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ISRN Mathematical Physics
VolumeΒ 2012Β (2012), Article IDΒ 782306, 30 pages
http://dx.doi.org/10.5402/2012/782306
Research Article

The Inviscid Limits to Piecewise Smooth Solutions for a General Parabolic System

School of Mathematical Sciences, South China Normal University, Guang Zhou 510631, China

Received 13 October 2011; Accepted 26 October 2011

Academic Editors: U.Β Kulshreshtha and M.Β Znojil

Copyright Β© 2012 Shixiang Ma. 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 study the viscous limit problem for a general system of conservation laws. We prove that if the solution of the underlying inviscid problem is piecewise smooth with finitely many noninteracting shocks satisfying the entropy condition, then there exist solutions to the corresponding viscous system which converge to the inviscid solutions away from shock discontinuities at a rate of πœ€1 as the viscosity coefficient πœ€ vanishes.

1. Introduction

We consider the relation between the solutions, π‘’πœ€, of the system of viscous conservation lawsπ‘’πœ€π‘‘+𝑓(π‘’πœ€)π‘₯𝐡=πœ€(π‘’πœ€)π‘’πœ€π‘₯ξ€Έπ‘₯,π‘’πœ€βˆˆπ‘…π‘›,π‘₯βˆˆπ‘…,𝑑β‰₯0,πœ€>0,(1.1) and the distributional solution, 𝑒, of the corresponding system of conservation laws without viscosity𝑒𝑑+𝑓(𝑒)π‘₯=0,π‘’βˆˆπ‘…π‘›,π‘₯βˆˆπ‘…,𝑑>0.(1.2)

We assume that (1.2) is strictly hyperbolic, then by normalization, we have the decomposition𝐴≑𝐷𝑒𝑓(𝑒)=𝑅Λ𝐿,𝑅𝐿=𝐼,(1.3) where Ξ›=diag(πœ†1,πœ†2,…,πœ†π‘›) with πœ†1<πœ†2<β‹―<πœ†π‘›,𝐿=(𝑙1,…,𝑙𝑛)𝑑 is a matrix whose rows are left eigenvectors of 𝐴, and 𝑅=(π‘Ÿ1,…,π‘Ÿπ‘›) is a matrix whose columns are right eigenvectors of 𝐴.

For the zero dissipation limit problem, there are many significant works. When the Euler flow contains a single shock, Hoff and Liu [1] studied the isentropic case, they established the limit process from the solutions of the compressible Navier-Stokes equations to the single shock-wave solution of the corresponding compressible Euler system (so-called p-system). They show that the solutions to the isentropic Navier-Stokes equations with shock data exist and converge to the inviscid shocks as the viscosity vanishes, uniformly away from the shocks. Ignoring the initial layers, Goodman and Xin [2] gave a very detailed description of the asymptotic behavior of solutions for the general viscous systems as the viscosity tends to zero, via a method of matching asymptotics. This method can be applied to the Navier-Stokes equations (1.1), such as [3–5]. Later, Yu [6] revealed the rich structure of nonlinear wave interactions due to the presence of shocks and initial layers by a detailed pointwise analysis. As far as rarefaction wave is concerned, Xin in [7] has obtained that the solutions for the isentropic Navier-Stokes equations with weak centered rarefaction wave data exist for all time and converge to the weak centered rarefaction wave solution of the corresponding Euler system, as the viscosity tends to zero, uniformly away from the initial discontinuity. Moreover, in the case that either the initial layers are ignored or the rarefaction waves are smooth, he also obtains a rate of convergence which is valid uniformly for all time. Recently, Jiang et al. [8] improve the first part with weak centered srarefaction waves data and Zeng [9] improve the other results, respectively, in [7] to the full compressible Navier-Stokes equations, provided that the viscosity and heat-conductivity coefficients are in the same order. Furthermore, by a spectral analysis and Evans function method, Kevin Zumbrun and his collaborators have obtained many important results even for large amplitude and multidimensional case [10–14], and so forth. The case that the solutions to the Euler system containing contact discontinuity is much more subtle, there are few results in this respect[15–17].

In this paper, motivated by Goodman and Xin’s work [2], we establish that the piecewise smooth solutions, 𝑒, of (1.2), with finitely many noninteracting shocks satisfying the entropy condition, are strong limits as πœ€β†’0 of solutions, π‘’πœ€, of (1.1) when the matrix 𝐿𝐡𝑅 is positive definite.

For simplicity of presentation, we only consider the case in which 𝑒 is a single-shock solution.

Definition 1.1. A function 𝑒(π‘₯,𝑑) is called a single-shock solution of (1.2) up to time 𝑇 if:(i)𝑒(π‘₯,𝑑) is a distributional solution of the hyperbolic system (1.2) in the region 𝑅1Γ—[0,𝑇];(ii)there is a smooth curve, the shock, π‘₯=𝑠(𝑑),0≀𝑑≀𝑇, so that 𝑒(π‘₯,𝑑) is sufficiently smooth at any point π‘₯≠𝑠(𝑑);(iii)the limits πœ•π‘˜π‘₯𝑒(𝑠(𝑑)βˆ’0,𝑑)=limπ‘₯→𝑠(𝑑)βˆ’πœ•π‘˜π‘₯π‘’πœ•(π‘₯,𝑑),π‘˜π‘₯𝑒(𝑠(𝑑)+0,𝑑)=limπ‘₯→𝑠(𝑑)+πœ•π‘˜π‘₯𝑒(π‘₯,𝑑),(1.4)exist and are finite for 𝑑≀𝑇 and 0β‰€π‘˜β‰€5;(iv)the Lax geometrical entropy condition [18] is satisfied at π‘₯=𝑠(𝑑), that is,πœ†1(𝑒(𝑠(𝑑)βˆ’0,𝑑))<β‹…β‹…β‹…<πœ†π‘πœ†(𝑒(𝑠(𝑑)βˆ’0,𝑑)),𝑝𝑑(𝑒(𝑠(𝑑)+0,𝑑))<𝑑𝑑𝑠(𝑑)<πœ†π‘πœ†(𝑒(𝑠(𝑑)βˆ’0,𝑑)),𝑝(𝑒(𝑠(𝑑)+0,𝑑))<β‹…β‹…β‹…<πœ†π‘›(𝑒(𝑠(𝑑)+0,𝑑)).(1.5)

The main results of this paper are as follows.

Theorem 1.2. Suppose that the system (1.2) is strictly hyperbolic and that the 𝑝th characteristic family is genuinely nonlinear. There exist positive constants, πœ‡0 and πœ€0, such that if 𝑒(π‘₯,𝑑) is a single-shock solution up to time 𝑇 with 1β‰€π‘˜β‰€6ξ€œπ‘‡0ξ€œ||πœ•π‘˜π‘₯||𝑒(π‘₯,𝑑)2𝑑π‘₯𝑑𝑑<∞,πœ‡β‰‘sup0≀𝑑≀𝑇||||𝑒(𝑠(𝑑)+0,𝑑)βˆ’π‘’(𝑠(𝑑)βˆ’0,𝑑)β‰€πœ‡0,(1.6) then for each πœ€βˆˆ[0,πœ€0], there is a smooth solution, π‘’πœ€(π‘₯,𝑑), of (1.1) with π‘’πœ€π‘₯∈𝐢1ξ€·[]0,𝑇,𝐻1ξ€Έ.(1.7) Moreover, for any given πœ‚βˆˆ(0,1), sup0β‰€π‘‘β‰€π‘‡ξ€œ||π‘’πœ€(||π‘₯,𝑑)βˆ’π‘’(π‘₯,𝑑)2𝑑π‘₯β‰€πΆπœ‚πœ€πœ‚,(1.8)sup0≀𝑑≀𝑇,|π‘₯βˆ’π‘ (𝑑)|β‰₯πœ€πœ‚||π‘’πœ€||(π‘₯,𝑑)βˆ’π‘’(π‘₯,𝑑)β‰€πΆπœ‚πœ€,(1.9) where πΆπœ‚ is a positive constant depending only on πœ‚.

Notation 1. In this paper, we use 𝐻𝑙(𝑙β‰₯1) to denote the usual Sobolev space with the norm ‖⋅‖𝑙, and β€–β‹…β€–=β€–β‹…β€–0 denotes the usual 𝐿2-norm. We also use 𝑂(1) to denote any positive bounded function which is independent of πœ€.

2. Construction of the Approximate Solution

In this section, following the method of Goodman and Xin, in [2], we construct the approximate solution π‘£πœ€ through different scaling and asymptotic expansions in the region near and away from the shock respectively, such that π‘£πœ€ approximate the piecewise smooth inviscid solution 𝑒 away from the shock and has a sharp change near the shock.

2.1. Outer and Inner Expansions and the Matching Conditions

In the region away from the shock, π‘₯=𝑠(𝑑), we approximate the solution of (1.1) by truncation of the formal seriesπ‘’πœ€(π‘₯,𝑑)βˆΌπ‘’0(π‘₯,𝑑)+πœ€π‘’1(π‘₯,𝑑)+πœ€2𝑒2(π‘₯,𝑑)+β‹―.(2.1) Substituting this into (1.1) and comparing the coefficients of powers of πœ€, we get, for π‘₯≠𝑠(𝑑), that𝑂(1)βˆΆπ‘’0𝑑𝑒+𝑓0ξ€Έπ‘₯=0,(2.2)𝑂(πœ€)βˆΆπ‘’1𝑑+ξ€·π‘“ξ…žξ€·π‘’0𝑒1ξ€Έπ‘₯=𝐡𝑒0𝑒0π‘₯ξ€Έπ‘₯π‘‚ξ€·πœ€,(2.3)2ξ€ΈβˆΆπ‘’2𝑑+ξ€·π‘“ξ…žξ€·π‘’0𝑒2ξ€Έπ‘₯=𝐡𝑒0𝑒1π‘₯ξ€Έπ‘₯+ξ€·π΅ξ…žξ€·π‘’0𝑒1,𝑒0π‘₯ξ€Έξ€Έπ‘₯βˆ’12ξ€·π‘“ξ…žξ…žξ€·π‘’0𝑒1,𝑒1ξ€Έξ€Έπ‘₯,(2.4) and so forth. The outer functions 𝑒0,𝑒1,…, are generally discontinuous at the shock, π‘₯=𝑠(𝑑), but smooth up to the shock. The leading term, 𝑒0, is the single-shock solution of (1.2) which is given in the theorem.

