝑢(𝑡,𝑥)}𝜕_𝜚𝑓(𝑡,𝑥,𝜚)𝑑𝜚²≤(𝜆−𝑢(𝑡,𝑥))^{𝜆𝑢(𝑡,𝑥)}𝜕_𝜚𝑓(𝑡,𝑥,𝜚)²𝑑𝜚𝑑𝜈_(𝑡,𝑥)(𝜆),(4.28) 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 (𝜆−𝑢(𝑡,𝑥))^{𝜆𝑢(𝑡,𝑥)}𝜕_𝜚𝑓(𝑡,𝑥,𝜚)²𝑑𝜚𝑑𝜈_(𝑡,𝑥)(𝜆)=0,(4.29) that is, that 𝜈_(𝑡,𝑥)=𝛿_{𝑢(𝑡,𝑥)} a.e. on 𝐑⁺×𝐑 implying strong 𝐿^1loc convergence of (𝑢_𝜀)_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 𝛿=𝛿(𝜀)=𝒪𝜀²,𝜚=𝒪(𝜀),𝜀→0,(4.30) and 𝑢₀∈𝐻¹(𝐑).
Assume that the flux function 𝑓 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 (𝑢_𝜀𝑘)⊂(𝑢_𝜀) of problem (3.5)–(3.6) converges in the sense of distributions to a weak solution of problem (3.1).
If the flux function 𝑓∈𝐶²(𝐑;𝐵𝑉(𝐑⁺×𝐑_𝑥))∩𝐿^∞(𝐑×𝐑⁺×𝐑_𝑥), and if it is genuinely nonlinear in the sense of (4.12), then a subsequence of solutions (𝑢_𝜀𝑘)⊂(𝑢_𝜀) of problem (3.5)–(3.6) converges strongly in 𝐿¹(𝐑⁺×𝐑) to a weak solution of (3.1).

