Asymptotic stability of traveling waves for viscous scalar conservation laws

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Asymptotic stability of traveling waves for viscous scalar conservation laws

Hyun Ki Kim

Department of Mathematical Sciences

Graduate School of UNIST

Asymptotic stability of traveling waves for viscous scalar conservation laws

A dissertation

submitted to the Graduate School of UNIST in partial fulfillment of the

requirements for the degree of Master of Science

Hyun Ki Kim

17.12.2016 Approved by

Advisor

Bongsuk Kwon

Asymptotic stability of traveling waves for viscous scalar conservation laws

Hyun Ki Kim

This certifies that the dissertation of Hyun Ki Kim is approved.

17.12.2016

Advisor: Bongsuk Kwon

Hantaek Bae: Thesis Committee Member #1

Kyudong Choi : Thesis Committee Member #2

I dedicate this dissertation to my parents

Abstract

There are many research about Viscous scalar conservation law u_t+f(u)_x =µu_xx. In this paper, we study stability and asymptotic stability of travelling solutions for viscous scalar conservation law. The major result shows that if Rankine Hugoniot condition, the generalized shock condition and some assumptions hold, there exist a solution approaches to the travelling solution, satisfy stability and asymptotic stability problems at corresponding rate. The important feature of this paper is to employ an appropriate weight function to show the stability and asymptotic behavior of the viscous shock waves.

Contents

1 Introduction 1

2 Existence of travelling wave solution 4

3 Reformulation of the Problem 7

4 Stability of travelling wave for convex flux case 9 5 Stability of travelling wave for non convex flux case 14 6 Asymptotic stability of travelling wave for non convex flux case 20

7 Conclusion 28

8 Appendix 30

References 31

1

Introduction

We consider the Cauchy problem for viscous scalar conservation laws:

ut+f(u)x=µuxx, (x, t)∈R×R⁺, (1.0.1)

u(x,0) =u₀(x), x∈R, (1.0.2)

whereµ >0 is called the coefficient of viscosity, f ∈C² and the initial data u0(x) satisfies

u₀(x)→u± as x→ ±∞. (1.0.3)

Let this scalar viscous conservation law admits smooth travelling wave solutions with shock profile

u(x, t) =U(ξ) , ξ =x−st, (1.0.4)

U(ξ)→u± as ξ → ±∞. (1.0.5)

ssatisfies the Rankine Hugoniot condition

s(u+−u−) =f(u+)−f(u−), (1.0.6) and the generalized shock condition

h(u) =−s(u−u±) +f(u)−f(u±)

<0 (u₊< u < u−),

>0 (u−< u < u₊). (1.0.7) The generalized shock condition means

f⁰(u+)≤s≤f⁰(u−). (1.0.8)

About travelling waves for viscous scalar conservation law, Il’in and Oleinik [1], Nishihara [2], Kawashima and Matsumura[3] investigated the asymptotic stability of travelling wave for viscous scalar conservation law with convex function f. However, we need to generalize the condition about f, because the convex function f is limited case. So, there were many re- searches about the non convex function with some assumptions. Matsumura and Kawashima [4], investigated stability of travelling waves with convex and concave function that has a one inflection point. Jones, Gardner and Kapitula[7] investigated stability of travelling wave and decay rate withC² function.

Using these results, Nishihara and Matsumura [6] generalized the asymptotic stability of travel- ling wave with non convex functionf. So, our purpose is to survey on their results on stability and decay rate for any C² function.

In section 2, we consider the existence of travelling wave using Lipschitz condition. In section 3, we reformulate the viscous scalar conservation law using section 2 results. In section 4, we check the stability of travelling wave for convex flux case using local existence and a priori estimate.

In section 5, we check stability of travelling wave for non convex flux case using local existence and a priori estimate. In section 6 and 7, we show asymptotic decay rate of the viscous shock waves for C² functionf.