Near the shock, π‘’πœ€ should be represented by an inner expansion:π‘’πœ€(π‘₯,𝑑)βˆΌπ‘ˆ0(πœ‰,𝑑)+πœ€π‘ˆ1(πœ‰,𝑑)+πœ€2π‘ˆ2(πœ‰,𝑑)+β‹―,(2.5) whereπœ‰=π‘₯βˆ’π‘ (𝑑)πœ€+𝛿(𝑑,πœ€),(2.6) and 𝛿(𝑑,πœ€) is the perturbation of the shock position to be determined later.

We assume that 𝛿(𝑑,πœ€) has the form𝛿(𝑑,πœ€)=𝛿0(𝑑)+πœ€π›Ώ1(𝑑)+πœ€2𝛿2(𝑑)+β‹―.(2.7) Substitute (2.5)–(2.7) into (1.1) to obtain𝑂1πœ€ξ‚βˆΆξ€·π΅ξ€·π‘ˆ0ξ€Έπ‘ˆ0πœ‰ξ€Έπœ‰+Μ‡π‘ π‘ˆ0πœ‰βˆ’ξ€·π‘“ξ€·π‘ˆ0ξ€Έξ€Έπœ‰π‘‚ξ€½π΅ξ€·π‘ˆ=0,(2.8)(1)∢0ξ€Έπ‘ˆ1πœ‰+π΅ξ…žξ€·π‘ˆ0π‘ˆξ€Έξ€·1,π‘ˆ0πœ‰ξ€Έξ€Ύπœ‰+Μ‡π‘ π‘ˆ1πœ‰βˆ’ξ€·π‘“ξ…žξ€·π‘ˆ0ξ€Έπ‘ˆ1ξ€Έπœ‰=π‘ˆ0𝑑+̇𝛿0π‘ˆ0πœ‰ξ€½π΅ξ€·π‘ˆ,(2.9)𝑂(πœ€)∢0ξ€Έπ‘ˆ2πœ‰+π΅ξ…žξ€·π‘ˆ0π‘ˆξ€Έξ€·2,π‘ˆ0πœ‰ξ€Έξ€Ύπœ‰+Μ‡π‘ π‘ˆ2πœ‰βˆ’ξ€·π‘“ξ…žξ€·π‘ˆ0ξ€Έπ‘ˆ2ξ€Έπœ‰,s=π‘ˆ1𝑑+̇𝛿1π‘ˆ0πœ‰+̇𝛿0π‘ˆ1πœ‰+12ξ€·π‘“ξ…žξ…žξ€·π‘ˆ0π‘ˆξ€Έξ€·1,π‘ˆ1ξ€Έξ€Έπœ‰βˆ’ξ€·π΅ξ…žξ€·π‘ˆ0π‘ˆξ€Έξ€·1,π‘ˆ1πœ‰ξ€Έξ€Έπœ‰,(2.10) and so forth, where ̇𝛿̇𝑠=𝑑𝑠/𝑑𝑑,0=𝑑𝛿0/𝑑𝑑, and so forth. The inner approximation is supposed to be valid in a small zone of size 𝑂(πœ€) around π‘₯=𝑠(𝑑).

In a matching zone, we expect the outer and the inner expansion agree with each other. Using the Taylor series to express the outer solutions in terms of πœ‰, we obtain the following β€œmatching conditions” as πœ‰β†’Β±βˆž:π‘ˆ0(πœ‰,𝑑)=𝑒0π‘ˆ(𝑠(𝑑)Β±0,𝑑)+π‘œ(1),(2.11)1(πœ‰,𝑑)=𝑒1ξ€·(𝑠(𝑑)Β±0,𝑑)+πœ‰βˆ’π›Ώ0ξ€Έπœ•π‘₯𝑒0π‘ˆ(𝑠(𝑑)Β±0,𝑑)+π‘œ(1),(2.12)2(πœ‰,𝑑)=𝑒2ξ€·(𝑠(𝑑)Β±0,𝑑)+πœ‰βˆ’π›Ώ0ξ€Έπœ•π‘₯𝑒1(𝑠(𝑑)Β±0,𝑑)βˆ’π›Ώ1πœ•π‘₯𝑒0+1(𝑠(𝑑)Β±0,𝑑)2ξ€·πœ‰βˆ’π›Ώ0ξ€Έ2πœ•2π‘₯𝑒0(𝑠(𝑑)Β±0,𝑑)+π‘œ(1),(2.13) and so forth.

2.2. The Structure of Viscous Shock Profiles

Our construction of the approximate solution depends on the properties of the viscous shock profiles, which are the solutions of the ordinary differential equation 𝐡(πœ™)πœ™πœ‰ξ€Έπœ‰=βˆ’πœŽπœ™πœ‰+𝑓(πœ™)πœ‰,(2.14) satisfying the boundary conditions πœ™(πœ‰)βŸΆπ‘’π‘™asπœ‰βŸΆβˆ’βˆž,πœ™(πœ‰)βŸΆπ‘’π‘Ÿasπœ‰βŸΆ+∞,(2.15) and moving with speed 𝜎: πœŽξ€·π‘’π‘™βˆ’π‘’π‘Ÿξ€Έξ€·π‘’=π‘“π‘™ξ€Έξ€·π‘’βˆ’π‘“π‘Ÿξ€Έ.(2.16) Integrate the differential equation to reduce that 𝐡(πœ™)πœ™πœ‰ξ€·πœ™=βˆ’πœŽ(πœ‰)βˆ’π‘’π‘™ξ€Έξ€·π‘’+𝑓(πœ™(πœ‰))βˆ’π‘“π‘™ξ€Έ.(2.17) It is well known that for a given state 𝑒 and the 𝑝 wave family, if |π‘’π‘™βˆ’π‘’|+|πœŽβˆ’πœ†π‘(𝑒)| is sufficiently small, then there exists a shock profile πœ™=πœ™(πœ‰,𝑒𝑙,𝜎), which connects 𝑒𝑙 and π‘’π‘Ÿ from left to right. Using the genuine nonlinearity, by similar arguments in [2], we can obtainπœ•πœ‰πœ†π‘||πœ•(πœ™)<0,βˆ€πœ‰,πœ‰πœ™||||πœ•β‰€π‘πœ‰πœ†π‘||||𝑒(πœ™)β‰€π‘π‘™βˆ’π‘’π‘Ÿ||.(2.18) And as πœ‰β†’βˆ’βˆž,πœ™ξ€·πœ‰,𝑒𝑙,πœŽβˆ’π‘’π‘™||𝑒=𝑂(1)π‘™βˆ’π‘’π‘Ÿ||π‘’βˆ’π›Ό|πœ‰|,πœ•πœ™πœ•π‘’π‘™βˆ’πΌ=𝑂(1)π‘’βˆ’π›Ό|πœ‰|,πœ•πœ™πœ•πœŽ=𝑂(1)π‘’βˆ’π›Ό|πœ‰|.(2.19) As πœ‰β†’+∞, πœ™ξ€·πœ‰,𝑒𝑙,πœŽβˆ’π‘’π‘Ÿ||𝑒=𝑂(1)π‘™βˆ’π‘’π‘Ÿ||π‘’βˆ’π›Ό|πœ‰|,πœ•πœ™πœ•π‘’π‘™βˆ’πœ•π‘’π‘Ÿπœ•π‘’π‘™=𝑂(1)π‘’βˆ’π›Ό|πœ‰|,πœ•πœ™βˆ’πœ•πœŽπœ•π‘’π‘Ÿπœ•πœŽ=𝑂(1)π‘’βˆ’π›Ό|πœ‰|.(2.20)

2.3. Solutions of the Outer and Inner Problems

Now we construct 𝑒𝑗 and π‘ˆπ‘— order by order.

The leading order outer function, 𝑒0, is the single-shock solution of the theorem. For any fixed 𝑑, the leading order inner solution π‘ˆ0(πœ‰,𝑑) is exactly the viscous shock profile with 𝑒𝑙(𝑑)≑𝑒(𝑠(𝑑)βˆ’0,𝑑),π‘’π‘Ÿ(𝑑)≑𝑒(𝑠(𝑑)+0,𝑑), and 𝜎=̇𝑠(𝑑). Soπ‘ˆ0ξ€·(πœ‰,𝑑)=πœ™πœ‰,𝑒𝑙(𝑑),̇𝑠(𝑑).(2.21) Here we take the shift to be zero since it can be absorbed into 𝛿0(𝑑).