Proof. Assume that 𝜂(𝑡,𝑥,𝜆), (𝑡,𝑥,𝜆)∈𝐑⁺×𝐑² is a function such that 𝜂∈𝐶²(𝐑;𝐿^∞∩𝐵𝑉(𝐑^+𝑡×𝐑_𝑥)). As usual, denote by 𝜂_𝑛 the truncation given by (4.10), and let the entropy flux corresponding to 𝜂_𝑛 and 𝑓 be 𝑞_𝑛(𝑡,𝑥,𝑢)=^𝑢𝜕_𝑣𝜂_𝑛(𝑡,𝑥,𝑣)𝜕_𝑣𝑓(𝑡,𝑥,𝑣)𝑑𝑣.(4.31)
According to Lemma 4.4, it is enough to prove that for every fixed 𝑛∈𝐍 the expression div(𝜂_𝑛(𝑡,𝑥,𝑢_𝜀(𝑡,𝑥)),𝑞_𝑛(𝑡,𝑥,𝑢_𝜀(𝑡,𝑥))) is precompact in 𝐻^−1loc(𝐑⁺×𝐑).
In order to prove the latter, take the following mollifier 𝜂_𝑛,𝜀(𝑡,𝑥,𝑢)=𝜂_𝑛(⋅,⋅,𝑢)⋆(1/𝜀^1/2)𝜔(𝑡/𝜀^1/4)𝜔(𝑥/𝜀^1/4), where 𝜔 is a nonnegative real function with unit mass. Denote the entropy flux corresponding to 𝜂_𝑛 and 𝑓 by 𝑞_𝑛,𝜀(𝑡,𝑥,𝑢)=^𝑢𝜕_𝑣𝜂_𝑛,𝜀(𝑡,𝑥,𝑣)𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝑑𝑣.(4.32) 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 on 𝜂 and the choice of the mollifier 𝜂_𝑛,𝜀 we have _𝐑+×𝐑||𝜕_𝑡𝜂_𝑛,𝜀(𝑡,𝑥,_𝐑+×𝐑𝑢)||+||𝜕_𝑥𝜂_𝑛,𝜀(𝑡,𝑥,𝑢)||+||𝜕_𝑥𝑣𝜂_𝑛,𝜀(𝑡,𝑥,𝑢)||𝑑𝑥𝑑𝑡≤𝐶₄,||𝜕^𝑥𝜂_𝑛,𝜀(𝑡,𝑥,𝑢)||²+||𝜕^2𝑥𝑣𝜂_𝑛,𝜀(𝑡,𝑥,𝑢)||²𝑑𝑥𝑑𝑡≤𝐶𝜀⁴,(4.33) for a constant 𝐶₄.
Then, applying (3.17) with 𝑆 replaced by 𝜂_𝑛,𝜀, we find 𝐷_𝑡𝜂=_𝑛𝑡,_𝑢𝜀𝑥,𝑢_𝜀+𝐷_𝑥𝑞_𝑛𝑡,𝑥,𝑢_𝜀−𝜕_𝑣𝜕^2𝑥𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝜕_𝑣𝜂_𝑛,𝜀(𝑡,𝑥,𝑣)+𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝜕^2𝑥𝑣𝜂_𝑛,𝜀(𝑡,𝑥,𝑣)𝑑𝑣+𝜀𝐷𝜂_{𝑛,𝜀𝑥}𝑡,𝑥,𝑢_𝜀𝜕_𝑥𝑓_𝜚𝑡,𝑥,𝑢_𝜀+𝜕_𝑡𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀𝜕_𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀𝑏𝜕_𝑥𝑢_𝜀−𝜀𝜕^2𝑣𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀𝑏𝜕_𝑥𝑢_𝜀𝜕_𝑥𝑢_𝜀−+𝛿𝜀𝑏𝐷_𝑥𝜕_𝑥𝑢_𝜀𝜕^2𝑥𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀−𝛿𝜕^2𝑥𝑥𝑢_𝜀𝜕^2𝑥𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀𝜕_𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀𝜕^2𝑥𝑥𝑢_𝜀−𝛿2𝜕^2𝑣𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀𝐷_𝑥𝜕_𝑥𝑢_𝜀²++𝐷𝐷_𝑥𝑡−𝑞_𝑛,𝜀𝑡,𝑥,𝑢_𝜀+𝑞_𝑛𝑡,𝑥,𝑢_𝜀−𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀+𝜂_𝑛𝑡,𝑥,𝑢_𝜀.(4.34) Now, we apply a similar procedure as in the multidimensional case.
Combining (H4b') and (4.33), we get _𝐑+×𝐑||||^𝑢_𝜀𝜕^2𝑥𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝜕_𝑣𝜂_𝑛,𝜀(𝑡,𝑥,𝑣)+𝜕_𝑣𝑓_𝜚(𝑡,𝑥,𝑣)𝜕^2𝑥𝑣𝜂_𝑛,𝜀(𝑡,𝑥,𝑣)𝑑𝑣||||𝑑𝑥𝑑𝑡≤𝐶₆,(4.35) for a constant 𝐶₆, implying boundedness of the subintegral expression in the sense of measures.
Similarly, for a constant 𝐶₇_𝐑+×𝐑||−𝜕_𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀𝜕_𝑥𝑓_𝜚𝑡,𝑥,𝑢_𝜀−𝜕_𝑡𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀||𝑑𝑥𝑑𝑡≤𝐶₇,(4.36) 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 Γ_6𝜀) that −𝜀𝑏𝜕_𝑥𝑢_𝜀𝜕^2𝑥𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀−𝛿𝜕^2𝑥𝑥𝑢_𝜀𝜕^2𝑥𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀(4.37) is bounded in ℳ(𝐑⁺×𝐑).
Next, 𝐷_𝑥𝜀𝜕_𝑣𝜂_𝑛𝑡,𝑥,𝑢_𝜀𝑏𝜕_𝑥𝑢_𝜀+𝛿𝜕_𝑣𝜂_𝑛𝑡,𝑥,𝑢_𝜀𝑘𝜕^2𝑥𝑥𝑢_𝜀𝑘(4.38) is precompact in 𝐻⁻¹(𝐑⁺×𝐑) since |𝜂^𝑛|<𝐶, 𝛿=𝒪(𝜀²), 𝜚=𝒪(𝜀), and from (3.8) and (3.9) (see also Remark 4.1) we have 𝜀b𝜕_xu_𝜀+𝛿𝜕²_xxu_𝜀⟶0as𝜀⟶0in𝐿²𝐑⁺×𝐑.(4.39)
Similarly, by (3.8) and (3.9) (see the estimation of Γ_6𝜀 again) 𝜀𝜕_𝑣𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀𝑏𝜕_𝑥𝑢_𝜀𝜕_𝑥𝑢_𝜀+𝛿2𝜕^𝑣𝑣𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀𝐷_𝑥𝜕_𝑥𝑢_𝜀²∈ℳ_loc,𝐵𝐑⁺×𝐑.(4.40)
Next, due to (H4b) and the well-known properties of the convolution, it holds for a constant 𝐶 independent on 𝜀: ||𝑞_𝑛,𝜀𝑡,𝑥,𝑢_𝜀−𝑞_𝑛𝑡,𝑥,𝑢_𝜀||≤𝐶𝜀⟶0in𝐿^2loc𝐑⁺×𝐑as𝜀⟶0(4.41) for arbitrary 𝑝>0 implying 𝐷_𝑥𝑞_𝑛,𝜚𝑡,𝑥,𝑢_𝜀−𝑞_𝑛𝑡,𝑥,𝑢_𝜀∈𝐻^−1loc,𝑐.(4.42)
Similarly, it is easy to see that max_{−2𝑛<𝑣<2𝑛}−𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀+𝜂_𝑛𝑡,𝑥,𝑢_𝜀⟶0in𝐿²𝐑⁺×𝐑,(4.43) and thus 𝐷_𝑡−𝜂_𝑛,𝜀𝑡,𝑥,𝑢_𝜀+𝜂_𝑛𝑡,𝑥,𝑢_𝜀∈𝐻^𝑐−1𝐑⁺×𝐑.(4.44)
From (4.35)–(4.44) and the fact that (𝜂_𝑛(𝑡,𝑥,𝑢_𝜀),𝑞_𝑛(𝑡,𝑥,𝑢_𝜀))∈𝐿^∞(𝐑⁺×𝐑), we conclude using Murat's lemma that div𝜂_𝑛𝑡,𝑥,𝑢_𝜀,𝑞_𝑛𝑡,𝑥,𝑢_𝜀∈𝐻^−1loc,𝑐𝐑⁺×𝐑.(4.45)
Finally, relying on Lemma 4.4 we conclude the proof of the theorem.

Acknowledgments

The work is supported in part by the Research Council of Norway. The work was supported by the Research Council of Norway through the Projects Nonlinear Problems in Mathematical Analysis, Waves In Fluids and Solids, and an Outstanding Young Investigators Award (KHK). This paper was written as part of the the International Research Program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008–2009.