To find the a priori estimates for each case, we use the section 3 and combine them with local existence. Then, we show that stability and asymptotic decay rate for each section. In these progresses, we use an Energy method with appropriate weight function.

Notation 1.

1.0.1 We denote the constant C_a,b,.. depending ona, b, .. by C_a,b,.. or only by C.

1.0.2 We denote f(x)∼g(x) as x→awhen C⁻¹g < f < Cg in a neighbourhood ofa.

1.0.3 We denote byL² space with the norm

||f||² =R

R|f(x)|²dx.

1.0.4 H^l is the sobolev space ofl th order with the norm

||f||²_l = Σ^l_j=0||(_∂x^∂ )^jf(x)||².

1.0.5 L²_w is the weighted L² space. f ∈L²_w meansw^1/2f ∈L² with the norm

|f|²_w =R

Rw(x)|f(x)|²dx.

1.0.6 Ifw(x) =hxi^a= (1 +x²)^α/2, we writeL²_w=L²_α and | · |_w=| · |_α.

1.0.7 If a weighted function ishxi^αw, we denote byf ∈L²_α,w with the norm

|f|_α,w = (R

R< x >^αw(x)|f(x)|²dx)^1/2.

For example, ifC⁻¹ ≤w(x)≤C, we know thatL²=H⁰=L²₀=L²_w with|| · ||=|| · ||₀ =| · |₀∼

| · |_w and thatL²_α,w=L²_α with| · |_α,w ∼| · |_α.

2

Existence of travelling wave solution

The first thing we need to check is the existence of travelling wave solutions of viscous scalar conservation law. If f is convex, we get the 2 Lemma.

Lemma 2.0.1. If (1.0.1) admits a travelling wave U(x−st), satisfies U(±∞) = u± and f is convex, then u± and s must satisfy the Rankine Hugoniot condition and the generalized shock condition.

Proof. Since we admitu(x, t) =U(x−st) =U(ξ), (1.0.1) satisfy −sU⁰+f(U)⁰=µU⁰⁰. Integrating over (±∞, ξ), we get

µU⁰ =−sU+f(U)−c=h(U). (2.0.1)

Since U(±∞) =u± ,U⁰(±∞) = 0 and using ξ→ ±∞ in (2.0.1) we know

h(u±) = 0 and c=−su_±+f(u±). (2.0.2) It equals to the Rankine Hugoniot condition.

The equation (2.0.1) withh(u±) = 0 admits a smooth solution U(ξ) satisfyingU(±∞) =u± if and only if

h(u)<0, if u₊< u−, h(u)>0, if u+> u−.

(2.0.3) is done. So, we know that the condition ofh(U) must be satisfied becausef is convex. Since the generalized shock condition is proved that it’s equivalent to the condition of h(U), we proved that the Rankine Hugoniot condition and the generalized shock condition are necessary for the existence of a travelling waveU(x−st) satisfying U(±∞) =u±.

Lemma 2.0.2. (Existence of travelling solutions with convex function f) Suppose the Rankine Hugoniot condition and the generalized shock condition hold. Then, there exist a travelling wave U(x−st), satisfyingU(±∞) =u±. The U(ξ) is a monotone function of ξ.

Proof. We have

Differential equation: Uξ=h(U, ξ) , U(±∞) =u±

Initial condition: U(ξ∗) =U∗

To prove Lemma 2.0.2, we use Lipschitz condition.

Since h(U_k(ξ)) uniformly converge to h(U·(ξ)) = U·(ξ) by the Pichard iteration, the global solution U(ξ) exist, ifh(U) is global lipschitz ofU.

So, we need to showh(U) is global lipschitz.

Since h is convex,U can’t beU ≥u− and U ≤u+. It meansU ∈[u+, u−].