Next we determine 𝑒1,π‘ˆ1, and 𝛿0(𝑑) together. Substituting (2.21) into (2.9) gives𝐡(πœ™)π‘ˆ1πœ‰+π΅ξ…žξ€·π‘ˆ(πœ™)1,πœ™πœ‰ξ€Έξ€Ύπœ‰+Μ‡π‘ π‘ˆ1πœ‰βˆ’ξ€·π‘“ξ…ž(πœ™)π‘ˆ1ξ€Έπœ‰=̇𝛿0(𝑑)πœ™πœ‰+πœ•πœ™πœ•π‘’π‘™Μ‡π‘’π‘™+πœ•πœ™πœ•Μ‡π‘ Μˆπ‘ .(2.22) By the matching condition (2.12), we expect that π‘ˆ1(πœ‰,𝑑)=πœ‰β‹…πœ•π‘₯𝑒0(𝑠(𝑑)Β±0,𝑑)+𝑂(1)asπœ‰βŸΆΒ±βˆž.(2.23) So we setπ‘ˆ1(πœ‰,𝑑)=𝑉1(πœ‰,𝑑)+𝐷1(πœ‰,𝑑),(2.24) where 𝐷1(πœ‰,𝑑) is a smooth function satisfying 𝐷1ξ‚»(πœ‰,𝑑)=πœ‰β‹…πœ•π‘₯𝑒0(𝑠(𝑑)βˆ’0,𝑑),πœ‰<βˆ’1,πœ‰β‹…πœ•π‘₯𝑒0(𝑠(𝑑)+0,𝑑),πœ‰>1.(2.25) Then inserting (2.24) into (2.22) and using (2.19)–(2.20) and the identity 𝑑𝑒𝑑𝑑0ξ€·(𝑠(𝑑)Β±0,𝑑)=Μ‡π‘ πΌβˆ’π‘“ξ…žξ€·π‘’0𝑒(𝑠(𝑑)Β±0,𝑑)ξ€Έξ€Έ0π‘₯(𝑠(𝑑)Β±0,𝑑),(2.26) we obtain 𝐡(πœ™)𝑉1πœ‰+π΅ξ…žξ€·π‘‰(πœ™)1,πœ™πœ‰ξ€Έξ€Ύπœ‰+̇𝑠𝑉1πœ‰βˆ’ξ€·π‘“ξ…ž(πœ™)𝑉1ξ€Έπœ‰=̇𝛿0(𝑑)πœ™πœ‰+𝑔(πœ‰,𝑑),(2.27) where |𝑔(πœ‰,𝑑)|≀𝑐exp{βˆ’π›Ό|πœ‰|} for large |πœ‰|. Define ∫𝐺(πœ‰,𝑑)=πœ‰0𝑔(πœ‚,𝑑)π‘‘πœ‚. Then we have𝐡(πœ™)𝑉1πœ‰+π΅ξ…žξ€·π‘‰(πœ™)1,πœ™πœ‰ξ€Έ+̇𝑠𝑉1βˆ’π‘“ξ…ž(πœ™)𝑉1=̇𝛿0(𝑑)πœ™+𝐺(πœ‰,𝑑)+𝑐(𝑑),(2.28) where 𝑐(𝑑)βˆˆπ‘…π‘› are integration constants to be determined later. We express 𝑉1 in terms of the basis, π‘Ÿ1(πœ™),π‘Ÿ2(πœ™),…,π‘Ÿπ‘›(πœ™), of the right eigenvectors of 𝑓′(πœ™). We write𝑉1(πœ‰,𝑑)=𝑛𝑗=1𝛼𝑗(πœ‰,𝑑)π‘Ÿπ‘—π‘’(πœ™(πœ‰,𝑑)),1(𝑠(𝑑)Β±0,𝑑)=𝑛𝑗=1𝛽𝑗±(𝑑)π‘Ÿπ‘—ξ€·π‘’0ξ€Έ,πœ•(𝑠(𝑑)Β±0,𝑑)π‘₯𝑒0(𝑠(𝑑)Β±0,𝑑)=𝑛𝑗=1𝛾𝑗±(𝑑)π‘Ÿπ‘—ξ€·π‘’0ξ€Έ.(𝑠(𝑑)Β±0,𝑑)(2.29) Here the π›½π‘—βˆ’ are for 𝑒1(𝑠(𝑑)βˆ’0,𝑑) and the 𝛽𝑗+ are for 𝑒1(𝑠(𝑑)+0,𝑑), and so forth. Then the matching conditions (2.12) are transformed intolimπœ‰β†’Β±βˆžπ›Όπ‘—(πœ‰,𝑑)=𝛽𝑗±(𝑑)βˆ’π›Ώ0(𝑑)𝛾𝑗±(𝑑),𝑗=1,…,𝑛.(2.30) Define πœŽπ‘—(πœ™)≑𝑙𝑗(πœ™)𝐡(πœ™)π‘Ÿπ‘—(πœ™)>0. Multiplying (2.28) by 𝑙𝑗(πœ™), and using (2.29), we obtainπ›Όπ‘—πœ‰+πœŽπ‘—(πœ™)βˆ’1𝑙𝑗(πœ™)π΅ξ…žξ€·π‘Ÿ(πœ™)𝑗(πœ™),πœ™πœ‰ξ€Έ+ξ€·Μ‡π‘ βˆ’πœ†π‘—π›Ό(πœ™)𝑗=πœŽπ‘—(πœ™)βˆ’1𝑙𝑗̇𝛿(πœ™)0ξ€Έβˆ’(𝑑)πœ™+𝐺(πœ‰,𝑑)+𝑐(𝑑)𝑛𝑖=1πœŽπ‘—(πœ™)βˆ’1𝛼𝑖(πœ‰,𝑑)𝑙𝑗(πœ™)𝐡(πœ™)π‘Ÿπ‘–πœ‰(πœ™),𝑗=1,…,𝑛,(2.31) and then we have the following result.

Lemma 2.1. There is a smooth solution, 𝛼(πœ‰,𝑑), to (2.31) with the following property: 𝛼𝑗(πœ‰,𝑑)=Μ‡π‘ βˆ’πœ†π‘—ξ€·π‘’π‘™ξ€Έξ€Έβˆ’1𝑙𝑗𝑒𝑙̇𝛿0𝑒𝑙+πΊβˆ’ξ€»ξ€½||πœ‰||ξ€Ύξ€·+𝑐(𝑑)+𝑂(1)expβˆ’π›Ό,πœ‰βŸΆβˆ’βˆž,Μ‡π‘ βˆ’πœ†π‘—ξ€·π‘’π‘Ÿξ€Έξ€Έβˆ’1π‘™π‘—ξ€·π‘’π‘ŸΜ‡π›Ώξ€Έξ€Ί0π‘’π‘Ÿ+𝐺+ξ€»ξ€½||πœ‰||ξ€Ύ+𝑐(𝑑)+𝑂(1)expβˆ’π›Ό,πœ‰βŸΆ+∞,(2.32) for 𝑗=1,…,𝑛, where 𝐺±=limπœ‰β†’Β±βˆžπΊ(πœ‰,𝑑), and 𝛼0 is a positive constant.

