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

Authors: Shixiang Ma
Journal: ISRN Mathematical Physics
 (2012)

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## 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 ofε1 as the viscosity coefficient ε vanishes.


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## Body

## 1. Introduction

We consider the relation between the solutions,uɛ, of the system of viscous conservation laws(1.1)utɛ+f(uɛ)x=ɛ(B(uɛ)uxɛ)x,uɛ∈Rn,x∈R,t≥0,ɛ>0,
and the distributional solution, u, of the corresponding system of conservation laws without viscosity(1.2)ut+f(u)x=0,u∈Rn,x∈R,t>0.We assume that (1.2) is strictly hyperbolic, then by normalization, we have the decomposition(1.3)A≡Duf(u)=RΛL,RL=I,
where Λ=diag(λ1,λ2,…,λn) with λ1<λ2<⋯<λn,L=(l1,…,ln)t is a matrix whose rows are left eigenvectors of A, and R=(r1,…,rn) 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, u, of (1.2), with finitely many noninteracting shocks satisfying the entropy condition, are strong limits as ɛ→0 of solutions, uɛ, of (1.1) when the matrix LBR is positive definite.For simplicity of presentation, we only consider the case in whichu is a single-shock solution.Definition 1.1.
A functionu(x,t) is called a single-shock solution of (1.2) up to time T if:(i)
u(x,t) is a distributional solution of the hyperbolic system (1.2) in the region R1×[0,T];(ii)
there is a smooth curve, the shock,x=s(t),0≤t≤T, so that u(x,t) is sufficiently smooth at any point x≠s(t);(iii)
the limits(1.4)∂xku(s(t)-0,t)=lim⁡x→s(t)-∂xku(x,t),∂xku(s(t)+0,t)=lim⁡x→s(t)+∂xku(x,t),exist and are finite for t≤T and 0≤k≤5;(iv)
the Lax geometrical entropy condition [18] is satisfied at x=s(t), that is,(1.5)λ1(u(s(t)-0,t))<⋅⋅⋅<λp(u(s(t)-0,t)),λp(u(s(t)+0,t))<ddts(t)<λp(u(s(t)-0,t)),λp(u(s(t)+0,t))<⋅⋅⋅<λn(u(s(t)+0,t)).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, μ0 and ɛ0, such that if u(x,t) is a single-shock solution up to time T with
(1.6)∑1≤k≤6∫0T∫|∂xku(x,t)|2dxdt<∞,μ≡sup⁡0≤t≤T|u(s(t)+0,t)-u(s(t)-0,t)|≤μ0,
then for each ɛ∈[0,ɛ0], there is a smooth solution, uɛ(x,t), of (1.1) with
(1.7)uxɛ∈C1([0,T],H1).
Moreover, for any given η∈(0,1),
(1.8)sup⁡0≤t≤T∫|uɛ(x,t)-u(x,t)|2dx≤Cηɛη,(1.9)sup⁡0≤t≤T,|x-s(t)|≥ɛη|uɛ(x,t)-u(x,t)|≤Cηɛ,
where Cη is a positive constant depending only on η.Notation 1.
In this paper, we useHl(l≥1) to denote the usual Sobolev space with the norm ∥·∥l, and ∥·∥=∥·∥0 denotes the usual L2-norm. We also use O(1) to denote any positive bounded function which is independent of ɛ.


## 2. Construction of the Approximate Solution

In this section, following the method of Goodman and Xin, in [2], we construct the approximate solution vɛ through different scaling and asymptotic expansions in the region near and away from the shock respectively, such that vɛ approximate the piecewise smooth inviscid solution u 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,x=s(t), we approximate the solution of (1.1) by truncation of the formal series(2.1)uɛ(x,t)~u0(x,t)+ɛu1(x,t)+ɛ2u2(x,t)+⋯.
Substituting this into (1.1) and comparing the coefficients of powers of ɛ, we get, for x≠s(t), that(2.2)O(1):u0t+f(u0)x=0,(2.3)O(ɛ):u1t+(f′(u0)u1)x=(B(u0)u0x)x,(2.4)O(ɛ2):u2t+(f′(u0)u2)x=(B(u0)u1x)x+(B′(u0)(u1,u0x))x-12(f′′(u0)(u1,u1))x,
and so forth. The outer functions u0,u1,…, are generally discontinuous at the shock, x=s(t), but smooth up to the shock. The leading term, u0, is the single-shock solution of (1.2) which is given in the theorem.Near the shock,uɛ should be represented by an inner expansion:(2.5)uɛ(x,t)~U0(ξ,t)+ɛU1(ξ,t)+ɛ2U2(ξ,t)+⋯,
where(2.6)ξ=x-s(t)ɛ+δ(t,ɛ),
and δ(t,ɛ) is the perturbation of the shock position to be determined later.We assume thatδ(t,ɛ) has the form(2.7)δ(t,ɛ)=δ0(t)+ɛδ1(t)+ɛ2δ2(t)+⋯.
Substitute (2.5)–(2.7) into (1.1) to obtain(2.8)O(1ɛ):(B(U0)U0ξ)ξ+ṡU0ξ-(f(U0))ξ=0,(2.9)O(1):{B(U0)U1ξ+B′(U0)(U1,U0ξ)}ξ+ṡU1ξ-(f′(U0)U1)ξ=U0t+δ̇0U0ξ,(2.10)O(ɛ):{B(U0)U2ξ+B′(U0)(U2,U0ξ)}ξ+ṡU2ξ-(f′(U0)U2)ξ,s=U1t+δ̇1U0ξ+δ̇0U1ξ+12(f′′(U0)(U1,U1))ξ-(B′(U0)(U1,U1ξ))ξ,
and so forth, where ṡ=ds/dt,δ̇0=dδ0/dt, and so forth. The inner approximation is supposed to be valid in a small zone of size O(ɛ) around x=s(t).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 ofξ, we obtain the following “matching conditions” as ξ→±∞:(2.11)U0(ξ,t)=u0(s(t)±0,t)+o(1),(2.12)U1(ξ,t)=u1(s(t)±0,t)+(ξ-δ0)∂xu0(s(t)±0,t)+o(1),(2.13)U2(ξ,t)=u2(s(t)±0,t)+(ξ-δ0)∂xu1(s(t)±0,t)-δ1∂xu0(s(t)±0,t)+12(ξ-δ0)2∂x2u0(s(t)±0,t)+o(1),
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(2.14)(B(ϕ)ϕξ)ξ=-σϕξ+f(ϕ)ξ,
satisfying the boundary conditions
(2.15)ϕ(ξ)⟶ulasξ⟶-∞,ϕ(ξ)⟶urasξ⟶+∞,
and moving with speed σ:
(2.16)σ(ul-ur)=f(ul)-f(ur).
Integrate the differential equation to reduce that
(2.17)B(ϕ)ϕξ=-σ(ϕ(ξ)-ul)+f(ϕ(ξ))-f(ul).
It is well known that for a given state u¯ and the p wave family, if |ul-u¯|+|σ-λp(u¯)| is sufficiently small, then there exists a shock profile ϕ=ϕ(ξ,ul,σ), which connects ul and ur from left to right. Using the genuine nonlinearity, by similar arguments in [2], we can obtain(2.18)∂ξλp(ϕ)<0,∀ξ,|∂ξϕ|≤c|∂ξλp(ϕ)|≤c|ul-ur|.
And as ξ→-∞,(2.19)ϕ(ξ,ul,σ)-ul=O(1)|ul-ur|e-α|ξ|,∂ϕ∂ul-I=O(1)e-α|ξ|,∂ϕ∂σ=O(1)e-α|ξ|.
As ξ→+∞,
(2.20)ϕ(ξ,ul,σ)-ur=O(1)|ul-ur|e-α|ξ|,∂ϕ∂ul-∂ur∂ul=O(1)e-α|ξ|,∂ϕ∂σ-∂ur∂σ=O(1)e-α|ξ|.