So, we get ∀(ξ, U)∈(R,(u₊, u−))

|_∂U^∂h|=|h⁰(U)|=|f⁰(U)−s| ≤sup_{U∈[u+,u−]}|f⁰(U)|+s≤max(|f⁰(u+)|,|f⁰(u−)|) +s≤K.

It means h(U) is Lipschitz. So, the global solutionU(ξ) exist.

By Lemma 2.0.1 and Lemma 2.0.2, we know:

there exists a travelling waveU(x−st) and it satisfiesU(±∞) =u±if and only ifu±and shock speedssatisfy the Rankine-Hugoniot condition and the generalized shock condition.

However, we can’t use Lemma 2.0.2 for the non-convex function f. So, to prove existence of travelling wave with non convex function f, we need some condition ofh(U).

Lemma 2.0.3. (Existence of travelling wave withf ∈C²) Assume Rankine-Hugoniot condition, the generalized shock condition and

|h(U)|∼|U −u±|^1+k± , asU →u±, (2.0.4)

with k± ≥ 0. Then there exists a travelling wave solution U(ξ) of viscous scalar conservation law with U(±∞) =u±. Also,

|U(ξ)−u±|∼e^−c|ξ| , if f⁰(u+)< s < f⁰(u−), (2.0.5)

|U(ξ)−u±|∼|ξ|^−1/k+ , if s=f⁰(u+), (2.0.6)

|U(ξ)−u±|∼|ξ|^−1/k− , if s=f⁰(u−), (2.0.7) is done as ξ→ ±∞, where k± is denoted to h⁽ⁿ⁾(u±) = 0 if 1≤n≤k± and h⁽ⁿ⁺¹⁾(u±)6= 0.

Proof. We have

Differential equation: U_ξ=h(U, ξ) , U(±∞) =u±

Initial condition: U(ξ∗) =U∗

To prove Lemma 2.0.3, we use Lipschitz condition.

Since h(U_k(ξ)) uniformly converge to h(U·(ξ)) = U·(ξ) by the Pichard iteration, the global solutionU(ξ) exist, ifh(U) is global lipschitz ofU. So, we need to showh(U) is global lipschitz.

Since h < 0 is non convex, |h(U)| ∼ |U −u±|^1+k± help U → u± as ξ → ∞. It means U is bounded.

So, we get ∀(ξ, U)∈(R,(u₊, u−))

|_∂U^∂h|=|h⁰(U)|=|f⁰(U)−s| ≤sup_{U∈[u+,u−]}|f⁰(U)|+s≤max(|f⁰(u+)|,|f⁰(u−)|) +s≤K.

It means h(U) is Lipschitz. So, the global solutionU(ξ) exist.

By the implicit formula, we get 1. f⁰(u+)< s < f⁰(u−) case,

Since h⁰(u±) =f⁰(U)−s6= 0 , |h(U)|∼|U −u±|asU →u± . Therefore, we get RU(ξ)

u++u− 2

1

|u−u±|du=ln(u−u±)|^u=U(ξ)

u=^u++₂^u− =ξ+c.

So, we know|U(ξ)−u±|∼e^−c|ξ| asξ→ ±∞ for some constantc.

2. s=f⁰(u₊) orf⁰(u−) case,

In these cases,|h(U)|∼|U−u±|^1+k± as U →u±. Therefore, RU(ξ)

u++u− 2

1

|u−u±|^1+k±du= ^c

(u−u±)^k±|^u=U(ξ)

u=^u++₂^u− =ξ+c.

So, we know

|U(ξ)−u₊|∼ _|ξ|1/k^c + as ξ→ ∞ for some constantcifs=f⁰(u₊),

|U(ξ)−u−|∼ _|ξ|1/k^c − asξ → −∞for some constantc ifs=f⁰(u−).

3

Reformulation of the Problem

For the stability and asymptotic stability of travelling wave solution U, we need a condition about U and the initial data u0. So, we add the assumption u0 −U is integrable. Since U(±∞) =u±, we get

R

R(u₀(x)−U(±∞))dx=R

R(u₀(x)−u±)dxT0.

