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International Journal of Differential Equations
VolumeΒ 2009, Article IDΒ 279818, 33 pages
http://dx.doi.org/10.1155/2009/279818
Research Article

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

1Department of Mathematical Sciences, Norwegian University of Science and Technology, Alfred Getz vei 1, 7491 Trondheim, Norway
2Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
3Department of Mathematics, Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway

Received 2 April 2009; Revised 24 August 2009; Accepted 24 September 2009

Academic Editor: Philippe G.Β LeFloch

Copyright Β© 2009 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.

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 𝑒=π‘’πœ€(𝑑,π‘₯) with (𝑑,π‘₯)βˆˆπ‘+×𝐑𝑑 of the nonlinear partial differential equation

πœ•π‘‘π‘’+divπ‘₯π‘“πœš(𝑑,π‘₯,𝑒)=πœ€divπ‘₯𝑏(βˆ‡π‘’)+𝛿𝑑𝑗=1πœ•3π‘₯𝑗π‘₯𝑗π‘₯𝑗𝑒(1.1) as πœ€β†’0 and 𝛿=𝛿(πœ€),𝜚=𝜚(πœ€)β†’0. Here π‘“βˆˆπΆ(𝐑;𝐡𝑉(𝐑+𝑑×𝐑𝑑π‘₯)) is the Caratheodory flux vector such that

max|𝑒|≀𝑙||π‘“πœš||(𝑑,π‘₯,𝑒)βˆ’π‘“(𝑑,π‘₯,𝑒)⟢0,𝜚⟢0,in𝐿𝑝loc𝐑+×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:

πœ•π‘‘π‘’+divπ‘₯𝑓(𝑑,π‘₯,𝑒)=0,𝑒=𝑒(𝑑,π‘₯),π‘₯βˆˆπ‘π‘‘,𝑑β‰₯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 KruΕΎhkov [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 (π‘’πœ€,𝛿)πœ€,𝛿 of solutions of

πœ•π‘‘π‘’+divπ‘₯𝑓(𝑑,π‘₯,𝑒)=πœ€divπ‘₯𝑏(βˆ‡π‘’)+𝛿𝑑𝑗=1πœ•3π‘₯𝑗π‘₯𝑗π‘₯𝑗𝑒,𝑒(π‘₯,0)=𝑒0(π‘₯),π‘₯βˆˆπ‘π‘‘.(2.1)

We assume that (π‘’πœ€,𝛿)πœ€,𝛿 has enough regularity so that all formal computations below are correct. So, following Schonbek [13], we assume that for every πœ€,𝛿>0 we have π‘’πœ€,π›ΏβˆˆπΏβˆž([0,𝑇];𝐻4(𝐑𝑑)).

Later on, we will assume that the initial data 𝑒0 depends on πœ€. In this section, we will determine a priori inequalities for the solutions of problem (2.1).

To simplify the notation we will write π‘’πœ€ instead of π‘’πœ€,𝛿.

We will need the following assumptions on the diffusion term 𝑏(πœ†)=(𝑏1(πœ†),…,𝑏𝑛(πœ†)).

(H1) For some positive constants 𝐢1,𝐢2 we have

𝐢1||πœ†||2β‰€πœ†β‹…π‘(πœ†)≀𝐢2||πœ†||2βˆ€πœ†βˆˆπ‘π‘‘.(2.2)

(H2) The gradient matrix 𝐷𝑏(πœ†) is a positive definite matrix, uniformly in πœ†βˆˆπ‘π‘‘, that is, for every πœ†,πœšβˆˆπ‘π‘‘, there exists a positive constant 𝐢3 such that we have

πœšπ‘‡π·π‘(πœ†)𝜚β‰₯𝐢3||𝜚||2.(2.3)

We use the following notation:

||𝐷2𝑒||2=𝑑𝑖,π‘˜=1||πœ•2π‘₯𝑖π‘₯π‘˜π‘’||2.(2.4) In the sequel, for a vector valued function 𝑔=(𝑔1,…,𝑔𝑑) defined on 𝐑+×𝐑𝑑×𝐑, we denote

||𝑔||2=𝑑𝑖=1||𝑔𝑖||2.(2.5) The partial derivative πœ•π‘₯𝑖 in the point (𝑑,π‘₯,𝑒), where 𝑒 possibly depends on (𝑑,π‘₯), is defined by the formula

πœ•π‘₯𝑖𝑔𝐷(𝑑,π‘₯,𝑒(𝑑,π‘₯))=π‘₯π‘–π‘”ξ€Έβˆ£(𝑑,π‘₯,πœ†)πœ†=𝑒(𝑑,π‘₯).(2.6) In particular, the total derivative 𝐷π‘₯𝑖 and the partial derivative πœ•π‘₯𝑖 are connected by the identity

𝐷π‘₯𝑖𝑔(𝑑,π‘₯,𝑒)=πœ•π‘₯𝑖𝑔(𝑑,π‘₯,𝑒)+πœ•π‘’π‘”(𝑑,π‘₯,𝑒)πœ•π‘₯𝑖𝑒.(2.7) Finally we use

divπ‘₯𝑔(𝑑,π‘₯,𝑒)=𝑑𝑖=1𝐷π‘₯𝑖𝑔𝑖𝑔(𝑑,π‘₯,𝑒),𝑔=1,…,𝑔𝑑,Ξ”π‘₯π‘ž(𝑑,π‘₯,𝑒)=𝑑𝑖=1𝐷2π‘₯𝑖π‘₯π‘–π‘ž(𝑑,π‘₯,𝑒),π‘žβˆˆπΆ2𝐑+×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 𝑒 and the spatial derivative of the flux 𝑓 are such that for some 𝐢,𝛼>0, 𝑝β‰₯1, and every 𝑙>0, we have

max||πœ†||<𝑙||𝑓𝑖||(𝑑,π‘₯,πœ†)βˆˆπΏπ‘ξ€·π‘+×𝐑𝑑,𝑖=1,…,𝑑,𝑑𝑖=1||πœ•π‘’π‘“π‘–||(𝑑,π‘₯,𝑒)≀𝐢,𝑑𝑖,𝑗=1||πœ•π‘₯𝑖𝑓𝑗||≀(𝑑,π‘₯,𝑒)πœ‡(𝑑,π‘₯)1+|𝑒|1+𝛼,(2.9) where πœ‡βˆˆβ„³(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(𝑑,π‘₯,𝑒) satisfies (H3) and that it is Lipschitz continuous on 𝐑+×𝐑𝑑×𝐑. Assume also that initial data 𝑒0 belongs to 𝐿2(𝐑𝑑). Under conditions (H1)-(H2) the sequence of solutions (π‘’πœ€)πœ€>0 of (2.1) for every π‘‘βˆˆ[0,𝑇] satisfies the following inequalities: ξ€œπ‘π‘‘||π‘’πœ€(||𝑑,π‘₯)2ξ€œπ‘‘π‘₯+πœ€π‘‘0ξ€œπ‘π‘‘||βˆ‡π‘’πœ€ξ€·π‘‘ξ…žξ€Έ||,π‘₯2𝑑π‘₯π‘‘π‘‘ξ…žβ‰€πΆ4ξƒ©ξ€œπ‘π‘‘||𝑒0||(π‘₯)2ξ€œπ‘‘π‘₯βˆ’π‘‘0ξ€œπ‘π‘‘ξ€œπ‘’πœ€ξ€·π‘‘β€²ξ€Έ0,π‘₯divπ‘₯π‘“ξ€·π‘‘ξ…žξ€Έ,π‘₯,𝑣𝑑𝑣𝑑π‘₯π‘‘π‘‘ξ…žξƒͺ,πœ€(2.10)2ξ€œπ‘π‘‘||||βˆ‡π‘’πœ€(𝑑,π‘₯)|2𝑑π‘₯+πœ€3ξ€œπ‘‘0ξ€œπ‘π‘‘||||𝐷2π‘’πœ€ξ€·π‘‘ξ…žξ€Έ|,π‘₯2𝑑π‘₯π‘‘π‘‘ξ…žβ‰€πΆ5ξƒ©πœ€2ξ€œπ‘π‘‘||βˆ‡π‘’0||(π‘₯)2ξ€œπ‘‘π‘₯+πœ€π‘‘0ξ€œπ‘π‘‘π‘‘ξ“π‘˜=1||πœ•π‘₯π‘˜π‘“ξ€·π‘‘ξ…ž,π‘₯,π‘’πœ€ξ€·π‘‘ξ…ž||,π‘₯ξ€Έξ€Έ2𝑑π‘₯π‘‘π‘‘ξ…ž+β€–β€–πœ•π‘’π‘“β€–β€–2𝐿∞(𝐑+×𝐑𝑑×𝐑)ξƒͺ,(2.11) for some constants 𝐢4 and 𝐢5.