### 2.3. Solutions of the Outer and Inner Problems

Now we constructuj and Uj order by order.The leading order outer function,u0, is the single-shock solution of the theorem. For any fixed t, the leading order inner solution U0(ξ,t) is exactly the viscous shock profile with ul(t)≡u(s(t)-0,t),ur(t)≡u(s(t)+0,t), and σ=ṡ(t). So(2.21)U0(ξ,t)=ϕ(ξ,ul(t),ṡ(t)).
Here we take the shift to be zero since it can be absorbed into δ0(t).Next we determineu1,U1, and δ0(t) together. Substituting (2.21) into (2.9) gives(2.22){B(ϕ)U1ξ+B′(ϕ)(U1,ϕξ)}ξ+ṡU1ξ-(f′(ϕ)U1)ξ=δ̇0(t)ϕξ+∂ϕ∂ulu̇l+∂ϕ∂ṡs̈.
By the matching condition (2.12), we expect that (2.23)U1(ξ,t)=ξ⋅∂xu0(s(t)±0,t)+O(1)asξ⟶±∞.
So we set(2.24)U1(ξ,t)=V1(ξ,t)+D1(ξ,t),
where D1(ξ,t) is a smooth function satisfying (2.25)D1(ξ,t)={ξ⋅∂xu0(s(t)-0,t),ξ<-1,ξ⋅∂xu0(s(t)+0,t),ξ>1.
Then inserting (2.24) into (2.22) and using (2.19)–(2.20) and the identity (2.26)ddtu0(s(t)±0,t)=(ṡI-f′(u0(s(t)±0,t)))u0x(s(t)±0,t),
we obtain (2.27){B(ϕ)V1ξ+B′(ϕ)(V1,ϕξ)}ξ+ṡV1ξ-(f′(ϕ)V1)ξ=δ̇0(t)ϕξ+g(ξ,t),
where |g(ξ,t)|≤cexp{-α|ξ|} for large |ξ|. Define G(ξ,t)=∫0ξg(η,t)dη. Then we have(2.28)B(ϕ)V1ξ+B′(ϕ)(V1,ϕξ)+ṡV1-f′(ϕ)V1=δ̇0(t)ϕ+G(ξ,t)+c(t),
where c(t)∈Rn are integration constants to be determined later. We express V1 in terms of the basis, r1(ϕ),r2(ϕ),…,rn(ϕ), of the right eigenvectors of f′(ϕ). We write(2.29)V1(ξ,t)=∑j=1nαj(ξ,t)rj(ϕ(ξ,t)),u1(s(t)±0,t)=∑j=1nβj±(t)rj(u0(s(t)±0,t)),∂xu0(s(t)±0,t)=∑j=1nγj±(t)rj(u0(s(t)±0,t)).
Here the βj- are for u1(s(t)-0,t) and the βj+ are for u1(s(t)+0,t), and so forth. Then the matching conditions (2.12) are transformed into(2.30)lim⁡ξ→±∞αj(ξ,t)=βj±(t)-δ0(t)γj±(t),j=1,…,n.
Define σj(ϕ)≡lj(ϕ)B(ϕ)rj(ϕ)>0. Multiplying (2.28) by lj(ϕ), and using (2.29), we obtain(2.31)αjξ+σj(ϕ)-1{lj(ϕ)B′(ϕ)(rj(ϕ),ϕξ)+(ṡ-λj(ϕ))}αj=σj(ϕ)-1lj(ϕ)(δ̇0(t)ϕ+G(ξ,t)+c(t))-∑i=1nσj(ϕ)-1αi(ξ,t)lj(ϕ)B(ϕ)riξ(ϕ),j=1,…,n,
and then we have the following result.Lemma 2.1.
There is a smooth solution,α(ξ,t), to (2.31) with the following property:
(2.32)αj(ξ,t)={(ṡ-λj(ul))-1lj(ul)[δ̇0ul+G-+c(t)]+O(1)exp⁡{-α|ξ|},ξ⟶-∞,(ṡ-λj(ur))-1lj(ur)[δ̇0ur+G++c(t)]+O(1)exp⁡{-α|ξ|},ξ⟶+∞,
for j=1,…,n, where G±=limξ→±∞G(ξ,t), and α0 is a positive constant.Proof.
We use the standard iteration argument. Defineα0(ξ,t)≡0, and
(2.33)αjk+1(ξ,t)=∫-∞ξexp⁡{-∫ηξσj(ϕ)-1(lj(ϕ)B′(ϕ)(rj(ϕ),ϕξ)+ṡ-λj(ϕ))dζ}⋅{∑i=1nσj(ϕ)-1lj(ϕ)(δ̇0(t)ϕ+G(ξ,t)+c(t))-∑i=1nσj(ϕ)-1αi(ξ,t)lj(ϕ)B(ϕ)riξ(ϕ)}dη,j<p,αpk+1(ξ,t)=∫0ξexp⁡{-∫ηξσp(ϕ)-1(lp(ϕ)B′(ϕ)(rp(ϕ),ϕξ)+ṡ-λp(ϕ))dζ}⋅{∑i=1nσp(ϕ)-1lp(ϕ)(δ̇0(t)ϕ+G(ξ,t)+c(t))-∑i=1nσp(ϕ)-1αi(ξ,t)lp(ϕ)B(ϕ)riξ(ϕ)}dη,αjk+1(ξ,t)=-∫ξ+∞exp⁡{-∫ηξσj(ϕ)-1(lj(ϕ)B′(ϕ)(rj(ϕ),ϕξ)+ṡ-λj(ϕ))dζ}⋅{∑i=1nσj(ϕ)-1lj(ϕ)(δ̇0(t)ϕ+G(ξ,t)+c(t))-∑i=1nσj(ϕ)-1αi(ξ,t)lj(ϕ)B(ϕ)riξ(ϕ)}dη,j>p.
Set
(2.34)Mk=∑0≤i≤nsup⁡R×[0,T]|αik(ξ,t)|,k=0,1,2,…,Dk=∑0≤i≤nsup⁡R×[0,T]|αik+1(ξ,t)-αik(ξ,t)|,k=0,1,2,….
Then from the lax entropy condition (1.5), we can obtain
(2.35)Dk≤c¯μDk-1,k=1,2,…,
where c¯ is independent of μ. And then for suitably small μ, we have
(2.36)Mk≤c¯M1,k=0,1,…,
and αk(ξ,t) converges uniformly to a smooth bounded function, α(ξ,t), which is a solution to (2.31). The asymptotic behavior of the solution, α(ξ,t), follows from the formulas (2.33).
With Lemma2.1 and the matching condition (2.30) at hand, we can determine, completely the same as in [2], β±(t),δ0(t), and c(t), which guarantees the existence of U1(ξ,t) and u1(x,t). 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,(2.37)lj(ur)[δ̇0ur+c(t)]=(βj+(t)-δ0γj+(t))(ṡ-λj(ur))-lj(ur)G+,for1≤j≤p,(2.38)lj(ul)[ṡur+c(t)]=(βj-(t)-δ0γj-(t))(ṡ-λj(ul))+lj(ul)(ur-ul)δ̇0-lj(ul)G-,for(p+1)≤j≤n,(2.39)(lj(ul)(ul-ur))δ̇0=(lp(ur)-lp(ul))(δ̇0ur+c(t))+(ṡ-λp(ul))βp-(t)-(ṡ-λp(ur))βp+(t)+δ0[γp+(ṡ-λp(ur))-γp-(ṡ-λp(ul))]+lp(ur)G+-ll(ul)G-.
Then we can solve for δ̇0ur+c(t) from (2.37)-(2.38). Substituting the resulting expression into (2.39), by writing βin=(βp-,β(p+1)-,…,βn-,β1+,…,βp+), we arrive at an ordinary differential equation for δ0:
(2.40)δ̇0+E1(t)δ0=E2(t)⋅βin+G1(t),
provided that μ is suitably small. Here E1(t),E2(t), and G1(t) are smooth known functions, and E1(t) and E2(t) remain bounded even as μ→0+. Solving for δ0 from (2.37) up to a constant, we obtain c(t) uniquely in terms of βin. Then substitute the expression of δ0 and c(t) into the equation of the matching condition for outgoing indices to yield the linear relations
(2.41)βout=E(t)⋅βin+F(t),
where βout=(β1-,β2-,…,β(p-1)-,β(p+1)+,…,βn+), and F(t)∈Rn-1 is a smooth known function, E(t) is a smooth (n-1)×(n+1) matrix and remains bounded even as μ→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, u1(x,0), is chosen to satisfy the appropriate compatibility conditions at x=s(0). Thus u1(x,t) is completely determined, which in turn gives δ0 and c(t) by (2.37)-(2.38), and therefore U1(ξ,t).Now we summarize the above discussion to achieve the following.Proposition 2.2.
Ifμ is suitably small, then u1(x,t),U1(ξ,t) and δ0 can be established such that(i)
u1(x,t) and its derivatives are uniformly continuous up to x=s(t), and(2.42)∑0≤k≤6∫0T∫|∂xku1(x,t)|2dxdt<∞.(ii)
U1(ξ,t) and δ0 are smooth functions, and there are an α>0, such that(2.43)U1(ξ,t)=u1(s(t)±0,t)+(ξ-δ0)∂xu0(s(t)±0,t)+O(1)exp⁡{-α|ξ|},asξ⟶±∞.The above constructions can be carried out to any order. In particular, we can determineui,Ui and δi-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(2.44)I(x,t)=ϕ(x-s(t)ɛ+δ0+ɛδ1+ɛδ2,t)+∑i=13ɛiUi(x-s(t)ɛ+δ0+ɛδ1+ɛδ2,t),O(x,t)=u0(x,t)+ɛu1(x,t)+ɛ2u2(x,t)+ɛ3u3(x,t).
Let m∈C0∞(R) satisfy 0≤m(y)≤1, and (2.45)m(y)={1,|y|≤1,0,|y|≥2.
Let γ∈(5/7,1) be a constant. Then we define the approximate solution to (1.1) as(2.46)vɛ(x,t)=m(x-s(t)ɛγ)I(x,t)+(1-m(x-s(t)ɛγ))O(x,t)+d(x,t),
where d(x,t) is a higher-order correction term to be determined.Using the structures of the various orders of inner and outer solutions, we compute that(2.47)vtɛ+f(vɛ)x-ɛ(B(vɛ)vxɛ)x=∑i=3nqi(x,t)+dt-(B(mI+(1-m)O)dx)x-ɛq4x(x,t)+(f(vɛ)-f(vɛ-d))x,
where(2.48)q1(x,t)=(1-m){[f(O)-Γ(f(O))]x-ɛ[B(O)Ox-Γ(B(O)Ox)]x},q2(x,t)=m{[f(I)-Γ(f(I))]x-ɛ[B(I)Ix-Γ(B(I)Ix)]x+ɛ3U3t+ɛ4(δ̇2U1+δ̇1U2+δ̇0U3+ɛδ̇2U2+ɛδ̇1U3+ɛ2δ̇2U3)x},q3(x,t)=mt(I-O)+mx(f(I)-f(O))+{f(mI+(1-m)O)-(mf(I)+(1-m)f(O))}x-ɛmx(B(I)Ix-B(O)Ox),q4(x,t)=B(vɛ)(vɛ-d)x-mB(I)Ix-(1-m)B(O)Ox+(B(vɛ)-B(vɛ-d))dx,
where Γ(f(O)),Γ(B(O)Ox) denote the truncated Taylor's expansion of f(O),B(O)Ox, respectively, at u0, including all the terms of O(1),O(1)ɛ,O(1)ɛ2,O(1)ɛ3, and Γ(f(I)),Γ(B(I)Ix) denote the truncated Taylor's expansion of f(I),B(I)Ix, respectively, at ϕ, including all the terms of O(1),O(1)ɛ,O(1)ɛ2,O(1)ɛ3.In view of our construction, we have(i)
suppq1(x,t)⊆{(x,t):|x-s(t)|≥ɛγ,0≤t≤T}, and
(2.49)∂xlq1(x,t)=O(1)ɛ4-lγ,(∫0T‖q1(⋅,t)‖2dt)1/2≤O(1)ɛ4,(∫0T‖∂xlq1(⋅,t)‖2dt)1/2≤O(1)ɛ4-(l-(1/2))γ,l=1,2,3,(ii)
suppq2(x,t)⊆{(x,t):|x-s(t)|≤2ɛγ,0≤t≤T}, and
(2.50)∂xlq2(x,t)=O(1)ɛ(3-l)γ,l=0,1,2,3,(iii)
suppq3(x,t)⊆{(x,t):ɛγ≤|x-s(t)|≤2ɛγ,0≤t≤T}, and
(2.51)∂xlq3(x,t)=O(1)ɛ(3-l)γ,l=0,1,2,3,where we have used the estimates∂xl(I-O)=O(1)ɛ(4-l)γ on {(x,t):ɛγ≤|x-s(t)|≤2ɛγ},l=0,1,2,3.We now choosed(x,t) to satisfy(2.52)dt-(B(mI+(1-m)O)dx)x=-∑i=13qi(x,t),d(x,0)=0,
so that vɛ satisfies(2.53)vtɛ+f(vɛ)x=ɛ(B(vɛ)vxɛ)x-ɛq4x+(f(vɛ)-f(vɛ-d)).
Since B 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.
Letd(x,t) be the solution of (2.52). Then the following estimates hold for all t∈[0,T]:
(2.54)‖∂xld(⋅,t)‖L∞≤O(1)ɛ(7/2)γ-(l+(1/2)),forl=0,1,2,3,‖∂xld(⋅,t)‖≤O(1)ɛ(7/2)γ-l,l=0,1,2,3,4.
Then for q4, we have
(2.55)‖∂xlq4‖≤O(1)ɛ(7/2)γ-(l+1),l=0,1,2.And by our construction, we obtain the following.Lemma 2.4.
One has(2.56)vɛ(x,t)={u0(x,t)+O(1)ɛ,|x-s(t)|≥ɛγ,ϕ(ξ,t)+O(1)ɛγ,|x-s(t)|≤2ɛγ,∂vɛ∂x=1ɛm∂ξϕ+O(1),∂vɛ∂t=O(1).