It means we determineU which satisfies Z

R

(u₀(x)−U(x))dx= 0, (3.0.1)

and define

ψ0(x) :=R_x

−∞(u0(y)−U(y))dy.

Let U(ξ) be the travelling wave solution where satisfy Lemma 2.0.3 and (3.0.1) , then we get

u(x, t) =U(ξ) +ψ_ξ(ξ, t) , whereξ =x−st. (3.0.2) It means (1.0.1) with the initial data is

ψ_t−sψ_ξ+ (f(U +ψ_ξ)−f(U)) =µψ_ξξ. (3.0.3)

ψ(ξ,0) = Z ξ

−∞

(u₀−U)(η)dη. (3.0.4)

(3.0.3) is equivalent to the following form

ψ_t+h⁰(U)ψ_ξ−µψ_ξξ=F, (3.0.5)

F =−(f(U +ψξ)−f(U)−f⁰(U)ψξ). (3.0.6)

Now, let select the weight as

w=w(U) =|(U−u+)(U −u−)

h(U) |. (3.0.7)

Since |h(U)|∼|U −u±|^1+k± in Lemma 2.0.3, we get the following form

iff⁰(u₊)< s < f⁰(u−) , w(U)∼C asU →u± and L²_α,w(U)=L²_α, (3.0.8) iff⁰(u+) =s < f⁰(u−) ,w(U)∼|U−u+|^−k+ asU →u+, (3.0.9)

orw(U(ξ))∼hξi asξ → ∞and hence L²_w(U)=L²_hξi

+,

iff⁰(u₊)< s=f⁰(u−) ,w(U)∼|U−u−|^−k− asU →u−, (3.0.10) orw(U(ξ))∼hξi asξ→ −∞ and henceL²_w(U) =L²_hξi

−,

iff⁰(u₊) =s=f⁰(u−) ,w(U)∼|U−u±|^−k± asU →u±, (3.0.11) orw(U(ξ))∼hξi asξ→ ±∞ and henceL²_w(U) =L²_hξi

− =L²₁.

Remark 3.0.1. The reason why the weight functionw(U) is introduced and has the value that form is the key point of this thesis. We consider theC² functionf(u), it’s not always h⁰⁰(U) = f⁰⁰(u) > 0. It means there are problems just doing estimate without using other method.

However, with the weight function w(U), w(U)h(U) is convex function, we can estimate much easily. The details comes out later.

We define the solution space

X(0, T) ={ψ∈C⁰(0, T;H²∩L²_w(U)), ψξ∈L²(0, T;H²∩L²_w(U)}.

Then U be the solution globally in time about stability and asymptotic stability for (1.0.1).

To prove it, we suppose there is a local existence result.

Proposition 3.0.1. (local existence)For any0 >0, there exists a positive constant To depend- ing on ₀ such that if ψ₀∈H²∩L²_w(U) and ||ψ₀||₂≤₀, then the problem has a unique solution ψ∈X(0, T) satisfying ||ψ(t)||₂≤20 for 0≤t≤T0.

We check the case u₊ < u−, h(U) ≤0 for U ∈[u₊, u−] because the other case, u₊ > u−, h(U)≥0, is proved in the same method.

4

Stability of travelling wave for convex flux case

In this chapter, we check A priori estimate and Stability of travelling waves for convex flux using continuation argument.

Theorem 4.0.1. (A priori estimate) Letψbe a solution of the(3.0.5) and satisfying inX(0, T) for a constant T >0. Then there exists a constant 3 >0 such that if sup_0≤t≤T ||ψ(t)||₂ < 3, thenψ(t) satisfies

||ψ(t)||²₂+Rt

0||ψ_ξ(τ)||²₂dτ ≤C₂²(||ψ₀||²₂), for any0≤t≤T.

The key point of this section is to use the standard energy method.