Proof. We follow the procedure from [19]. Given a smooth function πœ‚=πœ‚(𝑒), π‘’βˆˆπ‘, we define π‘žπ‘–ξ€œ(𝑑,π‘₯,𝑒)=𝑒0πœ‚ξ…ž(𝑣)πœ•π‘£π‘“π‘–(𝑑,π‘₯,𝑣)𝑑𝑣,𝑖=1,…,𝑑.(2.12) If we multiply (2.1) by πœ‚ξ…ž(𝑒), it becomes πœ•π‘‘πœ‚ξ€·π‘’πœ€ξ€Έ+𝑑𝑖=1πœ•π‘₯π‘–π‘žπ‘–ξ€·π‘‘,π‘₯,π‘’πœ€ξ€Έβˆ’π‘‘ξ“π‘–=1ξ€œπ‘’πœ€0πœ•2π‘₯𝑖𝑣𝑓𝑖(𝑑,π‘₯,𝑣)πœ‚ξ…ž(𝑣)𝑑𝑣+𝑑𝑖=1πœ‚ξ…žξ€·π‘’πœ€ξ€Έπœ•π‘₯𝑖𝑓𝑖𝑑,π‘₯,π‘’πœ€ξ€Έ=πœ€π‘‘ξ“π‘–=1πœ•π‘₯π‘–ξ€·πœ‚ξ…žξ€·π‘’πœ€ξ€Έπ‘π‘–ξ€·βˆ‡π‘’πœ€ξ€Έξ€Έβˆ’πœ€πœ‚ξ…žξ…žξ€·π‘’πœ€ξ€Έπ‘‘ξ“π‘–=1π‘π‘–ξ€·βˆ‡π‘’πœ€ξ€Έπœ•π‘₯π‘–π‘’πœ€+𝛿𝑑𝑖=1πœ•π‘₯π‘–ξ€·πœ‚ξ…žξ€·π‘’πœ€ξ€Έπœ•2π‘₯𝑖π‘₯π‘–π‘’πœ€ξ€Έβˆ’π›Ώ2πœ‚ξ…žξ…žξ€·π‘’πœ€ξ€Έπ‘‘ξ“π‘–=1πœ•π‘₯π‘–ξ€·πœ•π‘₯π‘–π‘’πœ€ξ€Έ2.(2.13) Choosing here πœ‚(𝑒)=𝑒2/2 and integrating over [0,𝑑)×𝐑𝑑, we get ξ€œπ‘π‘‘||π‘’πœ€(||𝑑,π‘₯)2ξ€œπ‘‘π‘₯+πœ€π‘‘0ξ€œπ‘π‘‘βˆ‡π‘’πœ€ξ€·π‘‘ξ…žξ€Έξ€·,π‘₯β‹…π‘βˆ‡π‘’πœ€ξ€·π‘‘ξ…ž,π‘₯𝑑π‘₯π‘‘π‘‘ξ…ž=ξ€œπ‘π‘‘||𝑒0||(π‘₯)2𝑑π‘₯+𝑑𝑗=1ξ€œπ‘‘0ξ€œπ‘π‘‘ξ€œπ‘’πœ€ξ€·π‘‘β€²ξ€Έ0,π‘₯𝑣𝐷2π‘₯π‘—π‘£π‘“π‘—ξ€·π‘‘ξ…žξ€Έ,π‘₯,𝑣𝑑𝑣𝑑π‘₯π‘‘π‘‘ξ…žβˆ’π‘‘ξ“π‘–=1ξ€œπ‘‘0ξ€œπ‘π‘‘π‘’πœ€ξ€·π‘‘ξ…žξ€Έπœ•,π‘₯π‘₯π‘–π‘“π‘–ξ€·π‘‘ξ…ž,π‘₯,π‘’πœ€ξ€·π‘‘ξ…ž,π‘₯𝑑π‘₯π‘‘π‘‘ξ…ž=ξ€œπ‘π‘‘||𝑒0||(π‘₯)2𝑑π‘₯βˆ’π‘‘ξ“π‘–=1ξ€œπ‘‘0ξ€œπ‘π‘‘ξ€œπ‘’πœ€ξ€·π‘‘β€²ξ€Έ0,π‘₯πœ•π‘₯π‘–π‘“π‘–ξ€·π‘‘ξ…žξ€Έ,π‘₯,𝑣𝑑𝑣𝑑π‘₯π‘‘π‘‘ξ…ž,(2.14) where the second equality sign is justified by the following partial integration: ξ€œπ‘‘0ξ€œπ‘π‘‘ξ€œπ‘’πœ€0𝑣𝐷2π‘₯π‘—π‘£π‘“π‘—ξ€·π‘‘ξ…žξ€Έ,π‘₯,𝑣𝑑𝑣𝑑π‘₯π‘‘π‘‘ξ…ž=ξ€œπ‘‘0ξ€œπ‘π‘‘π‘’πœ€πœ•π‘₯π‘–π‘“π‘–ξ€·π‘‘ξ…ž,π‘₯,π‘’πœ€ξ€Έπ‘‘π‘₯π‘‘π‘‘ξ…žβˆ’ξ€œπ‘‘0ξ€œπ‘π‘‘ξ€œπ‘’πœ€0πœ•π‘₯π‘–π‘“π‘–ξ€·π‘‘ξ…žξ€Έ,π‘₯,𝑣𝑑𝑣𝑑π‘₯π‘‘π‘‘ξ…ž.(2.15)
Now inequality (2.10) follows from (2.14), using (H1).
As for inequality (2.11), we start by using (2.14), namely, ξ€œπ‘π‘‘||π‘’πœ€(||𝑑,π‘₯)2ξ€œπ‘‘π‘₯+πœ€π‘‘0ξ€œπ‘π‘‘βˆ‡π‘’πœ€ξ€·π‘‘ξ…žξ€Έξ€·,π‘₯β‹…π‘βˆ‡π‘’πœ€ξ€·π‘‘ξ…ž,π‘₯𝑑π‘₯π‘‘π‘‘ξ…ž=ξ€œπ‘π‘‘||𝑒0||(π‘₯)2𝑑π‘₯βˆ’π‘‘ξ“π‘–=1ξ€œπ‘‘0ξ€œπ‘π‘‘ξ€œπ‘’πœ€ξ€·π‘‘β€²ξ€Έ0,π‘₯πœ•π‘₯π‘–π‘“π‘–ξ€·π‘‘ξ…žξ€Έ,π‘₯,𝑣𝑑𝑣𝑑π‘₯π‘‘π‘‘ξ…žβ‰€ξ€œπ‘π‘‘||𝑒0||(π‘₯)2𝑑π‘₯+𝑑𝑖=1ξ€œπ‘‘0ξ€œπ‘π‘‘ξ€œπ‘’πœ€ξ€·π‘‘β€²ξ€Έ0,π‘₯||πœ•π‘₯π‘–π‘“π‘–ξ€·π‘‘ξ…žξ€Έ||,π‘₯,𝑣𝑑𝑣𝑑π‘₯π‘‘π‘‘ξ…žβ‰€ξ€œπ‘π‘‘||𝑒0||(π‘₯)2ξ€œπ‘‘π‘₯+𝑑0ξ€œπ‘π‘‘ξ€œπ‘πœ‡ξ€·π‘‘ξ…žξ€Έ,π‘₯1+|𝑣|1+𝛼𝑑𝑣𝑑π‘₯π‘‘π‘‘ξ…žβ‰€ξ€œπ‘π‘‘||𝑒0||(π‘₯)2ξ€œπ‘‘π‘₯+𝐢𝑑0ξ€œπ‘π‘‘πœ‡ξ€·π‘‘ξ…žξ€Έ,π‘₯𝑑π‘₯π‘‘π‘‘ξ…ž,(2.16) where ∫𝐢=𝐑(𝑑𝑣/(1+|𝑣|1+𝛼)).
From here, using (H3), we conclude in particular that πœ€ξ€œπ‘‘0ξ€œπ‘π‘‘||βˆ‡π‘’πœ€ξ€·π‘‘ξ…žξ€Έ||,π‘₯2𝑑π‘₯π‘‘π‘‘ξ…žβ‰€πΆ11,(2.17) for some constant 𝐢11 independent of πœ€.
Next, we differentiate (2.1) with respect to π‘₯π‘˜ and multiply the expression by πœ•π‘₯π‘˜π‘’. Integrating over 𝐑𝑑, using integration by parts and then summing over π‘˜=1,…,𝑑, we get: 12ξ€œπ‘π‘‘πœ•π‘‘||βˆ‡π‘’πœ€||2𝑑π‘₯βˆ’π‘‘ξ“π‘˜=1ξ€œπ‘π‘‘ξ€·βˆ‡πœ•π‘₯π‘˜π‘’πœ€ξ€Έβ‹…ξ€·πœ•π‘₯π‘˜π‘“π‘˜ξ€·π‘‘,π‘₯,π‘’πœ€ξ€Έ+πœ•π‘’π‘“π‘˜πœ•π‘₯π‘˜π‘’πœ€ξ€Έπ‘‘π‘₯=βˆ’πœ€π‘‘ξ“π‘˜=1ξ€œπ‘π‘‘ξ€·βˆ‡πœ•π‘₯π‘˜π‘’πœ€ξ€Έπ‘‡ξ€·π·π‘βˆ‡π‘’πœ€ξ€Έξ€·βˆ‡πœ•π‘₯π‘˜π‘’πœ€ξ€Έπ‘‘π‘₯.(2.18) Integrating this over [0,𝑑] and using the Cauchy-Schwarz inequality and condition (H2), we find 12ξ€œπ‘π‘‘||βˆ‡π‘’πœ€||(𝑑,β‹…)2𝑑π‘₯+πœ€πΆ3π‘‘ξ“π‘˜=1ξ€œπ‘‘0ξ€œπ‘π‘‘||βˆ‡πœ•π‘₯π‘˜π‘’πœ€||2𝑑π‘₯π‘‘π‘‘ξ…žβ‰€12ξ€œπ‘π‘‘||βˆ‡π‘’0||2𝑑π‘₯+π‘‘ξ“π‘˜=1β€–β€–βˆ‡ξ€·πœ•π‘₯π‘˜π‘’πœ€ξ€Έβ€–β€–πΏ2𝐑+Γ—π‘π‘‘ξ€Έβ€–β€–πœ•π‘₯π‘˜π‘“π‘˜ξ€·β‹…,β‹…,π‘’πœ€ξ€Έ+πœ•π‘’π‘“π‘˜πœ•π‘₯π‘˜π‘’πœ€β€–β€–πΏ2𝐑+×𝐑𝑑,(2.19) where 𝐢3 is independent of πœ€. Then, using Young's inequality (the constant 𝐢3 is the same as previously mentioned) πΆπ‘Žπ‘β‰€3πœ€2π‘Ž2+12𝐢3πœ€π‘2,π‘Ž,π‘βˆˆπ‘,(2.20) we obtain 12ξ€œπ‘π‘‘||βˆ‡π‘’πœ€||(𝑑,β‹…)2𝑑π‘₯+πœ€πΆ3π‘‘ξ“π‘˜=1ξ€œπ‘‘0ξ€œπ‘π‘‘||βˆ‡πœ•π‘₯π‘˜π‘’πœ€||2𝑑π‘₯π‘‘π‘‘ξ…žβ‰€12ξ€œπ‘π‘‘||βˆ‡π‘’0||2𝑑π‘₯+𝐢3πœ€2π‘‘ξ“π‘˜=1ξ€œπ‘‘0ξ€œπ‘π‘‘||βˆ‡πœ•π‘₯π‘˜π‘’πœ€||2𝑑π‘₯π‘‘π‘‘ξ…ž+12𝐢3πœ€ξ€œπ‘‘0ξ€œπ‘π‘‘π‘‘ξ“π‘˜=1||πœ•π‘₯π‘˜π‘“π‘˜ξ€·π‘‘ξ…ž,π‘₯,π‘’πœ€ξ€Έ+πœ•π‘’π‘“π‘˜πœ•π‘₯π‘˜π‘’πœ€||2𝑑π‘₯π‘‘π‘‘ξ…ž.(2.21) Multiplying this by πœ€2, using (π‘Ž+𝑏)2≀2π‘Ž2+2𝑏2, and applying (2.17), we conclude πœ€22ξ€œπ‘π‘‘||βˆ‡π‘’πœ€(||𝑑,β‹…)2𝑑π‘₯+𝐢3πœ€32ξ€œπ‘π‘‘ξ€œπ‘‘0||𝐷2π‘’πœ€||2𝑑π‘₯π‘‘π‘‘ξ…žβ‰€πœ€22ξ€œπ‘π‘‘||βˆ‡π‘’0||2𝑑π‘₯π‘‘π‘‘ξ…ž+πœ€πΆ3ξ€œπ‘‘0ξ€œπ‘π‘‘π‘‘ξ“π‘˜=1||πœ•π‘₯π‘˜π‘“π‘˜ξ€·π‘‘ξ…ž,π‘₯,π‘’πœ€ξ€·π‘‘ξ…ž||,π‘₯ξ€Έξ€Έ2𝑑π‘₯π‘‘π‘‘ξ…ž+𝐢11𝐢3β€–β€–πœ•π‘’π‘“π‘˜β€–β€–2πΏβˆžξ€·π‘+×𝐑𝑑×𝐑.(2.22) This inequality is actually inequality (2.11) when we take 𝐢5=2max{1,1/𝐢3,𝐢11/𝐢3}/min{1,𝐢3}.