## 2.1. Outer and Inner Expansions and the Matching Conditions

In the region away from the shock,x=s(t), we approximate the solution of (1.1) by truncation of the formal series(2.1)uɛ(x,t)~u0(x,t)+ɛu1(x,t)+ɛ2u2(x,t)+⋯.
Substituting this into (1.1) and comparing the coefficients of powers of ɛ, we get, for x≠s(t), that(2.2)O(1):u0t+f(u0)x=0,(2.3)O(ɛ):u1t+(f′(u0)u1)x=(B(u0)u0x)x,(2.4)O(ɛ2):u2t+(f′(u0)u2)x=(B(u0)u1x)x+(B′(u0)(u1,u0x))x-12(f′′(u0)(u1,u1))x,
and so forth. The outer functions u0,u1,…, are generally discontinuous at the shock, x=s(t), but smooth up to the shock. The leading term, u0, is the single-shock solution of (1.2) which is given in the theorem.Near the shock,uɛ should be represented by an inner expansion:(2.5)uɛ(x,t)~U0(ξ,t)+ɛU1(ξ,t)+ɛ2U2(ξ,t)+⋯,
where(2.6)ξ=x-s(t)ɛ+δ(t,ɛ),
and δ(t,ɛ) is the perturbation of the shock position to be determined later.We assume thatδ(t,ɛ) has the form(2.7)δ(t,ɛ)=δ0(t)+ɛδ1(t)+ɛ2δ2(t)+⋯.
Substitute (2.5)–(2.7) into (1.1) to obtain(2.8)O(1ɛ):(B(U0)U0ξ)ξ+ṡU0ξ-(f(U0))ξ=0,(2.9)O(1):{B(U0)U1ξ+B′(U0)(U1,U0ξ)}ξ+ṡU1ξ-(f′(U0)U1)ξ=U0t+δ̇0U0ξ,(2.10)O(ɛ):{B(U0)U2ξ+B′(U0)(U2,U0ξ)}ξ+ṡU2ξ-(f′(U0)U2)ξ,s=U1t+δ̇1U0ξ+δ̇0U1ξ+12(f′′(U0)(U1,U1))ξ-(B′(U0)(U1,U1ξ))ξ,
and so forth, where ṡ=ds/dt,δ̇0=dδ0/dt, and so forth. The inner approximation is supposed to be valid in a small zone of size O(ɛ) around x=s(t).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 ofξ, we obtain the following “matching conditions” as ξ→±∞:(2.11)U0(ξ,t)=u0(s(t)±0,t)+o(1),(2.12)U1(ξ,t)=u1(s(t)±0,t)+(ξ-δ0)∂xu0(s(t)±0,t)+o(1),(2.13)U2(ξ,t)=u2(s(t)±0,t)+(ξ-δ0)∂xu1(s(t)±0,t)-δ1∂xu0(s(t)±0,t)+12(ξ-δ0)2∂x2u0(s(t)±0,t)+o(1),
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(2.14)(B(ϕ)ϕξ)ξ=-σϕξ+f(ϕ)ξ,
satisfying the boundary conditions
(2.15)ϕ(ξ)⟶ulasξ⟶-∞,ϕ(ξ)⟶urasξ⟶+∞,
and moving with speed σ:
(2.16)σ(ul-ur)=f(ul)-f(ur).
Integrate the differential equation to reduce that
(2.17)B(ϕ)ϕξ=-σ(ϕ(ξ)-ul)+f(ϕ(ξ))-f(ul).
It is well known that for a given state u¯ and the p wave family, if |ul-u¯|+|σ-λp(u¯)| is sufficiently small, then there exists a shock profile ϕ=ϕ(ξ,ul,σ), which connects ul and ur from left to right. Using the genuine nonlinearity, by similar arguments in [2], we can obtain(2.18)∂ξλp(ϕ)<0,∀ξ,|∂ξϕ|≤c|∂ξλp(ϕ)|≤c|ul-ur|.
And as ξ→-∞,(2.19)ϕ(ξ,ul,σ)-ul=O(1)|ul-ur|e-α|ξ|,∂ϕ∂ul-I=O(1)e-α|ξ|,∂ϕ∂σ=O(1)e-α|ξ|.
As ξ→+∞,
(2.20)ϕ(ξ,ul,σ)-ur=O(1)|ul-ur|e-α|ξ|,∂ϕ∂ul-∂ur∂ul=O(1)e-α|ξ|,∂ϕ∂σ-∂ur∂σ=O(1)e-α|ξ|.