Proof. We use the standard iteration argument. Define 𝛼0(πœ‰,𝑑)≑0, and π›Όπ‘—π‘˜+1ξ€œ(πœ‰,𝑑)=πœ‰βˆ’βˆžξ‚»βˆ’ξ€œexpπœ‰πœ‚πœŽπ‘—(πœ™)βˆ’1𝑙𝑗(πœ™)π΅ξ…žξ€·π‘Ÿ(πœ™)𝑗(πœ™),πœ™πœ‰ξ€Έ+Μ‡π‘ βˆ’πœ†π‘—ξ€Έξ‚Όβ‹…ξƒ―πœŽ(πœ™)π‘‘πœπ‘—(πœ™)βˆ’1𝑙𝑗̇𝛿(πœ™)0ξ€Έβˆ’(𝑑)πœ™+𝐺(πœ‰,𝑑)+𝑐(𝑑)𝑛𝑖=1πœŽπ‘—(πœ™)βˆ’1𝛼𝑖(πœ‰,𝑑)𝑙𝑗(πœ™)𝐡(πœ™)π‘Ÿπ‘–πœ‰ξƒ°π›Ό(πœ™)π‘‘πœ‚,𝑗<𝑝,π‘π‘˜+1(ξ€œπœ‰,𝑑)=πœ‰0ξ‚»βˆ’ξ€œexpπœ‰πœ‚πœŽπ‘(πœ™)βˆ’1𝑙𝑝(πœ™)π΅ξ…ž(ξ€·π‘Ÿπœ™)𝑝(πœ™),πœ™πœ‰ξ€Έ+Μ‡π‘ βˆ’πœ†π‘(ξ€Έξ‚Όβ‹…ξƒ―πœŽπœ™)π‘‘πœπ‘(πœ™)βˆ’1𝑙𝑝̇𝛿(πœ™)0ξ€Έβˆ’(𝑑)πœ™+𝐺(πœ‰,𝑑)+𝑐(𝑑)𝑛𝑖=1πœŽπ‘(πœ™)βˆ’1𝛼𝑖(πœ‰,𝑑)𝑙𝑝(πœ™)𝐡(πœ™)π‘Ÿπ‘–πœ‰ξƒ°π›Ό(πœ™)π‘‘πœ‚,π‘—π‘˜+1ξ€œ(πœ‰,𝑑)=βˆ’πœ‰+βˆžξ‚»βˆ’ξ€œexpπœ‰πœ‚πœŽπ‘—(πœ™)βˆ’1𝑙𝑗(πœ™)π΅ξ…žξ€·π‘Ÿ(πœ™)𝑗(πœ™),πœ™πœ‰ξ€Έ+Μ‡π‘ βˆ’πœ†π‘—ξ€Έξ‚Όβ‹…ξƒ―πœŽ(πœ™)π‘‘πœπ‘—(πœ™)βˆ’1𝑙𝑗̇𝛿(πœ™)0ξ€Έβˆ’(𝑑)πœ™+𝐺(πœ‰,𝑑)+𝑐(𝑑)𝑛𝑖=1πœŽπ‘—(πœ™)βˆ’1𝛼𝑖(πœ‰,𝑑)𝑙𝑗(πœ™)𝐡(πœ™)π‘Ÿπ‘–πœ‰ξƒ°(πœ™)π‘‘πœ‚,𝑗>𝑝.(2.33) Set π‘€π‘˜=0≀𝑖≀𝑛sup[]𝑅×0,𝑇||π›Όπ‘˜π‘–(||π·πœ‰,𝑑),π‘˜=0,1,2,…,π‘˜=0≀𝑖≀𝑛sup[]𝑅×0,𝑇||π›Όπ‘–π‘˜+1(πœ‰,𝑑)βˆ’π›Όπ‘˜π‘–||(πœ‰,𝑑),π‘˜=0,1,2,….(2.34) Then from the lax entropy condition (1.5), we can obtain π·π‘˜β‰€π‘πœ‡π·π‘˜βˆ’1,π‘˜=1,2,…,(2.35) where 𝑐 is independent of πœ‡. And then for suitably small πœ‡, we have π‘€π‘˜β‰€π‘π‘€1,π‘˜=0,1,…,(2.36) and π›Όπ‘˜(πœ‰,𝑑) converges uniformly to a smooth bounded function, 𝛼(πœ‰,𝑑), which is a solution to (2.31). The asymptotic behavior of the solution, 𝛼(πœ‰,𝑑), follows from the formulas (2.33).
With Lemma 2.1 and the matching condition (2.30) at hand, we can determine, completely the same as in [2], 𝛽±(𝑑),𝛿0(𝑑), and 𝑐(𝑑), which guarantees the existence of π‘ˆ1(πœ‰,𝑑) and 𝑒1(π‘₯,𝑑). We give the sketch of this process. First, we use Lemma 2.1 and (2.30) for incoming indices to get a system of (𝑛+1) equations for 𝑛+1 unknowns, that is,π‘™π‘—ξ€·π‘’π‘ŸΜ‡π›Ώξ€Έξ€Ί0π‘’π‘Ÿξ€»=𝛽+𝑐(𝑑)𝑗+(𝑑)βˆ’π›Ώ0𝛾𝑗+(𝑑)ξ€Έξ€·Μ‡π‘ βˆ’πœ†π‘—ξ€·π‘’π‘Ÿξ€Έξ€Έβˆ’π‘™π‘—ξ€·π‘’π‘Ÿξ€ΈπΊ+𝑙,for1≀𝑗≀𝑝,(2.37)π‘—ξ€·π‘’π‘™ξ€Έξ€ΊΜ‡π‘ π‘’π‘Ÿξ€»=𝛽+𝑐(𝑑)π‘—βˆ’(𝑑)βˆ’π›Ώ0π›Ύπ‘—βˆ’(𝑑)ξ€Έξ€·Μ‡π‘ βˆ’πœ†π‘—ξ€·π‘’π‘™ξ€Έξ€Έ+π‘™π‘—ξ€·π‘’π‘™π‘’ξ€Έξ€·π‘Ÿβˆ’π‘’π‘™ξ€ΈΜ‡π›Ώ0βˆ’π‘™π‘—ξ€·π‘’π‘™ξ€ΈπΊβˆ’ξ€·π‘™,for(𝑝+1)≀𝑗≀𝑛,(2.38)π‘—ξ€·π‘’π‘™π‘’ξ€Έξ€·π‘™βˆ’π‘’π‘ŸΜ‡π›Ώξ€Έξ€Έ0=ξ€·π‘™π‘ξ€·π‘’π‘Ÿξ€Έβˆ’π‘™π‘ξ€·π‘’π‘™Μ‡π›Ώξ€Έξ€Έξ€·0π‘’π‘Ÿξ€Έ+ξ€·+𝑐(𝑑)Μ‡π‘ βˆ’πœ†π‘ξ€·π‘’π‘™π›½ξ€Έξ€Έπ‘βˆ’βˆ’ξ€·(𝑑)Μ‡π‘ βˆ’πœ†π‘ξ€·π‘’π‘Ÿπ›½ξ€Έξ€Έπ‘+(𝑑)+𝛿0𝛾𝑝+ξ€·Μ‡π‘ βˆ’πœ†π‘ξ€·π‘’π‘Ÿξ€Έξ€Έβˆ’π›Ύπ‘βˆ’ξ€·Μ‡π‘ βˆ’πœ†π‘ξ€·π‘’π‘™ξ€Έξ€Έξ€»+π‘™π‘ξ€·π‘’π‘Ÿξ€ΈπΊ+βˆ’π‘™π‘™ξ€·π‘’π‘™ξ€ΈπΊβˆ’.(2.39) Then we can solve for ̇𝛿0π‘’π‘Ÿ+𝑐(𝑑) from (2.37)-(2.38). Substituting the resulting expression into (2.39), by writing 𝛽in=(π›½π‘βˆ’,𝛽(𝑝+1)βˆ’,…,π›½π‘›βˆ’,𝛽1+,…,𝛽𝑝+), we arrive at an ordinary differential equation for 𝛿0: ̇𝛿0+𝐸1(𝑑)𝛿0=𝐸2(𝑑)⋅𝛽in+𝐺1(𝑑),(2.40) provided that πœ‡ is suitably small. Here 𝐸1(𝑑),𝐸2(𝑑), and 𝐺1(𝑑) are smooth known functions, and 𝐸1(𝑑) and 𝐸2(𝑑) remain bounded even as πœ‡β†’0+. Solving for 𝛿0 from (2.37) up to a constant, we obtain 𝑐(𝑑) uniquely in terms of 𝛽in. Then substitute the expression of 𝛿0 and 𝑐(𝑑) into the equation of the matching condition for outgoing indices to yield the linear relations 𝛽out=𝐸(𝑑)⋅𝛽in+𝐹(𝑑),(2.41) where 𝛽out=(𝛽1βˆ’,𝛽2βˆ’,…,𝛽(π‘βˆ’1)βˆ’,𝛽(𝑝+1)+,…,𝛽𝑛+), and 𝐹(𝑑)βˆˆπ‘…π‘›βˆ’1 is a smooth known function, 𝐸(𝑑) is a smooth (π‘›βˆ’1)Γ—(𝑛+1) matrix and remains bounded even as πœ‡β†’0+. Then the theory of linear hyperbolic equations [19, 20] shows that the problem (2.3), (2.41) has a solution smooth up to the shock provided that the initial value, 𝑒1(π‘₯,0), is chosen to satisfy the appropriate compatibility conditions at π‘₯=𝑠(0). Thus 𝑒1(π‘₯,𝑑) is completely determined, which in turn gives 𝛿0 and 𝑐(𝑑) by (2.37)-(2.38), and therefore π‘ˆ1(πœ‰,𝑑).

Now we summarize the above discussion to achieve the following.

Proposition 2.2. If πœ‡ is suitably small, then 𝑒1(π‘₯,𝑑),π‘ˆ1(πœ‰,𝑑) and 𝛿0 can be established such that(i)𝑒1(π‘₯,𝑑) and its derivatives are uniformly continuous up to π‘₯=𝑠(𝑑), and0β‰€π‘˜β‰€6ξ€œπ‘‡0ξ€œ||πœ•π‘˜π‘₯𝑒1||(π‘₯,𝑑)2𝑑π‘₯𝑑𝑑<∞.(2.42)(ii)π‘ˆ1(πœ‰,𝑑) and 𝛿0 are smooth functions, and there are an 𝛼>0, such thatπ‘ˆ1(πœ‰,𝑑)=𝑒1ξ€·(𝑠(𝑑)Β±0,𝑑)+πœ‰βˆ’π›Ώ0ξ€Έπœ•π‘₯𝑒0ξ€½||πœ‰||ξ€Ύ(𝑠(𝑑)Β±0,𝑑)+𝑂(1)expβˆ’π›Ό,asπœ‰βŸΆΒ±βˆž.(2.43)

The above constructions can be carried out to any order. In particular, we can determine 𝑒𝑖,π‘ˆπ‘– and π›Ώπ‘–βˆ’1(𝑖=2,3) simultaneously and similar results as in Proposition 2.2 hold for them.

2.4. Approximate Solutions

Now we can construct an approximate solution to (1.1) by patching the truncated outer and inner solutions in the previous discussion as in [2]. Define𝐼(π‘₯,𝑑)=πœ™π‘₯βˆ’π‘ (𝑑)πœ€+𝛿0+πœ€π›Ώ1+πœ€π›Ώ2ξ‚Ά+,𝑑3𝑖=1πœ€π‘–π‘ˆπ‘–ξ‚΅π‘₯βˆ’π‘ (𝑑)πœ€+𝛿0+πœ€π›Ώ1+πœ€π›Ώ2ξ‚Ά,,𝑑𝑂(π‘₯,𝑑)=𝑒0(π‘₯,𝑑)+πœ€π‘’1(π‘₯,𝑑)+πœ€2𝑒2(π‘₯,𝑑)+πœ€3𝑒3(π‘₯,𝑑).(2.44) Let π‘šβˆˆπΆβˆž0(𝑅) satisfy 0β‰€π‘š(𝑦)≀1, and ξ‚»||𝑦||||𝑦||π‘š(𝑦)=1,≀1,0,β‰₯2.(2.45) Let π›Ύβˆˆ(5/7,1) be a constant. Then we define the approximate solution to (1.1) asπ‘£πœ€ξ‚΅(π‘₯,𝑑)=π‘šπ‘₯βˆ’π‘ (𝑑)πœ€π›Ύξ‚Άξ‚΅ξ‚΅πΌ(π‘₯,𝑑)+1βˆ’π‘šπ‘₯βˆ’π‘ (𝑑)πœ€π›Ύξ‚Άξ‚Άπ‘‚(π‘₯,𝑑)+𝑑(π‘₯,𝑑),(2.46) where 𝑑(π‘₯,𝑑) is a higher-order correction term to be determined.

