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(||||