## 2.3. Solutions of the Outer and Inner Problems

Now we constructuj and Uj order by order.The leading order outer function,u0, is the single-shock solution of the theorem. For any fixed t, the leading order inner solution U0(ξ,t) is exactly the viscous shock profile with ul(t)≡u(s(t)-0,t),ur(t)≡u(s(t)+0,t), and σ=ṡ(t). So(2.21)U0(ξ,t)=ϕ(ξ,ul(t),ṡ(t)).
Here we take the shift to be zero since it can be absorbed into δ0(t).Next we determineu1,U1, and δ0(t) together. Substituting (2.21) into (2.9) gives(2.22){B(ϕ)U1ξ+B′(ϕ)(U1,ϕξ)}ξ+ṡU1ξ-(f′(ϕ)U1)ξ=δ̇0(t)ϕξ+∂ϕ∂ulu̇l+∂ϕ∂ṡs̈.
By the matching condition (2.12), we expect that (2.23)U1(ξ,t)=ξ⋅∂xu0(s(t)±0,t)+O(1)asξ⟶±∞.
So we set(2.24)U1(ξ,t)=V1(ξ,t)+D1(ξ,t),
where D1(ξ,t) is a smooth function satisfying (2.25)D1(ξ,t)={ξ⋅∂xu0(s(t)-0,t),ξ<-1,ξ⋅∂xu0(s(t)+0,t),ξ>1.
Then inserting (2.24) into (2.22) and using (2.19)–(2.20) and the identity (2.26)ddtu0(s(t)±0,t)=(ṡI-f′(u0(s(t)±0,t)))u0x(s(t)±0,t),
we obtain (2.27){B(ϕ)V1ξ+B′(ϕ)(V1,ϕξ)}ξ+ṡV1ξ-(f′(ϕ)V1)ξ=δ̇0(t)ϕξ+g(ξ,t),
where |g(ξ,t)|≤cexp{-α|ξ|} for large |ξ|. Define G(ξ,t)=∫0ξg(η,t)dη. Then we have(2.28)B(ϕ)V1ξ+B′(ϕ)(V1,ϕξ)+ṡV1-f′(ϕ)V1=δ̇0(t)ϕ+G(ξ,t)+c(t),
where c(t)∈Rn are integration constants to be determined later. We express V1 in terms of the basis, r1(ϕ),r2(ϕ),…,rn(ϕ), of the right eigenvectors of f′(ϕ). We write(2.29)V1(ξ,t)=∑j=1nαj(ξ,t)rj(ϕ(ξ,t)),u1(s(t)±0,t)=∑j=1nβj±(t)rj(u0(s(t)±0,t)),∂xu0(s(t)±0,t)=∑j=1nγj±(t)rj(u0(s(t)±0,t)).
Here the βj- are for u1(s(t)-0,t) and the βj+ are for u1(s(t)+0,t), and so forth. Then the matching conditions (2.12) are transformed into(2.30)lim⁡ξ→±∞αj(ξ,t)=βj±(t)-δ0(t)γj±(t),j=1,…,n.
Define σj(ϕ)≡lj(ϕ)B(ϕ)rj(ϕ)>0. Multiplying (2.28) by lj(ϕ), and using (2.29), we obtain(2.31)αjξ+σj(ϕ)-1{lj(ϕ)B′(ϕ)(rj(ϕ),ϕξ)+(ṡ-λj(ϕ))}αj=σj(ϕ)-1lj(ϕ)(δ̇0(t)ϕ+G(ξ,t)+c(t))-∑i=1nσj(ϕ)-1αi(ξ,t)lj(ϕ)B(ϕ)riξ(ϕ),j=1,…,n,
and then we have the following result.Lemma 2.1.
There is a smooth solution,α(ξ,t), to (2.31) with the following property:
(2.32)αj(ξ,t)={(ṡ-λj(ul))-1lj(ul)[δ̇0ul+G-+c(t)]+O(1)exp⁡{-α|ξ|},ξ⟶-∞,(ṡ-λj(ur))-1lj(ur)[δ̇0ur+G++c(t)]+O(1)exp⁡{-α|ξ|},ξ⟶+∞,
for j=1,…,n, where G±=limξ→±∞G(ξ,t), and α0 is a positive constant.Proof.
We use the standard iteration argument. Defineα0(ξ,t)≡0, and
(2.33)αjk+1(ξ,t)=∫-∞ξexp⁡{-∫ηξσj(ϕ)-1(lj(ϕ)B′(ϕ)(rj(ϕ),ϕξ)+ṡ-λj(ϕ))dζ}⋅{∑i=1nσj(ϕ)-1lj(ϕ)(δ̇0(t)ϕ+G(ξ,t)+c(t))-∑i=1nσj(ϕ)-1αi(ξ,t)lj(ϕ)B(ϕ)riξ(ϕ)}dη,j<p,αpk+1(ξ,t)=∫0ξexp⁡{-∫ηξσp(ϕ)-1(lp(ϕ)B′(ϕ)(rp(ϕ),ϕξ)+ṡ-λp(ϕ))dζ}⋅{∑i=1nσp(ϕ)-1lp(ϕ)(δ̇0(t)ϕ+G(ξ,t)+c(t))-∑i=1nσp(ϕ)-1αi(ξ,t)lp(ϕ)B(ϕ)riξ(ϕ)}dη,αjk+1(ξ,t)=-∫ξ+∞exp⁡{-∫ηξσj(ϕ)-1(lj(ϕ)B′(ϕ)(rj(ϕ),ϕξ)+ṡ-λj(ϕ))dζ}⋅{∑i=1nσj(ϕ)-1lj(ϕ)(δ̇0(t)ϕ+G(ξ,t)+c(t))-∑i=1nσj(ϕ)-1αi(ξ,t)lj(ϕ)B(ϕ)riξ(ϕ)}dη,j>p.
Set
(2.34)Mk=∑0≤i≤nsup⁡R×[0,T]|αik(ξ,t)|,k=0,1,2,…,Dk=∑0≤i≤nsup⁡R×[0,T]|αik+1(ξ,t)-αik(ξ,t)|,k=0,1,2,….
Then from the lax entropy condition (1.5), we can obtain
(2.35)Dk≤c¯μDk-1,k=1,2,…,
where c¯ is independent of μ. And then for suitably small μ, we have
(2.36)Mk≤c¯M1,k=0,1,…,
and αk(ξ,t) converges uniformly to a smooth bounded function, α(ξ,t), which is a solution to (2.31). The asymptotic behavior of the solution, α(ξ,t), follows from the formulas (2.33).
With Lemma2.1 and the matching condition (2.30) at hand, we can determine, completely the same as in [2], β±(t),δ0(t), and c(t), which guarantees the existence of U1(ξ,t) and u1(x,t). 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,(2.37)lj(ur)[δ̇0ur+c(t)]=(βj+(t)-δ0γj+(t))(ṡ-λj(ur))-lj(ur)G+,for1≤j≤p,(2.38)lj(ul)[ṡur+c(t)]=(βj-(t)-δ0γj-(t))(ṡ-λj(ul))+lj(ul)(ur-ul)δ̇0-lj(ul)G-,for(p+1)≤j≤n,(2.39)(lj(ul)(ul-ur))δ̇0=(lp(ur)-lp(ul))(δ̇0ur+c(t))+(ṡ-λp(ul))βp-(t)-(ṡ-λp(ur))βp+(t)+δ0[γp+(ṡ-λp(ur))-γp-(ṡ-λp(ul))]+lp(ur)G+-ll(ul)G-.
Then we can solve for δ̇0ur+c(t) from (2.37)-(2.38). Substituting the resulting expression into (2.39), by writing βin=(βp-,β(p+1)-,…,βn-,β1+,…,βp+), we arrive at an ordinary differential equation for δ0:
(2.40)δ̇0+E1(t)δ0=E2(t)⋅βin+G1(t),
provided that μ is suitably small. Here E1(t),E2(t), and G1(t) are smooth known functions, and E1(t) and E2(t) remain bounded even as μ→0+. Solving for δ0 from (2.37) up to a constant, we obtain c(t) uniquely in terms of βin. Then substitute the expression of δ0 and c(t) into the equation of the matching condition for outgoing indices to yield the linear relations
(2.41)βout=E(t)⋅βin+F(t),
where βout=(β1-,β2-,…,β(p-1)-,β(p+1)+,…,βn+), and F(t)∈Rn-1 is a smooth known function, E(t) is a smooth (n-1)×(n+1) matrix and remains bounded even as μ→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, u1(x,0), is chosen to satisfy the appropriate compatibility conditions at x=s(0). Thus u1(x,t) is completely determined, which in turn gives δ0 and c(t) by (2.37)-(2.38), and therefore U1(ξ,t).Now we summarize the above discussion to achieve the following.Proposition 2.2.
Ifμ is suitably small, then u1(x,t),U1(ξ,t) and δ0 can be established such that(i)
u1(x,t) and its derivatives are uniformly continuous up to x=s(t), and(2.42)∑0≤k≤6∫0T∫|∂xku1(x,t)|2dxdt<∞.(ii)
U1(ξ,t) and δ0 are smooth functions, and there are an α>0, such that(2.43)U1(ξ,t)=u1(s(t)±0,t)+(ξ-δ0)∂xu0(s(t)±0,t)+O(1)exp⁡{-α|ξ|},asξ⟶±∞.The above constructions can be carried out to any order. In particular, we can determineui,Ui and δi-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(2.44)I(x,t)=ϕ(x-s(t)ɛ+δ0+ɛδ1+ɛδ2,t)+∑i=13ɛiUi(x-s(t)ɛ+δ0+ɛδ1+ɛδ2,t),O(x,t)=u0(x,t)+ɛu1(x,t)+ɛ2u2(x,t)+ɛ3u3(x,t).
Let m∈C0∞(R) satisfy 0≤m(y)≤1, and (2.45)m(y)={1,|y|≤1,0,|y|≥2.
Let γ∈(5/7,1) be a constant. Then we define the approximate solution to (1.1) as(2.46)vɛ(x,t)=m(x-s(t)ɛγ)I(x,t)+(1-m(x-s(t)ɛγ))O(x,t)+d(x,t),
where d(x,t) is a higher-order correction term to be determined.Using the structures of the various orders of inner and outer solutions, we compute that(2.47)vtɛ+f(vɛ)x-ɛ(B(vɛ)vxɛ)x=∑i=3nqi(x,t)+dt-(B(mI+(1-m)O)dx)x-ɛq4x(x,t)+(f(vɛ)-f(vɛ-d))x,
where(2.48)q1(x,t)=(1-m){[f(O)-Γ(f(O))]x-ɛ[B(O)Ox-Γ(B(O)Ox)]x},q2(x,t)=m{[f(I)-Γ(f(I))]x-ɛ[B(I)Ix-Γ(B(I)Ix)]x+ɛ3U3t+ɛ4(δ̇2U1+δ̇1U2+δ̇0U3+ɛδ̇2U2+ɛδ̇1U3+ɛ2δ̇2U3)x},q3(x,t)=mt(I-O)+mx(f(I)-f(O))+{f(mI+(1-m)O)-(mf(I)+(1-m)f(O))}x-ɛmx(B(I)Ix-B(O)Ox),q4(x,t)=B(vɛ)(vɛ-d)x-mB(I)Ix-(1-m)B(O)Ox+(B(vɛ)-B(vɛ-d))dx,
where Γ(f(O)),Γ(B(O)Ox) denote the truncated Taylor's expansion of f(O),B(O)Ox, respectively, at u0, including all the terms of O(1),O(1)ɛ,O(1)ɛ2,O(1)ɛ3, and Γ(f(I)),Γ(B(I)Ix) denote the truncated Taylor's expansion of f(I),B(I)Ix, respectively, at ϕ, including all the terms of O(1),O(1)ɛ,O(1)ɛ2,O(1)ɛ3.In view of our construction, we have(i)
suppq1(x,t)⊆{(x,t):|x-s(t)|≥ɛγ,0≤t≤T}, and
(2.49)∂xlq1(x,t)=O(1)ɛ4-lγ,(∫0T‖q1(⋅,t)‖2dt)1/2≤O(1)ɛ4,(∫0T‖∂xlq1(⋅,t)‖2dt)1/2≤O(1)ɛ4-(l-(1/2))γ,l=1,2,3,(ii)
suppq2(x,t)⊆{(x,t):|x-s(t)|≤2ɛγ,0≤t≤T}, and
(2.50)∂xlq2(x,t)=O(1)ɛ(3-l)γ,l=0,1,2,3,(iii)
suppq3(x,t)⊆{(x,t):ɛγ≤|x-s(t)|≤2ɛγ,0≤t≤T}, and
(2.51)∂xlq3(x,t)=O(1)ɛ(3-l)γ,l=0,1,2,3,where we have used the estimates∂xl(I-O)=O(1)ɛ(4-l)γ on {(x,t):ɛγ≤|x-s(t)|≤2ɛγ},l=0,1,2,3.We now choosed(x,t) to satisfy(2.52)dt-(B(mI+(1-m)O)dx)x=-∑i=13qi(x,t),d(x,0)=0,
so that vɛ satisfies(2.53)vtɛ+f(vɛ)x=ɛ(B(vɛ)vxɛ)x-ɛq4x+(f(vɛ)-f(vɛ-d)).
Since B 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.
Letd(x,t) be the solution of (2.52). Then the following estimates hold for all t∈[0,T]:
(2.54)‖∂xld(⋅,t)‖L∞≤O(1)ɛ(7/2)γ-(l+(1/2)),forl=0,1,2,3,‖∂xld(⋅,t)‖≤O(1)ɛ(7/2)γ-l,l=0,1,2,3,4.
Then for q4, we have
(2.55)‖∂xlq4‖≤O(1)ɛ(7/2)γ-(l+1),l=0,1,2.And by our construction, we obtain the following.Lemma 2.4.
One has(2.56)vɛ(x,t)={u0(x,t)+O(1)ɛ,|x-s(t)|≥ɛγ,ϕ(ξ,t)+O(1)ɛγ,|x-s(t)|≤2ɛγ,∂vɛ∂x=1ɛm∂ξϕ+O(1),∂vɛ∂t=O(1).