3. The Multidimensional Case

Consider the following initial-value problem. Find 𝑒=𝑒(𝑑,π‘₯) such that

πœ•π‘‘π‘’+divπ‘₯𝑒𝑓(𝑑,π‘₯,𝑒)=0,(π‘₯,0)=𝑒0(π‘₯),π‘₯βˆˆπ‘π‘‘,(3.1) where 𝑒0∈𝐿2(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(𝑑,π‘₯,𝑒), (𝑑,π‘₯,𝑒)βˆˆπ‘+×𝐑𝑑×𝐑, we assume that π‘“βˆˆπΆ(𝐑;𝐡𝑉(𝐑+×𝐑𝑑)) and that for every π‘™βˆˆπ‘+ we have maxπ‘’βˆˆ[βˆ’π‘™,𝑙]|𝑓(𝑑,π‘₯,𝑒)|βˆˆπΏπ‘(𝐑+×𝐑𝑑), 𝑝>2 .

(H4b) There exists a sequence π‘“πœš=(𝑓1𝜚,…,π‘“π‘‘πœš), 𝜚∈(0,1), such that π‘“πœš=π‘“πœš(𝑑,π‘₯,𝑒)∈𝐢1(𝐑+×𝐑𝑑×𝐑), satisfying for some 𝑝>2 and every π‘™βˆˆπ‘+:

maxπ‘§βˆˆ[βˆ’π‘™,𝑙]||π‘“πœš||⟢(β‹…,β‹…,𝑧)βˆ’π‘“(β‹…,β‹…,𝑧)πœšβ†’00in𝐿𝑝𝐑+×𝐑𝑑=0,(3.2a)𝑑𝑖=1ξ€œπ‘+×𝐑𝑑||πœ•π‘₯π‘–π‘“π‘–πœš(||𝐢𝑑,π‘₯,𝑒)𝑑π‘₯𝑑𝑑≀11+|𝑒|1+π›Όπœš,(3.2b)𝑑𝑖=1,π‘˜ξ€œπ‘+×𝐑𝑑||πœ•π‘₯π‘˜π‘“π‘–πœš||(𝑑,π‘₯,𝑒)2𝐢𝑑π‘₯𝑑𝑑≀2,(3.2c)𝑑𝑖=1||πœ•π‘’π‘“π‘–πœš||≀𝐢(𝑑,π‘₯,𝑒)𝛽(𝜚),(3.2d)𝑑𝑖=1ξ€œπ‘+×𝐑𝑑||πœ•2π‘₯π‘–π‘’π‘“π‘–πœš||𝐢(𝑑,π‘₯,𝑒)𝑑π‘₯𝑑𝑑≀31+|𝑒|1+𝛼,(3.2e) where 𝐢𝑖, 𝑖=1,2,3, and 𝐢 are constants, while the function π›½βˆΆπ‘β†’π‘ is such that limπœŒβ†’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 πœ‘1∈𝐢∞0(𝐑+×𝐑𝑑) and πœ‘2∈𝐢∞0(𝐑) with total mass one denote πœ‘πœš1(𝑧,𝑒)=πœšπ‘‘+1πœ‘1ξ‚΅π‘§πœšξ‚Ά1πœ‘π›½(𝜚)2𝑒𝛽(𝜚),(3.3)π‘§βˆˆπ‘+×𝐑𝑑 and π‘’βˆˆπ‘, where 𝛽 is a positive function tending to zero as πœšβ†’0. In the case when the flux π‘“βˆˆπΆ(𝐑;𝐡𝑉(𝐑+×𝐑𝑑))βˆ©π΅π‘‰(𝐑×𝐑+×𝐑𝑑)) is bounded, straightforward computation shows that the sequence π‘“πœš=π‘“β‹†πœ‘πœš=(𝑓1𝜚,…,π‘“π‘‘πœš) satisfies (H4b) with 𝛽(𝜚)=𝜚.

We also need to assume that the flux 𝑓 is genuinely nonlinear, that is, for every (𝑑,π‘₯)βˆˆπ‘+×𝐑𝑑 and every πœ‰βˆˆπ‘π‘‘β§΅{0}, the mapping

π‘βˆ‹πœ†βŸΌπ‘‘ξ“π‘–=1π‘“π‘–πœ‰(𝑑,π‘₯,πœ†)𝑖||πœ‰||(3.4) is nonconstant on every nondegenerate interval of the real line.