Using the structures of the various orders of inner and outer solutions, we compute thatπ‘£πœ€π‘‘+𝑓(π‘£πœ€)π‘₯ξ€·βˆ’πœ€π΅(π‘£πœ€)π‘£πœ€π‘₯ξ€Έπ‘₯=π‘›βˆ‘π‘–=3π‘žπ‘–(π‘₯,𝑑)+π‘‘π‘‘βˆ’ξ€·π΅(π‘šπΌ+(1βˆ’π‘š)𝑂)𝑑π‘₯ξ€Έπ‘₯βˆ’πœ€π‘ž4π‘₯(π‘₯,𝑑)+(𝑓(π‘£πœ€)βˆ’π‘“(π‘£πœ€βˆ’π‘‘))π‘₯,(2.47) whereπ‘ž1ξ€½[𝑓](π‘₯,𝑑)=(1βˆ’π‘š)(𝑂)βˆ’Ξ“(𝑓(𝑂))π‘₯ξ€Ίπ΅βˆ’πœ€(𝑂)𝑂π‘₯ξ€·π΅βˆ’Ξ“(𝑂)𝑂π‘₯ξ€Έξ€»π‘₯ξ€Ύ,π‘ž2ξ€½[](π‘₯,𝑑)=π‘šπ‘“(𝐼)βˆ’Ξ“(𝑓(𝐼))π‘₯ξ€Ίβˆ’πœ€π΅(𝐼)𝐼π‘₯ξ€·βˆ’Ξ“π΅(𝐼)𝐼π‘₯ξ€Έξ€»π‘₯+πœ€3π‘ˆ3𝑑+πœ€4̇𝛿2π‘ˆ1+̇𝛿1π‘ˆ2+̇𝛿0π‘ˆ3̇𝛿+πœ€2π‘ˆ2̇𝛿+πœ€1π‘ˆ3+πœ€2̇𝛿2π‘ˆ3ξ€Έπ‘₯ξ€Ύ,π‘ž3(π‘₯,𝑑)=π‘šπ‘‘(πΌβˆ’π‘‚)+π‘šπ‘₯+(𝑓(𝐼)βˆ’π‘“(𝑂)){𝑓(π‘šπΌ+(1βˆ’π‘š)𝑂)βˆ’(π‘šπ‘“(𝐼)+(1βˆ’π‘š)𝑓(𝑂))}π‘₯βˆ’πœ€π‘šπ‘₯𝐡(𝐼)𝐼π‘₯βˆ’π΅(𝑂)𝑂π‘₯ξ€Έ,π‘ž4(π‘₯,𝑑)=𝐡(π‘£πœ€)(π‘£πœ€βˆ’π‘‘)π‘₯βˆ’π‘šπ΅(𝐼)𝐼π‘₯βˆ’(1βˆ’π‘š)𝐡(𝑂)𝑂π‘₯+(𝐡(π‘£πœ€)βˆ’π΅(π‘£πœ€βˆ’π‘‘))𝑑π‘₯,(2.48) where Ξ“(𝑓(𝑂)),Ξ“(𝐡(𝑂)𝑂π‘₯) denote the truncated Taylor's expansion of 𝑓(𝑂),𝐡(𝑂)𝑂π‘₯, respectively, at 𝑒0, including all the terms of 𝑂(1),𝑂(1)πœ€,𝑂(1)πœ€2,𝑂(1)πœ€3, and Ξ“(𝑓(𝐼)),Ξ“(𝐡(𝐼)𝐼π‘₯) denote the truncated Taylor's expansion of 𝑓(𝐼),𝐡(𝐼)𝐼π‘₯, respectively, at πœ™, including all the terms of 𝑂(1),𝑂(1)πœ€,𝑂(1)πœ€2,𝑂(1)πœ€3.

In view of our construction, we have(i)supp π‘ž1(π‘₯,𝑑)βŠ†{(π‘₯,𝑑)∢|π‘₯βˆ’π‘ (𝑑)|β‰₯πœ€π›Ύ,0≀𝑑≀𝑇}, and πœ•π‘™π‘₯π‘ž1(π‘₯,𝑑)=𝑂(1)πœ€4βˆ’π‘™π›Ύ,ξ‚΅ξ€œπ‘‡0β€–β€–π‘ž1β€–β€–(β‹…,𝑑)2𝑑𝑑1/2≀𝑂(1)πœ€4,ξ‚΅ξ€œπ‘‡0β€–β€–πœ•π‘™π‘₯π‘ž1β€–β€–(β‹…,𝑑)2𝑑𝑑1/2≀𝑂(1)πœ€4βˆ’(π‘™βˆ’(1/2))𝛾,𝑙=1,2,3,(2.49)(ii)supp π‘ž2(π‘₯,𝑑)βŠ†{(π‘₯,𝑑)∢|π‘₯βˆ’π‘ (𝑑)|≀2πœ€π›Ύ,0≀𝑑≀𝑇}, and πœ•π‘™π‘₯π‘ž2(π‘₯,𝑑)=𝑂(1)πœ€(3βˆ’π‘™)𝛾,𝑙=0,1,2,3,(2.50)(iii)supp π‘ž3(π‘₯,𝑑)βŠ†{(π‘₯,𝑑)βˆΆπœ€π›Ύβ‰€|π‘₯βˆ’π‘ (𝑑)|≀2πœ€π›Ύ,0≀𝑑≀𝑇}, and πœ•π‘™π‘₯π‘ž3(π‘₯,𝑑)=𝑂(1)πœ€(3βˆ’π‘™)𝛾,𝑙=0,1,2,3,(2.51)

where we have used the estimates πœ•π‘™π‘₯(πΌβˆ’π‘‚)=𝑂(1)πœ€(4βˆ’π‘™)𝛾 on {(π‘₯,𝑑)βˆΆπœ€π›Ύβ‰€|π‘₯βˆ’π‘ (𝑑)|≀2πœ€π›Ύ},𝑙=0,1,2,3.

We now choose 𝑑(π‘₯,𝑑) to satisfyπ‘‘π‘‘βˆ’ξ€·π΅(π‘šπΌ+(1βˆ’π‘š)𝑂)𝑑π‘₯ξ€Έπ‘₯=βˆ’3𝑖=1π‘žπ‘–(π‘₯,𝑑),𝑑(π‘₯,0)=0,(2.52) so that π‘£πœ€ satisfiesπ‘£πœ€π‘‘+𝑓(π‘£πœ€)π‘₯𝐡=πœ€(π‘£πœ€)π‘£πœ€π‘₯ξ€Έπ‘₯βˆ’πœ€π‘ž4π‘₯+(𝑓(π‘£πœ€)βˆ’π‘“(π‘£πœ€βˆ’π‘‘)).(2.53) Since 𝐡 is smooth and positive definite, by the standard energy estimates for the linear parabolic system and Sobolev’s inequalities, we have the following results.

Lemma 2.3. Let 𝑑(π‘₯,𝑑) be the solution of (2.52). Then the following estimates hold for all π‘‘βˆˆ[0,𝑇]: β€–β€–πœ•π‘™π‘₯‖‖𝑑(β‹…,𝑑)πΏβˆžβ‰€π‘‚(1)πœ€(7/2)π›Ύβˆ’(𝑙+(1/2))β€–β€–πœ•,for𝑙=0,1,2,3,𝑙π‘₯𝑑‖‖(β‹…,𝑑)≀𝑂(1)πœ€(7/2)π›Ύβˆ’π‘™,𝑙=0,1,2,3,4.(2.54) Then for π‘ž4, we have β€–β€–πœ•π‘™π‘₯π‘ž4‖‖≀𝑂(1)πœ€(7/2)π›Ύβˆ’(𝑙+1),𝑙=0,1,2.(2.55)

And by our construction, we obtain the following.

Lemma 2.4. One has π‘£πœ€ξ‚»π‘’(π‘₯,𝑑)=0||||(π‘₯,𝑑)+𝑂(1)πœ€,π‘₯βˆ’π‘ (𝑑)β‰₯πœ€π›Ύ,πœ™(πœ‰,𝑑)+𝑂(1)πœ€π›Ύ,||||π‘₯βˆ’π‘ (𝑑)≀2πœ€π›Ύ,πœ•π‘£πœ€=1πœ•π‘₯πœ€π‘šπœ•πœ‰πœ™+𝑂(1),πœ•π‘£πœ€πœ•π‘‘=𝑂(1).(2.56)

3. Stability Analysis

We now show that there exists an exact solution to (1.1) in a neighborhood of the approximate solution π‘£πœ€(π‘₯,𝑑), and that the asymptotic behavior of the viscous solution is given by π‘£πœ€ for small viscosity πœ€.

Suppose that π‘’πœ€(π‘₯,𝑑) is the exact solution to (1.1) with the initial data π‘’πœ€(π‘₯,0)=π‘£πœ€(π‘₯,0). We decompose the solution asπ‘’πœ€(π‘₯,𝑑)=π‘£πœ€(π‘₯,𝑑)+[].𝑀(π‘₯,𝑑),(π‘₯,𝑑)βˆˆπ‘…Γ—0,𝑇(3.1) Then using the relation (2.53) for π‘£πœ€, we compute that𝑀𝑑+ξ€·π‘“ξ…ž(π‘£πœ€)𝑀π‘₯𝑣+π‘„πœ€,𝑀π‘₯𝐡=πœ€(π‘’πœ€)π‘’πœ€π‘₯βˆ’π΅(π‘£πœ€)π‘£πœ€π‘₯ξ€Έπ‘₯+πœ€π‘ž4π‘₯+(𝑓(π‘£πœ€βˆ’π‘‘)βˆ’π‘“(π‘£πœ€))π‘₯,𝑀(π‘₯,0)=0,(3.2) where 𝑄(π‘£πœ€,𝑀)=𝑓(π‘’πœ€)βˆ’π‘“(π‘£πœ€)βˆ’π‘“ξ…ž(π‘£πœ€)𝑀 satisfies |𝑄|≀𝑂(1)𝑀 for small 𝑀.

Set 𝑀𝑀(π‘₯,𝑑)=π‘₯(π‘₯,𝑑) in (3.2) and integrate the resulting equation with respect to π‘₯ to give𝑀𝑑+ξ€·π‘“ξ…ž(π‘£πœ€)𝑀π‘₯𝑣+π‘„πœ€,𝑀𝐡=πœ€(π‘’πœ€)π‘’πœ€π‘₯βˆ’π΅(π‘£πœ€)π‘£πœ€π‘₯ξ€Έ+πœ€π‘ž4+𝑓(π‘£πœ€βˆ’π‘‘)βˆ’π‘“(π‘£πœ€ξ‚),𝑀(π‘₯,0)=0,(3.3) by making the following scalings,𝑀(π‘₯,𝑑)=πœ€π‘€(𝑦,𝜏),𝑦=π‘₯βˆ’π‘ (𝑑)πœ€π‘‘,𝜏=πœ€,(3.4) we transform (3.3) into π‘€πœβˆ’Μ‡π‘ π‘€π‘¦+π‘“ξ…ž(π‘£πœ€)𝑀𝑦𝑣+π‘„πœ€,𝑀𝑦=𝐡(π‘£πœ€)𝑀𝑦𝑦+(𝐡(π‘’πœ€)βˆ’π΅(π‘£πœ€))π‘’πœ€π‘¦+πœ€π‘ž4+𝑓(π‘£πœ€βˆ’π‘‘)βˆ’π‘“(π‘£πœ€),𝑀(𝑦,0)=0.(3.5) Then we only need to show that for suitably small πœ€, (3.5) has a unique β€œsmall” smooth solution up to 𝜏=𝑇/πœ€. By the standard existence and uniqueness theory, and the continuous induction argument for parabolic equations [21], this will follow from the following a priori estimate.