## 3. Stability Analysis

We now show that there exists an exact solution to (1.1) in a neighborhood of the approximate solution vɛ(x,t), and that the asymptotic behavior of the viscous solution is given by vɛ for small viscosity ɛ.Suppose thatuɛ(x,t) is the exact solution to (1.1) with the initial data uɛ(x,0)=vɛ(x,0). We decompose the solution as(3.1)uɛ(x,t)=vɛ(x,t)+w¯(x,t),(x,t)∈R×[0,T].
Then using the relation (2.53) for vɛ, we compute that(3.2)w¯t+(f′(vɛ)w¯)x+Q(vɛ,w¯)x=ɛ(B(uɛ)uxɛ-B(vɛ)vxɛ)x+ɛq4x+(f(vɛ-d)-f(vɛ))x,w¯(x,0)=0,
where Q(vɛ,w¯)=f(uɛ)-f(vɛ)-f′(vɛ)w¯ satisfies |Q|≤O(1)w¯ for small w¯.Setw¯(x,t)=w̃x(x,t) in (3.2) and integrate the resulting equation with respect to x to give(3.3)w̃t+(f′(vɛ)w̃)x+Q(vɛ,w̃)=ɛ(B(uɛ)uxɛ-B(vɛ)vxɛ)+ɛq4+f(vɛ-d)-f(vɛ),w̃(x,0)=0,
by making the following scalings,(3.4)w̃(x,t)=ɛw(y,τ),y=x-s(t)ɛ,τ=tɛ,
we transform (3.3) into
(3.5)wτ-ṡwy+f′(vɛ)wy+Q(vɛ,wy)=B(vɛ)wyy+(B(uɛ)-B(vɛ))uyɛ+ɛq4+f(vɛ-d)-f(vɛ),w(y,0)=0.
Then we only need to show that for suitably small ɛ, (3.5) has a unique “small” smooth solution up to τ=T/ɛ. 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 w∈C1([0,τ0]:H3(R1)) for some τ0∈(0,T/ɛ]. Then there exist positive constants μ1,ɛ1 and C, which are independent of ɛ and τ0, such that if
(3.6)0<ɛ<ɛ1,sup⁡0≤τ≤τ0‖w(⋅,τ)‖3+μ≤μ1,
then
(3.7)sup⁡0≤τ≤τ0‖w(⋅,τ)‖32+∫0τ0‖wy(⋅,τ)‖32dτ≤Cɛ7γ-3,
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). Define(3.8)θ(y,τ)=L(vɛ)w(y,τ),M(y,τ)=(∂yL(vɛ))⋅R(vɛ),N(y,τ)=(∂τL(vɛ))⋅R(vɛ).
Then we have(3.9)θy=Mθ+Lwy,θτ=Nθ+Lwτ,θyy=(Mθ)y+Mθy-M2θ+Lwyy.
Using the identity (3.9), we can rewrite (3.5) as(3.10)θτ+(Λ-ṡ)θy+(ṡ-Λ)Mθ-Nθ+LQ(vɛ,Rθy+Ryθ)=L(vɛ)B(vɛ)R(vɛ)(θyy-(Mθ)y-Mθy+M2θ)+L(vɛ)(B(uɛ)-B(vɛ))uyɛ+ɛL(vɛ)q4-ΛL(vɛ)d+L(vɛ)Q1(vɛ,d).In what follows, we usec to denote any positive constant which is independent of ɛ,y, and τ;c¯ to denote any positive constant which is independent of ɛ and μ. And we set ɛ≤1.Now we do the following estimates on transversal waves.Lemma 3.2.
There exist suitably small positive constantsμ2, ɛ2, independent of ɛ and τ0, such that
(3.11)∑k≠p∫|∂yλp(ϕ)|θk2dy≤ddτ[-∑k≠p∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy]+(c¯μ+cɛγ)∫m|∂yλp(ϕ)|θp2dy+(c¯μ+cɛ)‖θy(⋅,τ)‖2+c¯μ‖∂y2w(⋅,τ)‖2+cɛ‖θ(⋅,τ)‖2+cɛ7γ-2,
provided that ∥θ(·,τ)∥L∞ is bounded.Proof.
Using (3.10), we compute that for k≠p,
(3.12)∫|∂yλp(ϕ)|θk2dy=-∫∂yλp(ϕ)θk2dy=2∫λp(ϕ)θk∂yθkdy=2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk×[+L(vɛ)(B(uɛ)-B(vɛ))uyɛ+ɛL(vɛ)q4}k-(λk(vɛ)-λk(ϕ))θky-θkτ-{(ṡ-Λ)Mθ-Nθ+ΛLd+L(Q-Q1)}k+{L(vɛ)B(vɛ)R(vɛ)(θyy-(Mθ)y-Mθy+M2θ)+L(vɛ)(B(uɛ)-B(vɛ))uyɛ+ɛL(vɛ)q4}k]dy≡∑i=112Ji.
We now estimate Ji(1≤j≤12) separately as follows:
(3.13)J1=-2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)(λk(vɛ)-λk(ϕ))θkθkydy=∫∂∂y(λp(ϕ)-ṡλk(ϕ)-ṡ)(λk(vɛ)-λk(ϕ))θk2dy+∫(λp(ϕ)-ṡλk(ϕ)-ṡ)(∇λk(vɛ)∂yvɛ-∇λk(ϕ)∂yϕ)θk2dy≡I1+I2.
By Lemma 2.4,
(3.14)I1≤∫|∂∂y(λp(ϕ)-ṡλk(ϕ)-ṡ)||(λk(vɛ)-λk(ϕ))|θk2dy≤c∫|∂yϕ||(λk(vɛ)-λk(ϕ))|θk2dy≤c∫|y|≤ɛγ-1|∂yλp(ϕ)||(λk(vɛ)-λk(ϕ))|θk2dy+c∫|y|≥ɛγ-1|∂yϕ||(λk(vɛ)-λk(ϕ))|θk2dy≤cɛγ∫|∂yλp(ϕ)|θk2dy+cɛ‖θk(⋅,τ)‖2.
For the second term I2, since
(3.15)|∇λk(vɛ)∂yvɛ-∇λk(ϕ)∂yϕ|=|∇λk(vɛ)m∂yϕ-∇λk(ϕ)∂yϕ+O(1)ɛ|≤|∇λk(ϕ)(m-1)∂yϕ|+|(∇λk(vɛ)-∇λk(ϕ))m∂yϕ|+O(1)ɛ≤O(1)ɛ+O(1)ɛγm|∂yλp(ϕ)|,(3.16)∂yvɛ(y,τ)=m(ɛ1-γy)∂yϕ+O(1)ɛ,
which follows from Lemma 2.4. Then we have
(3.17)I2≤∫|λp(ϕ)-ṡλk(ϕ)-ṡ||∇λk(vɛ)∂yvɛ-∇λk(ϕ)∂yϕ|θk2dy≤cμɛγ∫|∂yλp(ϕ)|θk2dy+cμɛ‖θk(⋅,τ)‖.
Consequently, we obtain
(3.18)J1=I1+I2≤cɛγ∫|∂yλp(ϕ)|θk2dy+cɛ‖θk(⋅,τ)‖2.
Using the estimate (d/dτ)((λp(ϕ)-ṡ)/λk(ϕ)-ṡ)=O(1)ɛ, we find
(3.19)J2=-2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θkθkτdy=-ddτ∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy+∫ddτ(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy≤-ddτ∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy+cɛ‖θk(⋅,τ)‖2.
Notice that the facts
(3.20)M=m(ɛ1-γy)(∇L(vɛ)∂yϕ)R(vɛ)+O(1)ɛ=m(ɛ1-γy)(∇L(ϕ)∂yϕ)R(ϕ)+O(1)ɛγm(ɛ1-γy)∂yϕ+O(1)ɛ,(3.21)|ṡ-λp(ϕ)|≤c¯μ,
we arrive at
(3.22)J3=-2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk[(ṡ-Λ(vɛ))Mθ]kdy≤(c¯μ+cɛγ)∫m|∂yλp(ϕ)||θ|2dy+cɛ‖θ(⋅,τ)‖2,J4=2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk[Nθ]kdy≤cμɛ‖θ(⋅,τ)‖2,
where we have used the estimate |N(y,τ)|≤O(1)ɛ. From Lemma 2.3,
(3.23)J5=-2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk(ΛLd)kdy≤μɛ‖θk(⋅,τ)‖2+cɛ-1∫d2dy≤μɛ‖θk(⋅,τ)‖2+cɛ7γ-2.
Using Lemma 2.4 again, we have
(3.24)|L(vɛ)(Q-Q1)|≤(c¯+cɛγ)(|d|2+|R(vɛ)θy+Ry(vɛ)θ|2).
Then it follows from Lemma 2.3 and (3.16) that
(3.25)J6=-2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk[L(Q-Q1)]kdy≤μ(c¯+cɛγ){∫|θk||d|2dy+(c¯+cɛγ)∫|θk||θy|2dy+∫|θk||Ry(vɛ)θ|2dy}≤μ(c¯+cɛγ){12ɛ‖θk(⋅,τ)‖2+12ɛ-1∫|d|4dy+(c¯+cɛγ)‖θk‖L∞‖θy(⋅,τ)‖2+(c¯μ+cɛγ)‖θk‖L∞∫m|∂yλp(ϕ)||θ|2dy+cɛ2‖θk‖L∞‖θ(⋅,τ)‖212}≤cμɛ14γ-3+μ(c¯+cɛ2γ)‖θk(⋅,τ)‖L∞‖θy(⋅,τ)‖2+cμɛ(1+ɛ‖θk(⋅,τ)‖L∞)‖θ(⋅,τ)‖2+μ(c¯μ+cɛγ)‖θk(⋅,τ)‖L∞∫m|∂yλp(ϕ)|2|θ|2dy.
In view of (3.16)–(3.21),