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

πœ•π‘‘π‘’+divπ‘₯π‘“πœš(𝑑,π‘₯,𝑒)=πœ€divπ‘₯𝑏(βˆ‡π‘’)+𝛿𝑑𝑗=1πœ•3π‘₯𝑗π‘₯𝑗π‘₯𝑗𝑒,(3.5)𝑒(π‘₯,0)=𝑒0,πœ€(π‘₯),π‘₯βˆˆπ‘π‘‘,(3.6) where the flux π‘“πœš satisfies the conditions (H4b). We denote the solution of (3.5)-(3.6) by π‘’πœ€=π‘’πœ€(𝑑,π‘₯). We assume that

‖‖𝑒0,πœ€βˆ’π‘’0‖‖𝐿2ξ€·π‘π‘‘ξ€Έβ€–β€–π‘’βŸΆ0,0,πœ€β€–β€–πΏ2𝐑𝑑‖‖𝑒+πœ€0,πœ€β€–β€–π»1𝐑𝑑≀𝐢.(3.7) We also assume that 𝜚=𝜚(πœ€)β†’0 and 𝛿=𝛿(πœ€)β†’0 as πœ€β†’0. We want to prove that under certain conditions, a sequence of solutions (π‘’πœ€)πœ€>0 of (3.5)-(3.6) converges to a weak solution of problem (3.1) as πœ€β†’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(𝑑,π‘₯,𝑒) satisfies (H4). Also assume that the initial data 𝑒0 satisfies (3.7). Under these conditions the sequence of smooth solutions (π‘’πœ€)πœ€>0 of (3.5)-(3.6) satisfies the following inequalities for every π‘‘βˆˆ[0,𝑇]: ξ€œπ‘π‘‘||π‘’πœ€(||𝑑,π‘₯)2ξ€œπ‘‘π‘₯+πœ€π‘‘0ξ€œπ‘π‘‘||βˆ‡π‘’πœ€(||π‘₯,𝑠)2𝑑π‘₯𝑑𝑠≀𝐢4ξ‚΅ξ€œπ‘π‘‘||𝑒0,πœ€(||π‘₯)2𝑑π‘₯+𝐢10ξ‚Ά,πœ€(3.8)2ξ€œπ‘π‘‘||βˆ‡π‘’πœ€(||𝑑,π‘₯)2𝑑π‘₯+πœ€3ξ€œπ‘‘0ξ€œπ‘π‘‘||𝐷2π‘’πœ€ξ€·π‘‘ξ…žξ€Έ||,π‘₯2𝑑π‘₯π‘‘π‘‘ξ…žβ‰€πΆ5ξ‚΅πœ€2ξ€œπ‘π‘‘||βˆ‡π‘’0,πœ€(||π‘₯)2πœ€π‘‘π‘₯+𝜚𝐢11+𝐢12𝛽(𝜚)2ξ‚Ά,(3.9) for some constants 𝐢10,𝐢11,𝐢12 (the constants 𝐢4,𝐢5 are introduced in Theorem 2.1).

Proof. For every fixed 𝜚, the function π‘“πœš=(𝑓1𝜚,…,π‘“π‘‘πœš) is smooth, and, due to (H4), we see that π‘“πœš satisfies (H3). This means that we can apply Theorem 2.1.
Replacing the flux 𝑓 by π‘“πœš from (3.5) and 𝑒0 by 𝑒0,πœ€ from (3.6) in (2.10) and (2.11), we get ξ€œπ‘π‘‘||π‘’πœ€(||𝑑,π‘₯)2ξ€œπ‘‘π‘₯+πœ€π‘‘0ξ€œπ‘π‘‘||βˆ‡π‘’πœ€(||π‘₯,𝑠)2𝑑π‘₯𝑑𝑠≀𝐢3ξƒ©ξ€œπ‘π‘‘||𝑒0,πœ€||(π‘₯)2ξ€œπ‘‘π‘₯βˆ’π‘‘0ξ€œπ‘π‘‘ξ€œπ‘’πœ€ξ€·π‘‘β€²ξ€Έ0,π‘₯divπ‘₯π‘“πœšξ€·π‘‘ξ…žξ€Έ,π‘₯,𝑣𝑑𝑣𝑑π‘₯π‘‘π‘‘ξ…žξƒͺ,πœ€(3.10)2ξ€œπ‘π‘‘||βˆ‡π‘’πœ€(||𝑑,π‘₯)2𝑑π‘₯+πœ€3ξ€œπ‘‘0ξ€œπ‘π‘‘||𝐷2π‘’πœ€ξ€·π‘‘ξ…žξ€Έ||,π‘₯2𝑑π‘₯π‘‘π‘‘ξ…žβ‰€πΆ4ξƒ©πœ€2ξ€œπ‘π‘‘||βˆ‡π‘’0,πœ€||(π‘₯)2β€–β€–πœ•π‘‘π‘₯+π‘’π‘“πœšβ€–β€–2πΏβˆžξ€·π‘+Γ—π‘π‘‘ξ€ΈΓ—π‘ξ€œ+πœ€π‘‘0ξ€œπ‘π‘‘π‘‘ξ“π‘‘π‘˜=1𝑖=1ξ€Ίπœ•π‘₯π‘˜π‘“π‘–πœšξ€·π‘‘ξ…ž,π‘₯,π‘’πœ€ξ€·π‘‘ξ…ž,π‘₯ξ€Έξ€Έξ€»2𝑑π‘₯π‘‘π‘‘ξ…žξƒͺ.(3.11) To proceed, we use assumption (H4). We have ξ€œπ‘‘0ξ€œπ‘π‘‘ξ€œπ‘’πœ€ξ€·π‘‘β€²ξ€Έ0,π‘₯divπ‘“π‘–πœšξ€·π‘‘ξ…žξ€Έ,π‘₯,𝑣𝑑𝑣𝑑π‘₯π‘‘π‘‘ξ…žβ‰€ξ€œπ‘‘0ξ€œπ‘π‘‘ξ€œπ‘π‘‘ξ“π‘–=1||πœ•π‘₯π‘–π‘“π‘–πœšξ€·π‘‘ξ…žξ€Έ||β‰€ξ€œ,π‘₯,𝑣𝑑𝑣𝑑π‘₯𝑑𝑑𝐑𝐢11+|𝑣|1+𝛼𝑑𝑣≀𝐢10,(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 (π‘’πœ€)πœ€>0 of solutions to (3.5)-(3.6) in the case when

π‘ξ€·πœ†1,…,πœ†π‘‘ξ€Έ=ξ€·πœ†1,…,πœ†π‘‘ξ€Έ(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 πœƒ denote the Heaviside function.