Lemma 4.0.1. Let ψ∈X(0, T) be a solution of (3.0.5). Then, it satisfies

||ψ(t)||+Rt

0 µ||ψ_ξ(τ)||²dτ ≤C||ψ₀||². Proof. Multiplying 2ψon (3.0.5) we get

2ψψ_t+ 2ψψ_ξh⁰(U)−2µψψ_ξξ = 2ψF. (4.0.1) It means

(ψ²)t+ (ψ²h⁰(U))ξ−h⁰⁰(U)Uξψ²−(2µψψξ)ξ+ 2µψ_ξ²= 2ψF, (4.0.2) In (4.0.2), (·)_ξ is the term which disappears after integration over R.

||ψ(t)||²+Rt 0

R

R−(h(U))⁰⁰U_ξψ²+ 2µψ²_ξdξdτ =||ψ₀||+Rt 0

R

R2ψF dξdτ. Since U_ξ <0 andh⁰⁰(U) =f⁰⁰(U)≥0 such that we get

||ψ(t)||²+Rt

0 2µ||ψ_ξ||²dτ ≤ ||ψ₀||²+Rt 0

R

R2ψF dξdτ. Let

N(t) = sup_0≤τ≤t||ψ(τ)||₂, and assumeN(t)≤0.

By the taylor extension, |F|=|f(U +ψ_ξ)−f(U)−f⁰(U)ψ_ξ|=O(ψ²_ξ). It satisfies |F| →0 as ξ→ ±∞.

Then, for some ₃ satisfying N(t)< ₃< ₀,

||ψ(t)||²+Rt

0 ||ψ_ξ||²dτ ≤C₂²||ψ₀||².

Moreover, we need to check the similarly case when we apply ∂_ξ and∂_ξξ to rewritten form.

Then, we can get a new lemma.

Lemma 4.0.2. There is a positive constant 3 where is smaller than 0 such that if N(t) ≤ ₃< ₀,

||ψ_ξ(t)||²₁+Rt

0||ψ_ξξ(τ)||²₁dτ ≤C(||(ψ₀)_ξ||²₁).

Proof. For higher order estimate, we apply ∂_ξ to the (3.0.5). Then, ψξt+{(h(U))⁰ψξ}_ξ−µψξξξ=Fξ. Multiplying 2ψ_ξ on (3.0.5) we get

(ψ_ξ²)_t+{(h(U))⁰ψ²_ξ}_ξ− {2µψ_ξξξψ_ξ}_ξ+h⁰⁰(U)U_ξψ_ξ²+ 2µψ_ξξ² = 2ψ_ξF_ξ. (4.0.3) Integrating (4.0.3) over (0, t×R) we get

||ψ_ξ(t)||²+R_t

0

R

Rh⁰⁰(U)U_ξψ²_ξdξ+ 2µ||ψ_ξξ(τ)||dτ =||ψ_ξ(0)||²+R_t

0

R

R2F_ξψ_ξdξdτ. Then, for some₃ satisfyingN(t)< ₃ < ₀, there exist C such that

||ψ_ξ(t)||²+Rt 0

R

Rh⁰⁰(U)U_ξψ_ξ²dξ+µ||ψ_ξξ(τ)||dτ ≤C(||ψ_ξ(0)||²).

By, −Rt 0

R

Rh⁰⁰(U)U_ξψ²_ξdξ≥0, we have

||ψ_ξ(t)||²+Rt

0µ||ψ_ξξ(τ)||dτ ≤C(||ψ_ξ(0)||²)−Rt 0

R

Rh⁰⁰(U)Uξψ_ξ²dξdτ,

≤C(||ψ_ξ(0)||²) +cRt 0

R

Rψ²_ξdξdτ,

wherec= sup_ξ∈R{|h⁰⁰(U)U_ξ|}<∞.