(3.26)J7=2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk[L(vɛ)B(vɛ)R(vɛ)θyy]kdy=-2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θky[L(vɛ)B(vɛ)R(vɛ)θy]kdy-2∫θk[((λp(ϕ)-ṡλk(ϕ)-ṡ)L(vɛ)B(vɛ)R(vɛ))yθy]kdy≤μ(c¯+cɛγ)‖θy(⋅,τ)‖2+c¯∫(|∂yϕ|+|∂yvɛ|)|θk||θy|dy≤(c¯μ0+cɛγ)‖θy(⋅,τ)‖2+c¯∫|∂yλp(ϕ)||θk||θy|dy+cɛ∫|θk||θy|dy≤12∫|∂yλp(ϕ)||θk|2dy+(c¯μ+cɛ)‖θy(⋅,τ)‖2+cɛ‖θk(⋅,τ)‖2,J8=-2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk[L(vɛ)B(vɛ)R(vɛ)(Mθ)y]kdy=2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)yθk[L(vɛ)B(vɛ)R(vɛ)Mθ]kdy+2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θky[L(vɛ)B(vɛ)R(vɛ)Mθ]kdy≤(c¯+cɛγ)∫|∂yϕ||θk||Mθ|dy+μ‖θy(⋅,τ)‖2+μ(c¯+cɛγ)∫|Mθ|2dy≤μ‖θy(⋅,τ)‖2+(c¯μ+cɛγ)∫|∂yλp(ϕ)||θk|2dy+cɛ‖θ(⋅,τ)‖2.
Same bounds hold for J9 and J10.
Applying Cauchy inequality and (3.21), J11 can be estimated as(3.27)J11=2∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk[L(vɛ)(B(uɛ)-B(vɛ))uyɛ]kdy≤c¯μ∫|θk[L(vɛ)(B(uɛ)-B(vɛ))wyy]k|dy+c¯μ∫|θk[L(vɛ)(B(uɛ)-B(vɛ))vyɛ]k|dy≡K1+K2.
Lemma 2.4 yields,
(3.28)K1≤μ(c¯+cɛγ)∫|θk||wy||∂y2w|dy≤(c¯μ+cɛγ)‖∂y2w(⋅,τ)‖2+(c¯μ+cɛγ)‖θk‖L∞2∫|wy|2dy≤(c¯μ+cɛγ)‖∂y2w(⋅,τ)‖2+(c¯μ+cɛγ)‖θk‖L∞2(‖θy(⋅,τ)‖2+∫|Mθ|2dy)≤(c¯μ+cɛγ)‖∂y2w(⋅,τ)‖2+(c¯μ+cɛγ)‖θk‖L∞2×{‖θy(⋅,τ)‖2+(c¯μ+cɛγ)∫m|∂yλp(ϕ)||θ|2dy+cɛ‖θ(⋅,τ)‖2},
where we used the fact wy=Rθy-RMθ. Similarly,
(3.29)K2≤μ(c¯+cɛγ)∫|θk||wy||∂yvɛ|dy≤(c¯μ+cɛγ)∫|θk||θy||∂yvɛ|dy+(c¯μ+cɛγ)∫|M||θ|2|∂yvɛ|dy≤(c¯μ+cɛγ)‖θy(⋅,τ)‖2+(c¯μ+cɛγ)∫|θ|2|∂yvɛ|2dy+(c¯μ+cɛγ)∫|M||θ|2|∂yvɛ|dy≤(c¯μ+cɛγ)‖θy(⋅,τ)‖2+(c¯μ+cɛγ)∫m|∂yλp(ϕ)||θ|2dy+cɛ‖θ(⋅,τ)‖2.
Thus, combining the above two inequality together, we obtain
(3.30)J11=K1+K2≤(c¯μ+cɛγ)‖∂y2w(⋅,τ)‖2+(1+‖θk‖L∞2)×{(c¯μ+cɛγ)‖θy(⋅,τ)‖2+(c¯μ+cɛγ)∫m|∂yλp(ϕ)||θ|2dy+cɛ‖θ(⋅,τ)‖2}.
Finally,
(3.31)J12=2ɛ∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk[L(vɛ)q4]kdy≤ɛ‖θk(⋅,τ)‖2+cμ2ɛ∫|q4|2dy≤ɛ‖θk(⋅,τ)‖2+cɛ7γ-2.
Summing all the inequalities for k≠p, we arrive at
(3.32)∑k≠p∫|∂yλp(ϕ)|θk2dy≤ddτ[-∑k≠p∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy]+(c¯μ+cɛγ)∫m|∂yλp(ϕ)|θp2dy+(c¯μ+cɛγ)‖θy(⋅,τ)‖2+(c¯μ+cɛγ)‖∂y2w(⋅,τ)‖2+cɛ‖θ(⋅,τ)‖2+cɛ7γ-2,
provided that ∥θ(·,τ)∥L∞ is bounded.
We complete the proof of Lemma3.2.Lemma 3.3.
Suppose that the conditions in Proposition3.1 be satisfied. Then
(3.33)‖w(⋅,τ)‖12+∫0τ‖wy(⋅,τ)‖12dτ+∫0τ∫m|∂yλp(ϕ)||w|2dydτ≤cɛ7γ-3,
for all τ∈[0,τ0], where c is independent of τ0 and ɛ.Proof.
Multiplying (3.10) on the left by θt and integrating over R1, we obtain after integration by parts that
(3.34)12ddτ‖θ(⋅,τ)‖2=∫θtL(vɛ)B(vɛ)R(vɛ)θyydy-∫θt(Λ-ṡ)θydy-∫θt(ṡ-Λ)Mθdy-∫θtNθdy-∫θtLQ(vɛ,Rθy+Ryθ)dy+∫θtLQ1(vɛ,d)dy-∫θtΛLddy+∫θtL(vɛ)B(vɛ)R(vɛ)(-(Mθ)y-Mθy+M2θ)dy+∫θtL(vɛ)(B(uɛ)-B(vɛ))uyɛdy+ɛ∫θtL(vɛ)q4dy.
Next we estimate each term on the right hand side above. First, it follows from (3.16) that
(3.35)∫θtL(vɛ)B(vɛ)R(vɛ)θyydy=-∫θytL(vɛ)B(vɛ)R(vɛ)θydy-∫θt(L(vɛ)B(vɛ)R(vɛ))yθydy≤(-C0+cɛγ)‖θy(⋅,τ)‖2+(c¯+cɛγ)∫|∂yvɛ||θ||θy|dy≤(-C0+cɛγ)‖θy(⋅,τ)‖2+C010‖θy(⋅,τ)‖2+(c¯+cɛγ)∫|∂yvɛ|2|θ|2dy≤(-C0+cɛγ)‖θy(⋅,τ)‖2+C010‖θy(⋅,τ)‖2+(c¯μ+cɛγ)∫m|∂yλp(ϕ)||θ|2dy+cɛ‖θ(⋅,τ)‖2.
Here C0>0 is the minimum of the eigenvalues, valued at u0 and ϕ, of (1/2)(LBR+(LBR)t) and (1/2)(B+Bt). And
(3.36)-∫θt(Λ-ṡ)θydy=12∫∑i=1n∂yλi(vɛ)θi2dy=12∫m(∑i=1n∂yλi(ϕ)θi2)dy+12∫m(∑i=1n(∇λi(vɛ)-∇λi(ϕ))∂yϕθi2)dy+ɛ∫O(1)θ2dy≤-12∫m|∂yλp(ϕ)|θp2dy+cɛγ∫m|∂yλp(ϕ)|θp2dy+(c¯+cɛγ)∑k≠p∫m|∂yλp(ϕ)|θk2dy+cɛ‖θ(⋅,τ)‖2.
By virtue of (3.16)–(3.21), one finds
(3.37)-∫θt(ṡ-Λ(vɛ))Mθdy=∫θt(Λ(ϕ)-ṡ)Mθdy+∫θt(Λ(vɛ)-Λ(ϕ))Mθdy≤(18+c¯μ+cɛγ)∫m|∂yλp(ϕ)|θp2dy+(c¯+cɛγ)∑k≠p∫m|∂yλp(ϕ)|θk2dy+cɛ‖θ(⋅,τ)‖2.
Now the remaining terms on the right hand side of (3.34) can be estimated in a similar way to that in the proof of Lemma 3.2. We list them below:
(3.38)∫θtNθdy≤cɛ‖θ(⋅,τ)‖2,-∫θtL(Q-Q1)dy≤cɛ14γ-3+(c¯+cɛγ)‖θ‖L∞‖θy(⋅,τ)‖2+cɛ(1+ɛ‖θ‖L∞)‖θ(⋅,τ)‖2+(c¯μ+cɛγ)‖θ‖L∞∫m|∂yλp(ϕ)||θ|2dy.
Lemma 2.3 leads to
(3.39)-∫θtΛLddy≤ɛ‖θ(⋅,τ)‖2+cɛ-1∫d2dy≤ɛ‖θ(⋅,τ)‖2+cɛ7γ-2.
Continuing, we compute that
(3.40)-∫θtL(vɛ)B(vɛ)R(vɛ)(Mθ)ydy,=∫θytL(vɛ)B(vɛ)R(vɛ)Mθdy+∫θt(L(vɛ)B(vɛ)R(vɛ))yMθdy≤C010‖θy(⋅,τ)‖2+(c¯+cɛγ)∫|Mθ|2dy+(c¯μ0+cɛγ)∫|M||θ|2dy≤C010‖θy(⋅,τ)‖2+(c¯μ+cɛγ)∫m|∂yλp(ϕ)||θ|2dy+cɛ‖θ(⋅,τ)‖2.
Same bounds hold for ∫θtL(vɛ)B(vɛ)R(vɛ)(-Mθy+M2θ)dy. As before, using Lemma 2.4 and (3.16)-(3.20), we obtain
(3.41)∫θtL(vɛ)(B(uɛ)-B(vɛ))uyɛdy=∫θtL(vɛ)(B(uɛ)-B(vɛ))wyydy+∫θtL(vɛ)(B(uɛ)-B(vɛ))vyɛdy≤(C010+(c¯+cɛγ)‖θ‖L∞)‖θy(⋅,τ)‖2+‖θ‖L∞‖∂y2w(⋅,τ)‖2+(c¯μ+cɛγ)∫m|∂yλp(ϕ)||θ|2dy+cɛ‖θ(⋅,τ)‖2.
In view of (2.55),
(3.42)ɛ∫θtL(vɛ)q4dy≤ɛ‖θ(⋅,τ)‖2+cɛ∫|q4|2dy≤ɛ‖θ(⋅,τ)‖2+cɛ7γ-2.
Collecting all the estimates previously we have achieved, we get
(3.43)12ddτ‖θ(⋅,τ)‖2+35C0‖θy(⋅,τ)‖2+38∫m|∂yλp(ϕ)|θp2dy≤(c¯+cɛγ)‖θ‖L∞‖θy(⋅,τ)‖2+‖θ‖L∞‖∂y2w(⋅,τ)‖2+(c¯μ+cɛγ)∑k≠p∫m|∂yλp(ϕ)|θk2dy+cɛ‖θ(⋅,τ)‖2+cɛ7γ-2,
provided that ∥θ(·,τ)∥L∞ is bounded.