Theorem 3.3 (see [10]). Assume that the vector 𝑓(𝑑,π‘₯,𝑒) is genuinely nonlinear in the sense of (3.4). Then each family (π‘£πœ€(𝑑,π‘₯))πœ€>0βŠ‚πΏβˆž(𝐑+×𝐑𝑑) such that for every π‘βˆˆπ‘ the distribution πœ•π‘‘ξ€·πœƒξ€·π‘£πœ€π‘£βˆ’π‘ξ€Έξ€·πœ€βˆ’π‘ξ€Έξ€Έ+divπ‘₯ξ€·πœƒξ€·π‘£πœ€π‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘£πœ€ξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)ξ€Έξ€Έ(3.14) is precompact in π»βˆ’1loc 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=πœ€,𝛿=πœ€2𝜌2(πœ€)with𝜌(πœ€)=π’ͺ(𝛽(πœ€)),(3.15) and that 𝑒0,πœ€ satisfies (3.7). Then, there exists a subsequence of the family (π‘’πœ€)πœ€>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 (π‘’πœ€)πœ€>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 𝐢2 function 𝑆=𝑆(𝑒), π‘’βˆˆπ‘, and multiply the regularized equation (3.5) by π‘†ξ…ž(π‘’πœ€). As usual, put ξ€œπ‘ž(𝑑,π‘₯,𝑒)=𝑒0π‘†ξ…ž(𝑣)πœ•π‘’π‘“πœšξ€·π‘žπ‘‘π‘£,π‘ž=1,…,π‘žπ‘‘ξ€Έ.(3.16)
We easily find that πœ•π‘‘π‘†ξ€·π‘’πœ€ξ€Έ+divπ‘₯π‘žξ€·π‘‘,π‘₯,π‘’πœ€ξ€Έβˆ’divπ‘₯π‘ž(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€+π‘†ξ…žξ€·π‘’πœ€ξ€Έdivπ‘₯π‘“πœš(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€=πœ€divπ‘₯ξ€·π‘†ξ…žξ€·π‘’πœ€ξ€Έβˆ‡π‘’πœ€ξ€Έβˆ’πœ€π‘†ξ…žξ…žξ€·π‘’πœ€ξ€Έ||βˆ‡π‘’πœ€||2+𝛿𝑑𝑗=1𝐷π‘₯π‘—ξ‚€π‘†ξ…žξ€·π‘’πœ€ξ€Έπœ•2π‘₯𝑗π‘₯π‘—π‘’πœ€ξ‚βˆ’π›Ώπ‘‘ξ“π‘—=1π‘†ξ…žξ…žξ€·π‘’πœ€ξ€Έπœ•π‘₯π‘—π‘’πœ€πœ•2π‘₯𝑗π‘₯π‘—π‘’πœ€.(3.17) We will apply this formula repeatedly with different choices for 𝑆(𝑒).
In order to apply Theorem 3.3, we will consider a truncated sequence (𝑇𝑙(π‘’πœ€))πœ€>0, where the truncation function 𝑇𝑙 is defined for every fixed π‘™βˆˆπ as π‘‡π‘™βŽ§βŽͺ⎨βŽͺ⎩(𝑒)=βˆ’π‘™,π‘’β‰€βˆ’π‘™,𝑒,βˆ’π‘™β‰€π‘’β‰€π‘™,𝑙,𝑒β‰₯𝑙.(3.18) We will prove that the sequence (𝑇𝑙(π‘’πœ€))πœ€>0 is precompact for every fixed 𝑙. Denote by 𝑒𝑙 a subsequential limit (in 𝐿1loc) of the family (𝑇𝑙(π‘’πœ€))πœ€>0, which gives raise to a new sequence (𝑒𝑙)𝑙>1 that we prove converges to a weak solution of (3.1).
To carry out this plan, we must replace 𝑇𝑙 by a 𝐢2 regularization 𝑇𝑙,πœŽβˆΆπ‘β†’π‘. We define 𝑇𝑙,πœŽβˆΆπ‘β†’π‘ by 𝑇𝑙,𝜎(0)=0 and π‘‡ξ…žπ‘™,𝜎(⎧βŽͺ⎨βŽͺβŽ©π‘’)=1,|𝑒|<𝑙,π‘™βˆ’|𝑒|+𝜎𝜎,𝑙<|𝑒|<𝑙+𝜎,0,|𝑒|>𝑙+𝜎.(3.19)
Next, we want to estimate β€–π‘‡ξ…žξ…žπ‘™,𝜎(π‘’πœ€)βˆ‡π‘’πœ€β€–πΏ2(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,πœŽξ‚ξ…žβŽ§βŽͺ⎨βŽͺ⎩(𝑒)=1,𝑒<𝑙,𝑙+πœŽβˆ’π‘’πœŽξ€·π‘‡,𝑙<𝑒<𝑙+𝜎,0,𝑒>𝑙+𝜎,(3.20)βˆ’π‘™,πœŽξ€Έξ…žβŽ§βŽͺ⎨βŽͺ⎩(𝑒)=1,𝑒>βˆ’π‘™,𝑙+𝜎+π‘’πœŽ,βˆ’π‘™βˆ’πœŽ<𝑒<βˆ’π‘™,0,𝑒<βˆ’π‘™βˆ’πœŽ.(3.21) Notice that 𝑇±𝑙,πœŽξ‚(𝑒)ξ…ž||𝑇≀1,±𝑙,𝜎||𝜎(𝑒)≀|𝑒|+2,𝑇+𝑙,𝜎(𝑒)=π‘‡βˆ’π‘™,𝜎(𝑒)forβˆ’π‘™β‰€π‘’β‰€π‘™.(3.22) By inserting 𝑆(𝑒)=βˆ’π‘‡+𝑙,𝜎(𝑒), π‘ž=π‘ž+∫(𝑑,π‘₯,𝑒)=βˆ’π‘’0(𝑇+𝑙,𝜎)ξ…ž(𝑣)πœ•π‘’π‘“πœšπ‘‘π‘£ in (3.17) and integrating over Π𝑑=[0,𝑑]×𝐑𝑑, we get βˆ’ξ€œπ‘π‘‘π‘‡+𝑙,πœŽξ€·π‘’πœ€ξ€Έξ€œπ‘‘π‘₯+𝐑𝑑𝑇+𝑙,πœŽξ€·π‘’0ξ€Έπœ€π‘‘π‘₯+πœŽξ€Ξ π‘‘βˆ©ξ€½π‘™<π‘’πœ€ξ€Ύ<𝑙+𝜎||βˆ‡π‘’πœ€||2=𝑑π‘₯𝑑𝑑Π𝑑divπ‘₯π‘ž+(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€ξ€π‘‘π‘₯𝑑𝑑+Π𝑑𝑇+𝑙,πœŽξ‚ξ…žξ€·π‘’πœ€ξ€Έdivπ‘₯π‘“πœš(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€βˆ’π›Ώπ‘‘π‘₯π‘‘π‘‘πœŽξ€Ξ π‘‘βˆ©ξ€½π‘™<π‘’πœ€ξ€Ύπ‘‘<𝑙+πœŽξ“π‘—=1πœ•π‘₯π‘—π‘’πœ€πœ•2π‘₯𝑗π‘₯π‘—π‘’πœ€π‘‘π‘₯𝑑𝑑.(3.23) Similarly, for 𝑆(𝑒)=π‘‡βˆ’π‘™,𝜎(𝑒), π‘ž=π‘žβˆ’βˆ«(𝑑,π‘₯,𝑒)=𝑒0(π‘‡βˆ’π‘™,𝜎)ξ…ž(𝑣)πœ•π‘’π‘“πœšπ‘‘π‘£, we have from (3.17) ξ€œπ‘π‘‘π‘‡βˆ’π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€œπ‘‘π‘₯βˆ’π‘π‘‘π‘‡βˆ’π‘™,πœŽξ€·π‘’0ξ€Έπœ€π‘‘π‘₯+πœŽξ€Ξ π‘‘βˆ©ξ€½βˆ’π‘™βˆ’πœŽ<π‘’πœ€ξ€Ύ<βˆ’π‘™||βˆ‡π‘’πœ€||2=𝑑π‘₯𝑑𝑑Π𝑑divπ‘₯π‘žβˆ’(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€ξ€π‘‘π‘₯π‘‘π‘‘βˆ’Ξ π‘‘ξ€·π‘‡βˆ’π‘™,πœŽξ€Έξ…žξ€·π‘’πœ€ξ€Έdivπ‘₯π‘“πœš(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€+𝛿𝑑π‘₯π‘‘π‘‘πœŽξ€Ξ π‘‘βˆ©ξ€½βˆ’π‘™βˆ’πœŽ<π‘’πœ€ξ€Ύπ‘‘<βˆ’π‘™ξ“π‘—=1πœ•π‘₯π‘—π‘’πœ€πœ•2π‘₯𝑗π‘₯π‘—π‘’πœ€π‘‘π‘₯𝑑𝑑.(3.24) Adding (3.23) to (3.24), we get πœ€πœŽξ€Ξ π‘‘βˆ©ξ€½||𝑒𝑙<πœ€||ξ€Ύ<𝑙+𝜎||βˆ‡π‘’πœ€||2ξ€œπ‘‘π‘₯𝑑𝑑=βˆ’π‘π‘‘ξ‚€π‘‡βˆ’π‘™,πœŽξ€·π‘’πœ€ξ€Έβˆ’π‘‡+𝑙,πœŽξ€·π‘’πœ€ξ€Έξ‚ξ€œπ‘‘π‘₯+π‘π‘‘ξ‚€π‘‡βˆ’π‘™,πœŽξ€·π‘’0ξ€Έβˆ’π‘‡+𝑙,πœŽξ€·π‘’0+𝑑π‘₯Π𝑑divπ‘₯π‘žβˆ’(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€ξ€π‘‘π‘₯𝑑𝑑+Π𝑑divπ‘₯π‘ž+(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€βˆ’ξ€π‘‘π‘₯π‘‘π‘‘Ξ π‘‘ξ€·π‘‡βˆ’π‘™,πœŽξ€Έξ…žξ€·π‘’πœ€ξ€Έdivπ‘₯π‘“πœš(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€ξ€π‘‘π‘₯𝑑𝑑+Π𝑑𝑇+𝑙,πœŽξ‚ξ…žξ€·π‘’πœ€ξ€Έdivπ‘₯π‘“πœš(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€+𝛿𝑑π‘₯π‘‘π‘‘πœŽξ€Ξ π‘‘βˆ©ξ€½βˆ’π‘™βˆ’πœŽ<π‘’πœ€ξ€Ύπ‘‘<βˆ’π‘™ξ“π‘—=1πœ•π‘₯π‘—π‘’πœ€πœ•2π‘₯𝑗π‘₯π‘—π‘’πœ€π›Ώπ‘‘π‘₯π‘‘π‘‘βˆ’πœŽξ€Ξ π‘‘βˆ©ξ€½π‘™<π‘’πœ€ξ€Ύπ‘‘<𝑙+πœŽξ“π‘—=1πœ•π‘₯π‘—π‘’πœ€πœ•2π‘₯𝑗π‘₯π‘—π‘’πœ€π‘‘π‘₯𝑑𝑑.