Proposition 3.1. Suppose that the Cauchy problem (3.5) has a solution π‘€βˆˆπΆ1([0,𝜏0]∢𝐻3(𝑅1)) for some 𝜏0∈(0,𝑇/πœ€]. Then there exist positive constants πœ‡1,πœ€1 and 𝐢, which are independent of πœ€ and 𝜏0, such that if 0<πœ€<πœ€1,sup0β‰€πœβ‰€πœ0‖𝑀(β‹…,𝜏)β€–3+πœ‡β‰€πœ‡1,(3.6) then sup0β‰€πœβ‰€πœ0‖‖𝑀(β‹…,𝜏)23+ξ€œπœ00‖‖𝑀𝑦‖‖(β‹…,𝜏)23π‘‘πœβ‰€πΆπœ€7π›Ύβˆ’3,(3.7) where 𝛾 is defined in Section 2.4.

The proof of the proposition occupies the rest of this section. We separate it into several parts. First we diagonalize the system (3.5). Defineπœƒ(𝑦,𝜏)=𝐿(π‘£πœ€)ξ€·πœ•π‘€(𝑦,𝜏),𝑀(𝑦,𝜏)=𝑦𝐿(π‘£πœ€)⋅𝑅(π‘£πœ€ξ€·πœ•),𝑁(𝑦,𝜏)=𝜏𝐿(π‘£πœ€)⋅𝑅(π‘£πœ€).(3.8) Then we haveπœƒπ‘¦=π‘€πœƒ+𝐿𝑀𝑦,πœƒπœ=π‘πœƒ+πΏπ‘€πœ,πœƒπ‘¦π‘¦=(π‘€πœƒ)𝑦+π‘€πœƒπ‘¦βˆ’π‘€2πœƒ+𝐿𝑀𝑦𝑦.(3.9) Using the identity (3.9), we can rewrite (3.5) asπœƒπœ+(Ξ›βˆ’Μ‡π‘ )πœƒπ‘¦+𝑣(Μ‡π‘ βˆ’Ξ›)π‘€πœƒβˆ’π‘πœƒ+πΏπ‘„πœ€,π‘…πœƒπ‘¦+π‘…π‘¦πœƒξ€Έ=𝐿(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)ξ€·πœƒπ‘¦π‘¦βˆ’(π‘€πœƒ)π‘¦βˆ’π‘€πœƒπ‘¦+𝑀2πœƒξ€Έ+𝐿(π‘£πœ€)(𝐡(π‘’πœ€)βˆ’π΅(π‘£πœ€))π‘’πœ€π‘¦+πœ€πΏ(π‘£πœ€)π‘ž4βˆ’Ξ›πΏ(π‘£πœ€)𝑑+𝐿(π‘£πœ€)𝑄1(π‘£πœ€,𝑑).(3.10)

In what follows, we use 𝑐 to denote any positive constant which is independent of πœ€,𝑦, and 𝜏;𝑐 to denote any positive constant which is independent of πœ€ and πœ‡. And we set πœ€β‰€1.

Now we do the following estimates on transversal waves.

Lemma 3.2. There exist suitably small positive constants πœ‡2, πœ€2, independent of πœ€ and 𝜏0, such that ξ“π‘˜β‰ π‘ξ€œ||πœ•π‘¦πœ†π‘||πœƒ(πœ™)2π‘˜π‘‘π‘‘π‘¦β‰€ξƒ¬βˆ’ξ“π‘‘πœπ‘˜β‰ π‘ξ€œξ‚΅πœ†π‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ 2π‘˜ξƒ­+ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œπ‘š||πœ•π‘¦πœ†π‘||πœƒ(πœ™)2𝑝+ξ€·π‘‘π‘¦ξ€Έβ€–β€–πœƒπ‘πœ‡+π‘πœ€π‘¦(β€–β€–β‹…,𝜏)2+β€–β€–πœ•π‘πœ‡2𝑦‖‖𝑀(β‹…,𝜏)2+π‘πœ€β€–πœƒ(β‹…,𝜏)β€–2+π‘πœ€7π›Ύβˆ’2,(3.11) provided that β€–πœƒ(β‹…,𝜏)β€–πΏβˆž is bounded.