Insert (3.11) into (3.43) to arrive at(3.44)12ddτ‖θ(⋅,τ)‖2+35C0‖θy(⋅,τ)‖2+38∫m|∂yλp(ϕ)|θp2dy≤(c¯+cɛγ)‖θ‖L∞‖θy(⋅,τ)‖2+‖θ‖L∞‖∂y2w(⋅,τ)‖2+(c¯μ+cɛγ)∫m|∂yλp(ϕ)|θp2dy+(c¯+cɛγ){ddτ[-∑k≠p∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy]+(c¯μ+cɛγ)∫m|∂yλp(ϕ)|θp2dy+(c¯μ+cɛ)‖θy(⋅,τ)‖2+c¯μ‖∂y2w(⋅,τ)‖2+cɛ‖θ(⋅,τ)‖2+cɛ7γ-2[-∑k≠p∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy]}+cɛ‖θ(⋅,τ)‖2+cɛ7γ-2,
which gives
(3.45)12ddτ‖θ(⋅,τ)‖2+35C0‖θy(⋅,τ)‖2+38∫m|∂yλp(ϕ)|θp2dy≤(c¯+cɛγ)(‖θ‖L∞+c¯μ+cɛγ)‖θy(⋅,τ)‖2+(‖θ‖L∞+c¯μ+cɛγ)‖∂y2w(⋅,τ)‖2+(c¯+cɛγ){ddτ[-∑k≠p∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy]+(c¯μ+cɛγ)∫m|∂yλp(ϕ)|θp2dy+cɛ‖θ(⋅,τ)‖2+cɛ7γ-2[-∑k≠p∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy]}.
Differentiating (3.5) with respect to y, multiplying the resulting equation by ∂ywt on the left and integrating over R1, we obtain after integration by parts that
(3.46)12ddτ‖wy(⋅,τ)‖2=-∫∂y2wtB(vɛ)∂y2wdy+∫∂y2wt(f′(vɛ)wy+Q(vɛ,wy)-(f(vɛ-d)-f(vɛ))-ɛq4)dy-∫∂y2wt(B(uɛ)-B(vɛ))(∂y2w+vyɛ)dy≤-C0‖∂y2w(⋅,τ)‖2+C04‖∂y2w(⋅,τ)‖2+(c¯+cɛγ)∫(|wy|2+|wy|4+|d|2+ɛ2|q4|2)dy+c¯∫|(B(uɛ)-B(vɛ))∂y2w|2dy+c¯∫|(B(uɛ)-B(vɛ))vyɛ|2dy.
Using Sobolev's inequality, we have
(3.47)∫|wy|4dy≤‖wy‖L∞2‖wy‖2≤c¯‖wy‖3‖∂y2w‖≤C08‖∂y2w(⋅,τ)‖2+c¯‖wy(⋅,τ)‖6,
and so
(3.48)∫(|wy|2+|wy|4)dy≤C08‖∂y2w(⋅,τ)‖2+c¯(1+‖wy(⋅,τ)‖4)‖wy(⋅,τ)‖2≤C08‖∂y2w(⋅,τ)‖2+c¯(1+μ14)‖wy(⋅,τ)‖2.
Lemma 2.3 gives
(3.49)∫(|d|2+ɛ2|q4|2)dy≤cɛ7γ-1.
The last term can be estimated as
(3.50)c¯∫|(B(uɛ)-B(vɛ))∂y2w|2dy+c¯∫|(B(uɛ)-B(vɛ))vyɛ|2dy≤c∫|wy|2|∂y2w|2dy+(c¯+cɛγ)∫|wy|2|∂yvɛ|2dy≤c‖w(⋅,τ)‖22‖∂y2w(⋅,τ)‖2+(c¯μ+cɛγ)‖wy(⋅,τ)‖2≤c‖w(⋅,τ)‖22‖∂y2w(⋅,τ)‖2+(c¯μ+cɛγ)‖θy(⋅,τ)‖2+(c¯μ2+cɛγ)∫m|∂yλp(ϕ)||θ|2dy+cɛ‖θ(⋅,τ)‖2.
By taking ∥w∥2 to be sufficiently small, we get
(3.51)12ddτ‖wy(⋅,τ)‖2+C02‖∂y2w(⋅,τ)‖2≤c¯(1+μ14)‖θy(⋅,τ)‖2+(c¯μ+cɛγ)‖θy(⋅,τ)‖2+(c¯μ(1+μ)+cɛγ)∫m|∂yλp(ϕ)||θ|2dy+cɛ‖θ(⋅,τ)‖2.
Denote the constant c¯(1+μ14) in this inequality by c1¯. Insert (3.11) into (3.51) to give
(3.52)12ddτ‖wy(⋅,τ)‖2+C02‖∂y2w(⋅,τ)‖2-c1¯‖θy(⋅,τ)‖2≤(c¯μ(1+μ2)+cɛγ)‖θy(⋅,τ)‖2+(c¯μ2(1+μ2)+cɛγ)‖∂y2w(⋅,τ)‖2+(c¯μ(1+μ)+cɛγ)ddτ[-∑k≠p∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy]+(c¯μ(1+μ2)+cɛγ)∫m|∂yλp(ϕ)|θp2dy+cɛ‖θ(⋅,τ)‖2+cɛ7γ-2.
Multiplying suitably small constants to (3.11) and (3.52), respectively, then adding the resulting inequalities to (3.45) and taking ∥θ∥L∞,μ and ɛ sufficiently small, we can obtain the following inequality:
(3.53)ddτ(‖θ(⋅,τ)‖2+‖wy(⋅,τ)‖2)+‖θy(⋅,τ)‖2+‖∂y2w(⋅,τ)‖2+∫m|∂yλp(ϕ)||θ|2dy≤(c¯(1+μ2)+cɛγ)ddτ[-∑k≠p∫(λp(ϕ)-ṡλk(ϕ)-ṡ)θk2dy]+cɛ‖θ(⋅,τ)‖2+cɛ7γ-2.
Integrating the above inequality to give
(3.54)‖θ(⋅,τ)‖2+‖wy(⋅,τ)‖2+∫0τ(‖θy(⋅,τ)‖2+‖∂y2w(⋅,τ)‖2)dτ+∫0τ∫m|∂yλp(ϕ)||θ|2dydτ≤(c¯(1+μ2)+cɛγ)μ∑k≠p‖θk(⋅,τ)‖2+cɛ∫0τ‖θ(⋅,τ)‖2+cτɛ7γ-2.
Take μ suitably small to yield
(3.55)‖θ(⋅,τ)‖2+‖wy(⋅,τ)‖2+∫0τ(‖θy(⋅,τ)‖2+‖∂y2w(⋅,τ)‖2)dτ+∫0τ∫m|∂yλp(ϕ)||θ|2dydτ≤cɛ∫0τ‖θ(⋅,τ)‖2+cτɛ7γ-2.
It follows from the Gronwall's inequality that
(3.56)‖θ(⋅,τ)‖2+‖wy(⋅,τ)‖2+∫0τ(‖θy(⋅,τ)‖2+‖∂y2w(⋅,τ)‖2)dτ+∫0τ∫m|∂yλp(ϕ)||θ|2dydτ≤cɛ7γ-3.
In particular,
(3.57)ɛ∫0τ‖θ(⋅,τ)‖2dτ≤cɛ7γ-3.
By the fact w=Rθ and wy=Rθy-RMθ, we obtain
(3.58)‖w(⋅,τ)‖12+∫0τ‖wy(⋅,τ)‖12dτ+∫0τ∫m|∂yλp(ϕ)||w|2dydτ≤cɛ7γ-3.Lemma 3.4.
Let the conditions in Proposition3.1 be satisfied. Then
(3.59)‖∂y2w(⋅,τ)‖12+∫0τ‖∂y3w(⋅,τ)‖12dτ≤cɛ7γ-3,forτ∈[0,τ0],
with the constant c independent of τ0 and ɛ.Proof.
Applying∂yl to (3.5), multiplying on the left of the resulting equation by ∂ylwt, and integrating over R1, we compute that
(3.60)12ddτ‖∂ylw(⋅,τ)‖2=-∫∂yl+1wt∂yl-1(B(vɛ)∂y2w)dy+∫∂yl+1wt∂yl-1{f′(vɛ)wy+Q(vɛ,wy)-(f(vɛ-d)-f(vɛ))-ɛq4}dy-∫∂yl+1wt∂yl-1{(B(uɛ)-B(vɛ))∂y2w}dy-∫∂yl+1wt∂yl-1{(B(uɛ)-B(vɛ))vyɛ}dy.
In the case l=1, this gives
(3.61)12ddτ‖∂y2w(⋅,τ)‖2=-∫∂y3wt(B(vɛ)∂y2w)ydy+∫∂y3wt{(f′(vɛ)wy)y+Q(vɛ,wy)y-(f(vɛ-d)-f(vɛ))y-ɛq4y}dy-∫∂y3wt{(B(uɛ)-B(vɛ))∂y2w}ydy-∫∂y3wt{(B(uɛ)-B(vɛ))vyɛ}ydy≡∑i=14Ri.
Using Cauchy inequality, we obtain
(3.62)R1=-∫∂y3wt(B(vɛ)∂y2w)ydy=-∫∂y3wtB(vɛ)∂y3wdy-∫∂y3wt(B(vɛ))y∂y2wdy≤-C0‖∂y3w(⋅,τ)‖2+C08‖∂y3w(⋅,τ)‖2+c‖∂y2w(⋅,τ)‖2.
Lemma 2.3 and Sobolev’s inequality give
(3.63)R2≤C08‖∂y3w(⋅,τ)‖2+c∫(|wy|2+|∂y2w|2+|wy|4+|wy|2|∂y2w|2+|d|2+|dy|2+ɛ2|q4y|2)dy≤C08‖∂y3w(⋅,τ)‖2+c(1+‖wy‖L∞2)(‖wy‖2+‖∂y2w‖2)+cɛ7γ-1≤C08‖∂y3w(⋅,τ)‖2+c(1+‖w‖22)(‖wy‖2+‖∂y2w‖2)+cɛ7γ-1≤C08‖∂y3w(⋅,τ)‖2+c(‖wy‖2+‖∂y2w‖2)+cɛ7γ-1,
provided that ∥w∥2 is bounded. For the last two terms, we have
(3.64)R3=-∫∂y3wt(B(uɛ)-B(vɛ))∂y3wdy-∫∂y3wt(B(uɛ)-B(vɛ))y∂y2wdy≤c∫|wy||∂y3w|2dy+c∫|∂y3w|(|wy|+|∂y2w|)|∂y2w|dy≤c‖wy‖L∞‖∂y2w(⋅,τ)‖12+c‖∂y2w‖L∞‖∂y2w(⋅,τ)‖‖∂y3w(⋅,τ)‖≤c‖w‖2‖∂y2w(⋅,τ)‖12+‖∂y2w(⋅,τ)‖3/2‖∂y3w(⋅,τ)‖3/2≤c‖w‖2‖∂y2w(⋅,τ)‖12+‖∂y2w(⋅,τ)‖(‖∂y2w(⋅,τ)‖2+‖∂y3w(⋅,τ)‖2)≤c‖w‖2‖∂y3w(⋅,τ)‖2,R4≤C08‖∂y3w(⋅,τ)‖2+c‖wy(⋅,τ)‖12.
Choose ∥w∥2 sufficiently small to yield
(3.65)ddτ‖∂y2w(⋅,τ)‖2+‖∂y3w(⋅,τ)‖2≤c‖wy(⋅,τ)‖12+cɛ7γ-1.
Integrate the above inequality, and use Lemma 3.3 to give
(3.66)‖∂y2w(⋅,τ)‖2+∫0τ‖∂y3w(⋅,τ)‖2≤cɛ7γ-3.
Similarly, for the case l=2, we can obtain
(3.67)‖∂y3w(⋅,τ)‖2+∫0τ‖∂y4w(⋅,τ)‖2≤cɛ7γ-3.
This finishes the proof of Lemma 3.4.Combining the results of Lemmas3.3-3.4, we complete the proof of Proposition 3.1.