(3.25) From (3.22) and the definition of π‘žβˆ’ and π‘ž+, it follows πœ€πœŽξ€Ξ π‘‘βˆ©ξ€½||𝑒𝑙<πœ€||ξ€Ύ<𝑙+𝜎||βˆ‡π‘’πœ€||2ξ€œπ‘‘π‘₯𝑑𝑑≀||π‘’πœ€||>𝑙2||π‘’πœ€||ξ€œπ‘‘π‘₯+||𝑒0||>𝑙2||𝑒0||𝑑π‘₯+2Ξ π‘‘ξ€œπ‘π‘‘ξ“π‘–=1||𝐷2π‘₯π‘–π‘£π‘“π‘–πœš(||𝑑,π‘₯,𝑣)𝑑𝑣𝑑π‘₯𝑑𝑑+2Π𝑑𝑑𝑖=1||πœ•π‘₯π‘–π‘“π‘–πœšξ€·π‘‘,π‘₯,π‘’πœ€ξ€Έ||𝛿𝑑π‘₯𝑑𝑑+2πœŽξ€Ξ π‘‘βˆ©ξ€½||π‘’π‘™βˆ’πœŽ<πœ€||𝑑<𝑙𝑗=1|||πœ•π‘₯π‘—π‘’πœ€πœ•2π‘₯𝑗π‘₯π‘—π‘’πœ€|||𝑑π‘₯𝑑𝑑.(3.26) Without loss of generality, we can assume that 𝑙>1. Having this in mind, we get from (H4) and (3.26) πœ€πœŽξ€Ξ π‘‘βˆ©ξ€½||𝑒𝑙<πœ€||ξ€Ύ<𝑙+𝜎||βˆ‡π‘’πœ€||2β‰€ξ€œπ‘‘π‘₯𝑑𝑑||π‘’πœ€||>𝑙2||π‘’πœ€||2ξ€œπ‘‘π‘₯+||𝑒0||>𝑙2||𝑒0||2ξ€œπ‘‘π‘₯+2𝐑𝑑𝑖=1𝐢31+|𝑣|1+𝛼𝑑𝑣+2Π𝑑𝑑𝑖=1||πœ•π‘₯π‘–π‘“π‘–πœšξ€·π‘‘,π‘₯,π‘’πœ€ξ€Έ||𝛿𝑑π‘₯𝑑𝑑+2πœŽξ€Ξ π‘‘βˆ©ξ€½||𝑒𝑙<πœ€||𝑑<𝑙+πœŽξ“π‘—=1|||πœ•π‘₯π‘—π‘’πœ€πœ•2π‘₯𝑗π‘₯π‘—π‘’πœ€|||β‰€ξ€œπ‘‘π‘₯𝑑𝑑𝐑𝑑2ξ‚€||π‘’πœ€||(π‘₯,𝑑)2+||𝑒0||(π‘₯,𝑑)2𝑑π‘₯+𝐾1+𝐾2𝛿+2πœŽπœ€2𝑑𝑖=1β€–β€–πœ€1/2πœ•π‘₯π‘–π‘’πœ€β€–β€–πΏ2𝐑+Γ—π‘π‘‘ξ€ΈΓ—β€–β€–πœ€3/2πœ•2π‘₯𝑖π‘₯π‘–π‘’πœ€β€–β€–πΏ2(𝐑+×𝐑𝑑)≀𝐾5+𝛿2𝜎2πœ€4(𝛽(𝜚))2+𝛿2𝜎2πœ€4ξ‚Ά1/2𝐾3𝐾4,(3.27) where 𝐾𝑖, 𝑖=1,…,5, are constants such that (cf. (3.8) and (3.9)) 2ξ€œπ‘π‘‘ξ“π‘–=1𝐢31+|𝑣|1+𝛼𝑑𝑣≀𝐾1,2Π𝑑𝑑𝑖=1||πœ•π‘₯π‘–π‘“π‘–πœšξ€·π‘‘,π‘₯,π‘’πœ€ξ€Έ||𝑑π‘₯𝑑𝑑≀𝐾2,𝑑𝑖=1β€–β€–πœ€1/2πœ•π‘₯π‘–π‘’πœ€β€–β€–πΏ2𝐑+×𝐑𝑑≀𝐾3,𝑑𝑖=1β€–β€–πœ€3/2πœ•2π‘₯𝑖π‘₯π‘–π‘’πœ€β€–β€–πΏ2𝐑+×𝐑𝑑≀1(𝛽(𝜚))2+πœ€πœšξ‚Ά1/2𝐾4,ξ€œπ‘π‘‘2ξ‚€||π‘’πœ€(||π‘₯,𝑑)2+||𝑒0(||π‘₯,𝑑)2𝑑π‘₯+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 πœ€=𝜚 from (3.15), we conclude π›ΏπœŽπœ€2β€–πœ€1/2βˆ‡π‘’πœ€β€–πΏ2𝐑+×𝐑𝑑𝑑𝑖=1β€–πœ€3/2πœ•2π‘₯𝑖π‘₯π‘–π‘’πœ€β€–πΏ2𝐑+×𝐑𝑑≀𝛿2𝜎2πœ€4𝛽2+𝛿(πœ€)2𝜎2πœ€4ξ‚Ά1/2𝐾3𝐾4.(3.29) Thus, in view of (3.27), πœ€πœŽξ€Ξ π‘‘βˆ©ξ€½||𝑒𝑙<πœ€||ξ€Ύ<𝑙+𝜎||βˆ‡π‘’πœ€||2𝑑π‘₯𝑑𝑑≀𝐾5+𝛿2𝜎2πœ€4𝛽2+𝛿(πœ€)2𝜎2πœ€4ξ‚Ά1/2𝐾3𝐾4,(3.30) which is the sought for estimate for β€–π‘‡ξ…žξ…žπ‘™,𝜎(π‘’πœ€)βˆ‡π‘’πœ€β€–πΏ2(𝐑+×𝐑𝑑).
Next, take a function π‘ˆπœŒ(𝑧) satisfying π‘ˆπœŒ(0)=0 and π‘ˆξ…žπœŒβŽ§βŽͺ⎨βŽͺβŽ©π‘§(𝑧)=0,𝑧<0,𝜌,0<𝑧<𝜌,1,𝑧>𝜌.(3.31) Clearly, π‘ˆπœŒ is convex, and π‘ˆξ…žπœŒ(𝑧)β†’πœƒ(𝑧) in 𝐿𝑝loc(𝐑) as πœŒβ†’0, for any 𝑝<∞; as before, πœƒ denotes the Heaviside function.
Inserting 𝑆(π‘’πœ€)=π‘ˆπœŒ(𝑇𝑙,𝜎(π‘’πœ€)βˆ’π‘) in (3.17), we get πœ•π‘‘π‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘+divπ‘₯ξ€œπ‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έπ‘‡(𝑣)βˆ’π‘ξ…žπ‘™,𝜎(𝑣)πœ•π‘£π‘“πœš=ξ€œ(𝑑,π‘₯,𝑣)π‘‘π‘£π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έπ‘‡(𝑣)βˆ’π‘ξ…žπ‘™,𝜎(𝑣)divπ‘₯πœ•π‘£π‘“πœš(𝑑,π‘₯,𝑣)π‘‘π‘£βˆ’π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έπ‘‡βˆ’π‘ξ…žπ‘™,πœŽξ€·π‘’πœ€ξ€Έdivπ‘₯π‘“πœš(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€+πœ€Ξ”π‘₯π‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘βˆ’πœ€π·2π‘’π‘’ξ€Ίπ‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έ||βˆ’π‘ξ€Έξ€»βˆ‡π‘’πœ€||2+𝛿𝑑𝑖=1𝐷π‘₯π‘–ξ€·π·π‘’ξ€Ίπ‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έπœ•βˆ’π‘ξ€Έξ€»2π‘₯𝑖π‘₯π‘–π‘’πœ€ξ€Έβˆ’π›Ώπ‘‘ξ“π‘–=1𝐷2π‘’π‘’ξ€Ίπ‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έπœ•βˆ’π‘ξ€Έξ€»π‘₯π‘–π‘’πœ€πœ•2π‘₯𝑖π‘₯π‘–π‘’πœ€.(3.32) We rewrite the previous expression in the following manner: πœ•π‘‘ξ€·πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘‡βˆ’π‘ξ€Έξ€·π‘™ξ€·π‘’πœ€ξ€Έβˆ’π‘ξ€Έξ€Έ+divπ‘₯ξ€·πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)ξ€Έξ€Έ=Ξ“1,πœ€+Ξ“2,πœ€+Ξ“3,πœ€+Ξ“4,πœ€+Ξ“5,πœ€+Ξ“6,πœ€+Ξ“7,πœ€,(3.33) where Ξ“1,πœ€=πœ•π‘‘ξ€·πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘‡βˆ’π‘ξ€Έξ€·π‘™ξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘βˆ’π‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έ,Ξ“βˆ’π‘ξ€Έξ€Έ2,πœ€=divπ‘₯ξ‚΅πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έβˆ’ξ€œξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έπ‘‡(𝑣)βˆ’π‘ξ…žπ‘™,𝜎(𝑣)πœ•π‘£π‘“πœšξ‚Ά,Ξ“(𝑑,π‘₯,𝑣)𝑑𝑣3,πœ€=ξ€œπ‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έπ‘‡(𝑣)βˆ’π‘ξ…žπ‘™,𝜎(𝑣)divπ‘₯πœ•π‘£π‘“πœš(𝑑,π‘₯,𝑣)π‘‘π‘£βˆ’π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έπ‘‡βˆ’π‘ξ…žπ‘™,πœŽξ€·π‘’πœ€ξ€Έdivπ‘₯π‘“πœš(𝑑,π‘₯,𝑣)βˆ£π‘£=π‘’πœ€,Ξ“4,πœ€=πœ€Ξ”π‘₯π‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘+𝛿𝑑𝑖=1𝐷π‘₯π‘–ξ€·π·π‘’ξ€Ίπ‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έπœ•βˆ’π‘ξ€Έξ€»2π‘₯𝑖π‘₯π‘–π‘’πœ€ξ€Έ,Ξ“5,πœ€=βˆ’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έπ‘‡βˆ’π‘ξ…žξ…žπ‘™,𝜎||||(π‘’πœ€)βˆ‡π‘’πœ€2,Ξ“6,πœ€=βˆ’π›Ώπ‘‘ξ“π‘–=1𝐷2π‘’π‘’ξ€Ίπ‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έπœ•βˆ’π‘ξ€Έξ€»π‘₯π‘–π‘’πœ€πœ•2π‘₯𝑖π‘₯π‘–π‘’πœ€,Ξ“7,πœ€=βˆ’πœ€π‘ˆπœŒξ…žξ…žξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘‡βˆ’π‘ξ€Έξ€·ξ…žπ‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έ2||βˆ‡π‘’πœ€||2.(3.34)
To continue, we assume that 𝜎 depends on πœ€ in the following way: 𝜎=𝜌=π’ͺ(𝛽(πœ€)).(3.35)