Proof. Using (3.10), we compute that for π‘˜β‰ π‘, ξ€œ||πœ•π‘¦πœ†π‘(||πœƒπœ™)2π‘˜ξ€œπœ•π‘‘π‘¦=βˆ’π‘¦πœ†π‘(πœ™)πœƒ2π‘˜ξ€œπœ†π‘‘π‘¦=2𝑝(πœ™)πœƒπ‘˜πœ•π‘¦πœƒπ‘˜ξ€œξ‚΅πœ†π‘‘π‘¦=2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜Γ—ξ€Ίβˆ’ξ€·πœ†π‘˜(π‘£πœ€)βˆ’πœ†π‘˜ξ€Έπœƒ(πœ™)π‘˜π‘¦βˆ’πœƒπ‘˜πœβˆ’ξ€½ξ€·(Μ‡π‘ βˆ’Ξ›)π‘€πœƒβˆ’π‘πœƒ+Λ𝐿𝑑+πΏπ‘„βˆ’π‘„1ξ€Έξ€Ύπ‘˜+𝐿(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)ξ€·πœƒπ‘¦π‘¦βˆ’(π‘€πœƒ)π‘¦βˆ’π‘€πœƒπ‘¦+𝑀2πœƒξ€Έ+𝐿(π‘£πœ€)(𝐡(π‘’πœ€)βˆ’π΅(π‘£πœ€π‘’))πœ€π‘¦+πœ€πΏ(π‘£πœ€)π‘ž4ξ€Ύπ‘˜ξ€»β‰‘π‘‘π‘¦12𝑖=1𝐽𝑖.(3.12) We now estimate 𝐽𝑖(1≀𝑗≀12) separately as follows: 𝐽1ξ€œξ‚΅πœ†=βˆ’2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άξ€·πœ†(πœ™)βˆ’Μ‡π‘ π‘˜(π‘£πœ€)βˆ’πœ†π‘˜ξ€Έπœƒ(πœ™)π‘˜πœƒπ‘˜π‘¦=ξ€œπœ•π‘‘π‘¦ξ‚΅πœ†πœ•π‘¦π‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜(ξ‚Άξ€·πœ†πœ™)βˆ’Μ‡π‘ π‘˜(π‘£πœ€)βˆ’πœ†π‘˜ξ€Έπœƒ(πœ™)2π‘˜+ξ€œξ‚΅πœ†π‘‘π‘¦π‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜(ξ‚Άξ€·πœ™)βˆ’Μ‡π‘ βˆ‡πœ†π‘˜(π‘£πœ€)πœ•π‘¦π‘£πœ€βˆ’βˆ‡πœ†π‘˜(πœ™)πœ•π‘¦πœ™ξ€Έπœƒ2π‘˜π‘‘π‘¦β‰‘πΌ1+𝐼2.(3.13) By Lemma 2.4, 𝐼1β‰€ξ€œ||||πœ•ξ‚΅πœ†πœ•π‘¦π‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Ά||||||ξ€·πœ†(πœ™)βˆ’Μ‡π‘ π‘˜(π‘£πœ€)βˆ’πœ†π‘˜ξ€Έ||πœƒ(πœ™)2π‘˜ξ€œ||πœ•π‘‘π‘¦β‰€π‘π‘¦πœ™||||ξ€·πœ†π‘˜(π‘£πœ€)βˆ’πœ†π‘˜ξ€Έ||πœƒ(πœ™)2π‘˜ξ€œπ‘‘π‘¦β‰€π‘|𝑦|β‰€πœ€π›Ύβˆ’1||πœ•π‘¦πœ†π‘||||ξ€·πœ†(πœ™)π‘˜(π‘£πœ€)βˆ’πœ†π‘˜ξ€Έ||πœƒ(πœ™)2π‘˜ξ€œπ‘‘π‘¦+𝑐|𝑦|β‰₯πœ€π›Ύβˆ’1||πœ•π‘¦πœ™||||ξ€·πœ†π‘˜(π‘£πœ€)βˆ’πœ†π‘˜ξ€Έ||πœƒ(πœ™)2π‘˜π‘‘π‘¦β‰€π‘πœ€π›Ύξ€œ||πœ•π‘¦πœ†π‘(||πœƒπœ™)2π‘˜β€–β€–πœƒπ‘‘π‘¦+π‘πœ€π‘˜(β€–β€–β‹…,𝜏)2.(3.14) For the second term 𝐼2, since ||βˆ‡πœ†π‘˜(π‘£πœ€)πœ•π‘¦π‘£πœ€βˆ’βˆ‡πœ†π‘˜(πœ™)πœ•π‘¦πœ™||=||βˆ‡πœ†π‘˜(π‘£πœ€)π‘šπœ•π‘¦πœ™βˆ’βˆ‡πœ†π‘˜(πœ™)πœ•π‘¦||≀||πœ™+𝑂(1)πœ€βˆ‡πœ†π‘˜(πœ™)(π‘šβˆ’1)πœ•π‘¦πœ™||+||ξ€·βˆ‡πœ†π‘˜(π‘£πœ€)βˆ’βˆ‡πœ†π‘˜ξ€Έ(πœ™)π‘šπœ•π‘¦πœ™||+𝑂(1)πœ€β‰€π‘‚(1)πœ€+𝑂(1)πœ€π›Ύπ‘š||πœ•π‘¦πœ†π‘||,πœ•(πœ™)(3.15)π‘¦π‘£πœ€ξ€·πœ€(𝑦,𝜏)=π‘š1βˆ’π›Ύπ‘¦ξ€Έπœ•π‘¦πœ™+𝑂(1)πœ€,(3.16) which follows from Lemma 2.4. Then we have 𝐼2β‰€ξ€œ||||πœ†π‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜||||||(πœ™)βˆ’Μ‡π‘ βˆ‡πœ†π‘˜(π‘£πœ€)πœ•π‘¦π‘£πœ€βˆ’βˆ‡πœ†π‘˜(πœ™)πœ•π‘¦πœ™||πœƒ2π‘˜π‘‘π‘¦β‰€π‘πœ‡πœ€π›Ύξ€œ||πœ•π‘¦πœ†π‘||πœƒ(πœ™)2π‘˜β€–β€–πœƒπ‘‘π‘¦+π‘πœ‡πœ€π‘˜β€–β€–.(β‹…,𝜏)(3.17) Consequently, we obtain 𝐽1=𝐼1+𝐼2β‰€π‘πœ€π›Ύξ€œ||πœ•π‘¦πœ†π‘(||πœƒπœ™)2π‘˜β€–β€–πœƒπ‘‘π‘¦+π‘πœ€π‘˜(β€–β€–β‹…,𝜏)2.(3.18) Using the estimate (𝑑/π‘‘πœ)((πœ†π‘(πœ™)βˆ’Μ‡π‘ )/πœ†π‘˜(πœ™)βˆ’Μ‡π‘ )=𝑂(1)πœ€, we find 𝐽2ξ€œξ‚΅πœ†=βˆ’2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜πœƒπ‘˜πœπ‘‘π‘‘π‘¦=βˆ’ξ€œξ‚΅πœ†π‘‘πœπ‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜(ξ‚Άπœƒπœ™)βˆ’Μ‡π‘ 2π‘˜ξ€œπ‘‘π‘‘π‘¦+ξ‚΅πœ†π‘‘πœπ‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜(ξ‚Άπœƒπœ™)βˆ’Μ‡π‘ 2π‘˜π‘‘π‘‘π‘¦β‰€βˆ’ξ€œξ‚΅πœ†π‘‘πœπ‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜(ξ‚Άπœƒπœ™)βˆ’Μ‡π‘ 2π‘˜β€–β€–πœƒπ‘‘π‘¦+π‘πœ€π‘˜β€–β€–(β‹…,𝜏)2.(3.19) Notice that the facts ξ€·πœ€π‘€=π‘š1βˆ’π›Ύπ‘¦ξ€Έξ€·βˆ‡πΏ(π‘£πœ€)πœ•π‘¦πœ™ξ€Έπ‘…(π‘£πœ€ξ€·πœ€)+𝑂(1)πœ€=π‘š1βˆ’π›Ύπ‘¦ξ€Έξ€·βˆ‡πΏ(πœ™)πœ•π‘¦πœ™ξ€Έπ‘…(πœ™)+𝑂(1)πœ€π›Ύπ‘šξ€·πœ€1βˆ’π›Ύπ‘¦ξ€Έπœ•π‘¦||πœ™+𝑂(1)πœ€,(3.20)Μ‡π‘ βˆ’πœ†π‘||≀(πœ™)π‘πœ‡,(3.21) we arrive at 𝐽3ξ€œξ‚΅πœ†=βˆ’2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜ξ€Ί(Μ‡π‘ βˆ’Ξ›(π‘£πœ€ξ€»))π‘€πœƒπ‘˜β‰€ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œπ‘š||πœ•π‘¦πœ†π‘||||πœƒ||(πœ™)2𝑑𝑦+π‘πœ€β€–πœƒ(β‹…,𝜏)β€–2,𝐽4ξ€œξ‚΅πœ†=2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜[]π‘πœƒπ‘˜β€–π‘‘π‘¦β‰€π‘πœ‡πœ€β€–πœƒ(β‹…,𝜏)2,(3.22) where we have used the estimate |𝑁(𝑦,𝜏)|≀𝑂(1)πœ€. From Lemma 2.3, 𝐽5ξ€œξ‚΅πœ†=βˆ’2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜(Λ𝐿𝑑)π‘˜β€–β€–πœƒπ‘‘π‘¦β‰€πœ‡πœ€π‘˜β€–β€–(β‹…,𝜏)2+π‘πœ€βˆ’1ξ€œπ‘‘2β€–β€–πœƒπ‘‘π‘¦β‰€πœ‡πœ€π‘˜β€–β€–(β‹…,𝜏)2+π‘πœ€7π›Ύβˆ’2.(3.23) Using Lemma 2.4 again, we have ||𝐿(π‘£πœ€)ξ€·π‘„βˆ’π‘„1ξ€Έ||≀𝑐+π‘πœ€π›Ύξ€Έξ‚€||𝑑||2+||𝑅(π‘£πœ€)πœƒπ‘¦+𝑅𝑦(π‘£πœ€||)πœƒ2.(3.24) Then it follows from Lemma 2.3 and (3.16) that 𝐽6ξ€œξ‚΅πœ†=βˆ’2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜ξ€ΊπΏξ€·π‘„βˆ’π‘„1ξ€Έξ€»π‘˜ξ€·π‘‘π‘¦β‰€πœ‡π‘+π‘πœ€π›Ύξ€Έξ‚»ξ€œ||πœƒπ‘˜||||𝑑||2𝑑𝑦+𝑐+π‘πœ€π›Ύξ€Έξ€œ||πœƒπ‘˜||||πœƒπ‘¦||2ξ€œ||πœƒπ‘‘π‘¦+π‘˜||||𝑅𝑦(π‘£πœ€||)πœƒ2ξ‚Όξ€·π‘‘π‘¦β‰€πœ‡π‘+π‘πœ€π›Ύξ€Έξ‚»12πœ€β€–β€–πœƒπ‘˜β€–β€–(β‹…,𝜏)2+12πœ€βˆ’1ξ€œ||𝑑||4𝑑𝑦+𝑐+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘˜β€–β€–πΏβˆžβ€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘˜β€–β€–πΏβˆžξ€œπ‘š||πœ•π‘¦πœ†π‘||||πœƒ||(πœ™)2𝑑𝑦+π‘πœ€2β€–β€–πœƒπ‘˜β€–β€–πΏβˆžβ€–πœƒ(β‹…,𝜏)β€–2ξ‚Όβ‰€π‘πœ‡πœ€14π›Ύβˆ’3ξ€·+πœ‡π‘+π‘πœ€2π›Ύξ€Έβ€–β€–πœƒπ‘˜β€–β€–(β‹…,𝜏)πΏβˆžβ€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2ξ€·β€–β€–πœƒ+π‘πœ‡πœ€1+πœ€π‘˜β€–β€–(β‹…,𝜏)πΏβˆžξ€Έβ€–β€–πœƒ(β‹…,𝜏)2ξ€·+πœ‡π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘˜β€–β€–(β‹…,𝜏)πΏβˆžξ€œπ‘š||πœ•π‘¦πœ†π‘||(πœ™)2||πœƒ||2𝑑𝑦.(3.25) In view of (3.16)–(3.21), 𝐽7ξ€œξ‚΅πœ†=2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜ξ€ΊπΏ(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)πœƒπ‘¦π‘¦ξ€»π‘˜ξ€œξ‚΅πœ†π‘‘π‘¦=βˆ’2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜(ξ‚Άπœƒπœ™)βˆ’Μ‡π‘ π‘˜π‘¦ξ€ΊπΏ(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)πœƒπ‘¦ξ€»π‘˜ξ€œπœƒπ‘‘π‘¦βˆ’2π‘˜ξƒ¬πœ†ξ‚΅ξ‚΅π‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Ά(πœ™)βˆ’Μ‡π‘ πΏ(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)ξ‚Άπ‘¦πœƒπ‘¦ξƒ­π‘˜ξ€·π‘‘π‘¦β‰€πœ‡π‘+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+π‘ξ€œξ€·||πœ•π‘¦πœ™||+||πœ•π‘¦π‘£πœ€||ξ€Έ||πœƒπ‘˜||||πœƒπ‘¦||β‰€ξ€·π‘‘π‘¦π‘πœ‡0+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+π‘ξ€œ||πœ•π‘¦πœ†π‘||||πœƒ(πœ™)π‘˜||||πœƒπ‘¦||ξ€œ||πœƒπ‘‘π‘¦+π‘πœ€π‘˜||||πœƒπ‘¦||≀1𝑑𝑦2ξ€œ||πœ•π‘¦πœ†π‘||||πœƒ(πœ™)π‘˜||2𝑑𝑦+ξ€Έβ€–β€–πœƒπ‘πœ‡+π‘πœ€π‘¦β€–β€–(β‹…,𝜏)2β€–β€–πœƒ+π‘πœ€π‘˜β€–β€–(β‹…,𝜏)2,𝐽8ξ€œξ‚΅πœ†=βˆ’2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜ξ€ΊπΏ(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)(π‘€πœƒ)π‘¦ξ€»π‘˜ξ€œξ‚΅πœ†π‘‘π‘¦=2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Ά(πœ™)βˆ’Μ‡π‘ π‘¦πœƒπ‘˜ξ€ΊπΏ(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€ξ€»)π‘€πœƒπ‘˜ξ€œξ‚΅πœ†π‘‘π‘¦+2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜π‘¦ξ€ΊπΏ(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)ξ€»π‘€πœƒπ‘˜β‰€ξ€·π‘‘π‘¦π‘+π‘πœ€π›Ύξ€Έξ€œ||πœ•π‘¦πœ™||||πœƒπ‘˜||||||β€–β€–πœƒπ‘€πœƒπ‘‘π‘¦+πœ‡π‘¦(β€–β€–β‹…,𝜏)2ξ€·+πœ‡π‘+π‘πœ€π›Ύξ€Έξ€œ||||π‘€πœƒ2β€–β€–πœƒπ‘‘π‘¦β‰€πœ‡π‘¦β€–β€–(β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œ||πœ•π‘¦πœ†π‘||||πœƒ(πœ™)π‘˜||2𝑑𝑦+π‘πœ€β€–πœƒ(β‹…,𝜏)β€–2.(3.26) Same bounds hold for 𝐽9 and 𝐽10.
Applying Cauchy inequality and (3.21), 𝐽11 can be estimated as𝐽11ξ€œξ‚΅πœ†=2𝑝(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜ξ€ΊπΏ(π‘£πœ€)(𝐡(π‘’πœ€)βˆ’π΅(π‘£πœ€))π‘’πœ€π‘¦ξ€»π‘˜β‰€π‘‘π‘¦ξ€œ||πœƒπ‘πœ‡π‘˜ξ€ΊπΏ(π‘£πœ€)(𝐡(π‘’πœ€)βˆ’π΅(π‘£πœ€))π‘€π‘¦π‘¦ξ€»π‘˜||+π‘‘π‘¦ξ€œ||πœƒπ‘πœ‡π‘˜ξ€ΊπΏ(π‘£πœ€)(𝐡(π‘’πœ€)βˆ’π΅(π‘£πœ€))π‘£πœ€π‘¦ξ€»π‘˜||𝑑𝑦≑𝐾1+𝐾2.(3.27) Lemma 2.4 yields, 𝐾1ξ€·β‰€πœ‡π‘+π‘πœ€π›Ύξ€Έξ€œ||πœƒπ‘˜||||𝑀𝑦||||πœ•2𝑦𝑀||β‰€ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœ•2𝑦‖‖𝑀(β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘˜β€–β€–2πΏβˆžξ€œ||𝑀𝑦||2β‰€ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœ•2𝑦‖‖𝑀(β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘˜β€–β€–2πΏβˆžξ‚΅β€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+ξ€œ||||π‘€πœƒ2ξ‚Άβ‰€ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœ•2𝑦‖‖𝑀(β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘˜β€–β€–2πΏβˆžΓ—ξ‚»β€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œπ‘š||πœ•π‘¦πœ†π‘||||πœƒ||(πœ™)2‖𝑑𝑦+π‘πœ€β€–πœƒ(β‹…,𝜏)2ξ‚Ό,(3.28) where we used the fact 𝑀𝑦=π‘…πœƒπ‘¦βˆ’π‘…π‘€πœƒ. Similarly, 𝐾2ξ€·β‰€πœ‡π‘+π‘πœ€π›Ύξ€Έξ€œ||πœƒπ‘˜||||𝑀𝑦||||πœ•π‘¦π‘£πœ€||β‰€ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œ||πœƒπ‘˜||||πœƒπ‘¦||||πœ•π‘¦π‘£πœ€||𝑑𝑦+π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œ||𝑀||||πœƒ||2||πœ•π‘¦π‘£πœ€||β‰€ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œ||πœƒ||2||πœ•π‘¦π‘£πœ€||2+ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œ||𝑀||||πœƒ||2||πœ•π‘¦π‘£πœ€||β‰€ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œπ‘š||πœ•π‘¦πœ†π‘||||πœƒ||(πœ™)2‖𝑑𝑦+π‘πœ€β€–πœƒ(β‹…,𝜏)2.(3.29) Thus, combining the above two inequality together, we obtain 𝐽11=𝐾1+𝐾2β‰€ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœ•2𝑦‖‖𝑀(β‹…,𝜏)2+ξ‚€β€–β€–πœƒ1+π‘˜β€–β€–2πΏβˆžξ‚Γ—ξ‚»ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œπ‘š||πœ•π‘¦πœ†π‘||||πœƒ||(πœ™)2𝑑𝑦+π‘πœ€β€–πœƒ(β‹…,𝜏)β€–2ξ‚Ό.(3.30) Finally, 𝐽12ξ€œξ‚΅πœ†=2πœ€π‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ π‘˜ξ€ΊπΏ(π‘£πœ€)π‘ž4ξ€»π‘˜β€–β€–πœƒπ‘‘π‘¦β‰€πœ€π‘˜β€–β€–(β‹…,𝜏)2+π‘πœ‡2πœ€ξ€œ||π‘ž4||2β€–β€–πœƒπ‘‘π‘¦β‰€πœ€π‘˜β€–β€–(β‹…,𝜏)2+π‘πœ€7π›Ύβˆ’2.(3.31) Summing all the inequalities for π‘˜β‰ π‘, we arrive at ξ“π‘˜β‰ π‘ξ€œ||πœ•π‘¦πœ†π‘||πœƒ(πœ™)2π‘˜π‘‘π‘‘π‘¦β‰€ξƒ¬βˆ’ξ“π‘‘πœπ‘˜β‰ π‘ξ€œξ‚΅πœ†π‘(πœ™)βˆ’Μ‡π‘ πœ†π‘˜ξ‚Άπœƒ(πœ™)βˆ’Μ‡π‘ 2π‘˜ξƒ­+ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œπ‘š||πœ•π‘¦πœ†π‘||πœƒ(πœ™)2𝑝+ξ€·π‘‘π‘¦π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘¦(β€–β€–β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έβ€–β€–πœ•2𝑦‖‖𝑀(β‹…,𝜏)2+π‘πœ€β€–πœƒ(β‹…,𝜏)β€–2+π‘πœ€7π›Ύβˆ’2,(3.32) provided that β€–πœƒ(β‹…,𝜏)β€–πΏβˆž is bounded.
We complete the proof of Lemma 3.2.