## 4. Proof of Theorem1.2

Using Proposition3.1 and the standard continuous induction argument, we conclude the following.Proposition 4.1.
There exist positive constantsη0,μ0, and C, which are independent of ɛ such that if 0<ɛ<ɛ0, and 0≤μ≤μ0, then the Cauchy problem (3.5) has a unique solution w∈C1([0,T/ɛ]:H3(R1)). Furthermore, the following inequality holds
(4.1)sup⁡0≤t≤T/ɛ‖w(⋅,t)‖32+∫0T/ɛ‖wy(⋅,t)‖32dt≤Cɛ7γ-3.Now we chooseγ∈(η,1)∩((5/7),1). Then due to (4.1), we have (4.2)sup⁡0≤t≤T‖(uɛ-vɛ)(⋅,t)‖2=sup⁡0≤t≤T‖w̃x(⋅,t)‖2=ɛsup⁡0≤τ≤T/ɛ‖wy(⋅,τ)‖2≤Cɛ7γ-2≤Cɛ3.
On the other hand, it follows from Lemma 2.4 that (4.3)sup⁡0≤t≤T‖(vɛ-u0)(⋅,t)‖2≤Cɛγ≤Cɛη.
Consequently, (4.4)sup⁡0≤t≤T‖(uɛ-u0)(⋅,t)‖2≤sup⁡0≤t≤T‖(uɛ-vɛ)(⋅,t)‖2+sup⁡0≤t≤T‖(vɛ-u0)(⋅,t)‖2≤cɛη,
which gives (1.8). Finally, (4.5)‖(uɛ-vɛ)(⋅,t)‖L∞=‖wy(⋅,t)‖L∞≤c‖wy(⋅,t)‖1/2‖∂y2w(⋅,t)‖1/2≤cɛ(7γ-3)/2≤cɛ.
This yields (1.9) by using Lemma 2.4 again. We complete the proof of Theorem 1.2.



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*Source: 782306-2012-01-05.xml*