From here, we will prove that the sequence (𝑇𝑙(π‘’πœ€))πœ€>0 satisfies the assumptions of Theorem 3.3. Accordingly, we need to prove that the left-hand side of (3.33) is precompact in π»βˆ’1loc(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 divπ‘„πœ€, we have π‘„πœ€βˆˆπΏπ‘loc(𝐑+×𝐑𝑑) for 𝑝>2.
(ii) The right-hand side of (3.33) is of the form β„³loc,𝐡+π»βˆ’1loc,𝑐, where β„³loc,𝐡 denotes a set of families which are locally bounded in the space of measures, and π»βˆ’1loc,𝑐 is a set of families precompact in π»βˆ’1loc.
First, since 𝑇𝑙(π‘’πœ€) 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 Ξ“1,πœ€=πœ•π‘‘ξ€·πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘‡βˆ’π‘ξ€Έξ€·π‘™ξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘βˆ’π‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έβˆ’π‘ξ€Έξ€Έβˆˆπ»βˆ’1loc,𝑐.(3.36) We have πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘‡βˆ’π‘ξ€Έξ€·π‘™ξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘βˆ’π‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έξ€·π‘‡βˆ’π‘=πœƒπ‘™ξ€·π‘’πœ€ξ€Έπ‘‡βˆ’π‘ξ€Έξ€·π‘™ξ€·π‘’πœ€ξ€Έξ€Έξ€·π‘‡βˆ’π‘βˆ’πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘‡βˆ’π‘ξ€Έξ€·π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έξ€·π‘‡βˆ’π‘+πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘‡βˆ’π‘ξ€Έξ€·π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘βˆ’π‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έ.βˆ’π‘(3.37) Since the function πœƒ(π‘§βˆ’π‘)(π‘§βˆ’π‘) is Lipschitz continuous in 𝑧 with the Lipschitz constant one, and, according to definition of π‘ˆπœŒ, it holds |π‘ˆπœŒ(𝑧)βˆ’πœƒ(𝑧)𝑧|≀1/2𝜌, we conclude from the last expression ||πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘‡βˆ’π‘ξ€Έξ€·π‘™ξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘βˆ’π‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έ||≀||π‘‡βˆ’π‘π‘™ξ€·π‘’πœ€ξ€Έβˆ’π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έ||+π’ͺ(𝜌)≀π’ͺ(𝜎)+π’ͺ(𝜌).(3.38)
From this and assumptions (3.15) and (3.35) on 𝜎=𝜎(πœ€) and 𝜌=𝜌(πœ€), it follows that as πœ€β†’0πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘‡βˆ’π‘ξ€Έξ€·π‘™ξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘βˆ’π‘ˆπœŒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘βŸΆ0(3.39) in 𝐿𝑝loc for all 𝑝<∞. Thus, (since we can take 𝑝=2 as well) we see that Ξ“1,πœ€βˆˆπ»βˆ’1loc,𝑐.
Next, we will prove that Ξ“2,πœ€=divπ‘₯ξ‚΅πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έβˆ’ξ€œξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έπ‘‡(𝑣)βˆ’π‘ξ…žπ‘™,𝜎(𝑣)πœ•π‘£π‘“πœšξ‚Ά(𝑑,π‘₯,𝑣)π‘‘π‘£βˆˆπ»βˆ’1loc,𝑐+β„³loc,𝐡.(3.40) Indeed, πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έβˆ’ξ€œξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έπ‘‡(𝑣)βˆ’π‘ξ…žπ‘™,𝜎(𝑣)πœ•π‘£π‘“πœšξ€·π‘‡(𝑑,π‘₯,𝑣)𝑑𝑣=πœƒπ‘™ξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έξ€·π‘‡ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)βˆ’πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“βˆ’π‘ξ€Έξ€·πœšξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘“πœšξ€Έξ€·π‘‡(𝑑,π‘₯,𝑐)+πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έβˆ’ξ€œξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έπ‘‡(𝑣)βˆ’π‘ξ…žπ‘™(𝑣)πœ•π‘£π‘“πœšβˆ’ξ€œ(𝑑,π‘₯,𝑣)π‘‘π‘£π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽπ‘‡(𝑣)βˆ’π‘ξ€Έξ€·ξ…žπ‘™,𝜎(𝑣)βˆ’π‘‡ξ…žπ‘™ξ€Έπœ•(𝑣)π‘£π‘“πœš(𝑑,π‘₯,𝑣)𝑑𝑣.(3.41) Since 𝑇𝑙(𝑒)=𝑒 if |𝑒|≀𝑙 and π‘‡ξ…žπ‘™(𝑒)=0 if |𝑒|β‰₯𝑙, ξ€œπ‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έπ‘‡(𝑣)βˆ’π‘ξ…žπ‘™(𝑣)πœ•π‘£π‘“πœšξ€œ(𝑑,π‘₯,𝑣)𝑑𝑣=π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έπ‘‡(𝑣)βˆ’π‘ξ…žπ‘™(𝑣)πœ•π‘£π‘“πœšξ€·π‘‘,π‘₯,𝑇𝑙(𝑣)𝑑𝑣,(3.42) from which we conclude πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έβˆ’ξ€œξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έπ‘‡(𝑣)βˆ’π‘ξ…žπ‘™,𝜎(𝑣)πœ•π‘£π‘“πœšξ€·π‘‡(𝑑,π‘₯,𝑣)𝑑𝑣=πœƒπ‘™ξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έξ€·π‘‡ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)βˆ’πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έξ€·π‘‡ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)+πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έβˆ’ξ€œξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,𝜎(𝑇𝑣)βˆ’π‘ξ…žπ‘™(𝑣)πœ•π‘£π‘“πœšξ€·π‘‘,π‘₯,𝑇𝑙(ξ€Έβˆ’ξ€œπ‘£)π‘‘π‘£π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,𝜎(𝑇𝑣)βˆ’π‘ξ€Έξ€·ξ…žπ‘™,𝜎(𝑣)βˆ’π‘‡ξ…žπ‘™(ξ€Έπœ•π‘£)π‘£π‘“πœš(𝑇𝑑,π‘₯,𝑣)𝑑𝑣=πœƒπ‘™ξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έξ€·π‘‡ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)βˆ’πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έξ€·π‘‡ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)+πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έβˆ’ξ€œξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)π‘’πœ€πœƒξ€·π‘‡π‘™,πœŽξ€Έπ·(𝑣)βˆ’π‘π‘£ξ€Ίπ‘“πœšξ€·π‘‘,π‘₯,π‘‡π‘™βˆ’ξ€œ(𝑣)ξ€Έξ€»π‘‘π‘£π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽπ‘‡(𝑣)βˆ’π‘ξ€Έξ€·ξ…žπ‘™,𝜎(𝑣)βˆ’π‘‡ξ…žπ‘™ξ€Έπœ•(𝑣)π‘£π‘“πœšβˆ’ξ€œ(𝑑,π‘₯,𝑣)π‘‘π‘£π‘’πœ€ξ€·π‘ˆξ…žπœŒξ€·π‘‡π‘™,𝜎(𝑇𝑣)βˆ’π‘βˆ’πœƒπ‘™,𝜎(𝑇𝑣)βˆ’π‘ξ€Έξ€Έξ…žπ‘™(𝑣)πœ•π‘£π‘“πœšξ€·π‘‘,π‘₯,𝑇𝑙(𝑣)𝑑𝑣=Ξ“12,πœ€+Ξ“22,πœ€+Ξ“32,πœ€,(3.43) with