Lemma 3.3. Suppose that the conditions in Proposition 3.1 be satisfied. Then ‖‖𝑀(β‹…,𝜏)21+ξ€œπœ0‖‖𝑀𝑦‖‖(β‹…,𝜏)21ξ€œπ‘‘πœ+𝜏0ξ€œπ‘š||πœ•π‘¦πœ†π‘||(πœ™)|𝑀|2π‘‘π‘¦π‘‘πœβ‰€π‘πœ€7π›Ύβˆ’3,(3.33) for all 𝜏∈[0,𝜏0], where 𝑐 is independent of 𝜏0 and πœ€.

Proof. Multiplying (3.10) on the left by πœƒπ‘‘ and integrating over 𝑅1, we obtain after integration by parts that 12𝑑(π‘‘πœβ€–πœƒβ‹…,𝜏)β€–2=ξ€œπœƒπ‘‘πΏ(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)πœƒπ‘¦π‘¦ξ€œπœƒπ‘‘π‘¦βˆ’π‘‘(Ξ›βˆ’Μ‡π‘ )πœƒπ‘¦βˆ’ξ€œπœƒπ‘‘π‘¦π‘‘ξ€œπœƒ(Μ‡π‘ βˆ’Ξ›)π‘€πœƒπ‘‘π‘¦βˆ’π‘‘ξ€œπœƒπ‘πœƒπ‘‘π‘¦βˆ’π‘‘ξ€·π‘£πΏπ‘„πœ€,π‘…πœƒπ‘¦+π‘…π‘¦πœƒξ€Έ+ξ€œπœƒπ‘‘π‘¦π‘‘πΏπ‘„1(π‘£πœ€ξ€œπœƒ,𝑑)π‘‘π‘¦βˆ’π‘‘+ξ€œπœƒΞ›πΏπ‘‘π‘‘π‘¦π‘‘πΏ(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)ξ€·βˆ’(π‘€πœƒ)π‘¦βˆ’π‘€πœƒπ‘¦+𝑀2πœƒξ€Έ+ξ€œπœƒπ‘‘π‘¦π‘‘πΏ(π‘£πœ€)(𝐡(π‘’πœ€)βˆ’π΅(π‘£πœ€))π‘’πœ€π‘¦ξ€œπœƒπ‘‘π‘¦+πœ€π‘‘πΏ(π‘£πœ€)π‘ž4𝑑𝑦.(3.34) Next we estimate each term on the right hand side above. First, it follows from (3.16) that ξ€œπœƒπ‘‘πΏ(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)πœƒπ‘¦π‘¦ξ€œπœƒπ‘‘π‘¦=βˆ’π‘‘π‘¦πΏ(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€)πœƒπ‘¦ξ€œπœƒπ‘‘π‘¦βˆ’π‘‘(𝐿(π‘£πœ€)𝐡(π‘£πœ€)𝑅(π‘£πœ€))π‘¦πœƒπ‘¦β‰€ξ€·π‘‘π‘¦βˆ’πΆ0+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+𝑐+π‘πœ€π›Ύξ€Έξ€œ||πœ•π‘¦π‘£πœ€||||πœƒ||||πœƒπ‘¦||β‰€ξ€·π‘‘π‘¦βˆ’πΆ0+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+𝐢0β€–β€–πœƒ10𝑦‖‖(β‹…,𝜏)2+𝑐+π‘πœ€π›Ύξ€Έξ€œ||πœ•π‘¦π‘£πœ€||2||πœƒ||2β‰€ξ€·π‘‘π‘¦βˆ’πΆ0+π‘πœ€π›Ύξ€Έβ€–β€–πœƒπ‘¦β€–β€–(β‹…,𝜏)2+𝐢0β€–β€–πœƒ10𝑦‖‖(β‹…,𝜏)2+ξ€·π‘πœ‡+π‘πœ€π›Ύξ€Έξ€œπ‘š||πœ•π‘¦πœ†π‘||||πœƒ||(πœ™)2𝑑𝑦+π‘πœ€β€–πœƒ(β‹…,𝜏)β€–2.(3.35) Here 𝐢0>0 is the minimum of the eigenvalues, valued at 𝑒0 and πœ™, of (1/2)(𝐿𝐡𝑅+(𝐿𝐡𝑅)𝑑) and (1/2)(𝐡+𝐡𝑑). And βˆ’ξ€œπœƒπ‘‘(Ξ›βˆ’Μ‡π‘ )πœƒπ‘¦1𝑑𝑦=2ξ€œπ‘›ξ“π‘–=1πœ•π‘¦πœ†π‘–(π‘£πœ€)πœƒ2𝑖=1𝑑𝑦2ξ€œπ‘šξƒ©π‘›ξ“π‘–=1πœ•π‘¦πœ†π‘–(πœ™)πœƒ2𝑖ξƒͺ+1𝑑𝑦