Ξ“12,πœ€ξ€·π‘‡=πœƒπ‘™ξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έξ€·π‘‡ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)βˆ’πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έ,Ξ“ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)22,πœ€ξ€·π‘‡=πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έβˆ’ξ€œξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)π‘’πœ€πœƒξ€·π‘‡π‘™,πœŽξ€Έπ·(𝑣)βˆ’π‘π‘£ξ€Ίπ‘“πœšξ€·π‘‘,π‘₯,𝑇𝑙Γ(𝑣)𝑑𝑣,32,πœ€ξ€œ=βˆ’π‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽπ‘‡(𝑣)βˆ’π‘ξ€Έξ€·ξ…žπ‘™,𝜎(𝑣)βˆ’π‘‡ξ…žπ‘™ξ€Έπœ•(𝑣)π‘£π‘“πœšβˆ’ξ€œ(𝑑,π‘₯,𝑣)π‘‘π‘£π‘’πœ€ξ€·π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽξ€Έξ€·π‘‡(𝑣)βˆ’π‘βˆ’πœƒπ‘™,πœŽπ‘‡(𝑣)βˆ’π‘ξ€Έξ€Έξ…žπ‘™(𝑣)πœ•π‘£π‘“πœšξ€·π‘‘,π‘₯,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(𝑒)∈[βˆ’π‘™,𝑙], the function 𝑓(𝑑,π‘₯,𝑇𝑙(𝑒)) is uniformly continuous in π‘’βˆˆπ‘. Therefore, we have pointwise on 𝐑+×𝐑𝑑: ||Ξ“12,πœ€||=||πœƒξ€·π‘‡π‘™ξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έξ€·π‘‡ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)βˆ’πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έ||ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)⟢0aπ‘ πœŽβŸΆ0.(3.45) Since maxπ‘’βˆˆ[βˆ’π‘™,𝑙]𝑓(𝑑,π‘₯,𝑒)βˆˆπΏπ‘(𝐑+×𝐑𝑑), 𝑝>2, Lebesgue's dominated convergence theorem yields |Ξ“12,πœ€|=π‘œπœŽ,𝐿𝑝loc(1), where βˆ«π‘+×𝐑𝑑|π‘œπœŽ,𝐿𝑝(1)|𝑝𝑑π‘₯𝑑𝑑→0 as πœŽβ†’0. Thus, we conclude divπ‘₯Ξ“12,πœ€βˆˆπ»βˆ’1loc𝐑+×𝐑𝑑.(3.46)
We pass to Ξ“22,πœ€. We have to distinguish between different cases depending on the relative size of 𝑐 and 𝑙. Consider first the case when |𝑐|≀𝑙, in which case we have 𝑇𝑙(𝑐)=𝑐 and 𝑇𝑙,𝜎(𝑐)=𝑐. Thus, ||Ξ“22,πœ€||=||||πœƒξ€·π‘‡π‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έβˆ’ξ€œξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)π‘’πœ€πœƒξ€·π‘‡π‘™,𝜎(𝐷𝑣)βˆ’π‘π‘£ξ€Ίπ‘“πœšξ€·π‘‘,π‘₯,𝑇𝑙(||||=||||πœƒξ€·π‘‡π‘£)𝑑𝑣𝑙,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έξ€·π‘‡ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)βˆ’πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έξ€Έξ€œβˆ’π‘π‘’πœ€π‘π·π‘£ξ€Ίπ‘“πœšξ€·π‘‘,π‘₯,𝑇𝑙(||||=||πœƒξ€·π‘‡π‘£)𝑑𝑣𝑙,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έξ€·π‘‡ξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)βˆ’πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“βˆ’π‘ξ€Έξ€·πœšξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘“πœš(ξ€Έ||≀||πœƒξ€·π‘‡π‘‘,π‘₯,𝑐)𝑙,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘“πœšξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€||+||πœƒξ€·π‘‡ξ€Έξ€Έξ€Έπ‘™,πœŽξ€·π‘’πœ€ξ€Έβˆ’π‘ξ€Έξ€·π‘“(𝑑,π‘₯,𝑐)βˆ’π‘“πœšξ€Έ||≀||πœƒξ€·π‘‡(𝑑,π‘₯,𝑐)𝑙,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘“πœšξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€||+||πœƒξ€·π‘‡ξ€Έξ€Έξ€Έπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“βˆ’π‘ξ€Έξ€·πœšξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘“πœšξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€||+||πœƒξ€·π‘‡ξ€Έξ€Έξ€Έπ‘™,πœŽξ€·π‘’πœ€ξ€Έβˆ’π‘ξ€Έξ€·π‘“(𝑑,π‘₯,𝑐)βˆ’π‘“πœš(ξ€Έ||𝑑,π‘₯,𝑐)=π‘œπœš,𝐿𝑝locξ‚΅πœŽ(1)+π’ͺ𝛽(𝜚)+π‘œπœš,𝐿𝑝loc(1)=π’ͺ(1)+π‘œπœš,𝐿𝑝loc(1),(3.47) where π‘œπœš,𝐿𝑝loc(1) appears due to (3.2a), and π’ͺ(1) comes from (3.35).
For 𝑐>𝑙 we have 𝑐β‰₯𝑙+𝜎 for a 𝜎 small enough, and therefore πœƒ(𝑇𝑙,𝜎(π‘’πœ€)βˆ’π‘)≑0. On the other hand, for 𝑐<βˆ’π‘™ we have π‘β‰€βˆ’π‘™βˆ’πœŽ, and so πœƒ(𝑇𝑙,𝜎(π‘’πœ€)βˆ’π‘)≑1. Thus, the problematic case is when 𝑐<βˆ’π‘™. In this case, we have instead of (3.47) Ξ“22,πœ€ξ€·π‘‡=πœƒπ‘™,πœŽξ€·π‘’πœ€ξ€Έπ‘“ξ€·βˆ’π‘ξ€Έξ€·π‘‘,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έβˆ’ξ€œξ€Έξ€Έβˆ’π‘“(𝑑,π‘₯,𝑐)π‘’πœ€πœƒξ€·π‘‡π‘™,πœŽξ€Έπ·(𝑣)βˆ’π‘π‘£ξ€Ίπ‘“πœšξ€·π‘‘,π‘₯,𝑇𝑙(𝑣)𝑑𝑣=𝑓𝑑,π‘₯,𝑇𝑙,πœŽξ€·π‘’πœ€ξ€Έξ€Έβˆ’π‘“πœšξ€·π‘‘,π‘₯,π‘‡π‘™ξ€·π‘’πœ€ξ€Έξ€Έ+π‘“πœš(𝑑,π‘₯,βˆ’π‘™)βˆ’π‘“(𝑑,π‘₯,𝑐)(3.48) implying divπ‘₯Ξ“22,πœ€βˆˆπ»βˆ’1loc,𝑐+β„³loc,𝐡,(3.49) since 𝑓(𝑑,π‘₯,𝑇𝑙,𝜎(π‘’πœ€))βˆ’π‘“πœš(𝑑,π‘₯,𝑇𝑙(π‘’πœ€))β†’0 in 𝐿𝑝loc(𝐑+×𝐑𝑑) for 𝑝β‰₯2, and π‘“πœš(𝑑,π‘₯,βˆ’π‘™)βˆ’π‘“(𝑑,π‘₯,𝑐)βˆˆπ΅π‘‰(𝐑+×𝐑𝑑).
It remains to estimate Ξ“32,πœ€. Noticing that |π‘ˆξ…žπœŒ|,|π‘‡ξ…žπ‘™,𝜎|≀1, we get ||||ξ€œπ‘’πœ€π‘ˆξ…žπœŒξ€·π‘‡π‘™,πœŽπ‘‡(𝑣)βˆ’π‘ξ€Έξ€·ξ…žπ‘™,𝜎(𝑣)βˆ’π‘‡ξ…žπ‘™ξ€Έπœ•(𝑣)π‘£π‘“πœš||||≀𝐢(𝑑,π‘₯,𝑣)π‘‘π‘£ξ€œπ›½(𝜌)𝐑||π‘‡ξ…žπ‘™,𝜎(𝑣)βˆ’π‘‡ξ…žπ‘™||ξ‚΅πœŽ(𝑣)𝑑𝑣=π’ͺ𝛽(𝜚)