Applying ∂_ξξ to (3.0.5) we get

ψ_ξξt+{(h(U))⁰ψ_ξ}_ξξ−µψ_ξξξξ=F_ξξ, Multiplying 2ψ_ξξ we get

2ψ_ξξψ_ξξt+{(h(U))⁰ψ_ξ}_ξξ2ψ_ξξ−µ2ψ_ξξψ_ξξξξ=F_ξξ2ψ_ξξ (4.0.4) we changed the second term in (4.0.4) to the other form as follows

(h⁰(U)ψξ)ξξψξξ ={(h⁰(U)ψξ)ξψξξ}_ξ−(h⁰(U)ψξ)ξψξξξ

={(h⁰(U))_ξψ_ξψ_ξξ+h⁰(U)ψ_ξξ² }_ξ−(h⁰(U))_ξψ_ξψ_ξξξ−h⁰(U)ψ_ξξψ_ξξξ

={(h⁰(U))_ξψ_ξψ_ξξ+h⁰(U)ψ_ξξ² }_ξ− {¹₂h⁰(U)ψ_ξξ² }_ξ+ ¹₂(h⁰(U))_ξψ_ξξ²

−{((h⁰(U))_ξψ_ξ)ψ_ξξ}_ξ+ ((h⁰(U))_ξψ_ξ)_ξψ_ξξ

= ¹₂{h⁰(U)ψ²_ξξ}_ξ+ ³₂(h⁰(U))_ξψ_ξξ² + (h⁰(U))_ξξψ_ξψ_ξξ. It means

(ψ_ξξ² )_t+{(h(U))⁰ψ_ξξ² }_ξ−{2µψ_ξξξψ_ξξ}_ξ+3h⁰(U)_ξψ²_ξξ+2h⁰(U)_ξξψ_ξψ_ξξ+2µψ_ξξξ² = 2ψ_ξξF_ξξ. (4.0.5) Integrating (4.0.5) over (0, t)×R we get

||ψ_ξξ(t)||²+Rt 0

R

R+3h⁰(U)ξψ_ξξ² + 2h⁰(U)ξξψξψξξdξ+ 2µ||ψ_ξξξ(τ)||dτ =

||ψ_ξξ(0)||²+Rt 0

R

R2F_ξξψ_ξξdξdτ. Then, for some3 satisfyingN(t)< 3 < 0, there exist C such that

||ψ_ξξ(t)||²+Rt 0

R

Rµψ²_ξξξ(τ)dξdτ ≤C(||ψ_ξξ(0)||²)−Rt 0

R

R3h⁰(U)_ξψ_ξξ² + 2h⁰(U)_ξξψ_ξψ_ξξdξdτ. Since −Rt

0

R

R3h⁰(U)_ξψ_ξξ² + 2h⁰(U)_ξξψ_ξψ_ξξdξdτ ≤cRt 0

R

Rψ_ξ²+ψ_ξξ² dξdτ, wherec= 5 sup_ξ∈R{|(h⁰(U))_ξ+ (h⁰(U))_ξξ|}<∞, we have

||ψ_ξξ(t)||²+Rt 0

R

Rµψ²_ξξξ(τ)dξdτ ≤C(||ψ_ξξ(0)||²) +cRt

0||ψ_ξξ(τ)||²+||ψ_ξ(τ)||²dτ. Combining with these higher order cases, we get

||ψ_ξ(t)||²+||ψ_ξξ(t)||²+Rt

0 ||ψ_ξξ(τ)||²+||ψ_ξξξ(τ)||²dτ ≤C(||(ψ₀)_ξ||²+||(ψ₀)_ξξ||²) +cRt

0 ||ψ_ξξ(τ)||²+||ψ_ξ(τ)||²dτ.

Combining Lemma 4.0.1 and Lemma 4.0.2. we get

||ψ(t)||²₂+Rt

0||ψ_ξ(τ)||²₂dτ ≤C(||ψ₀||²₂).

where constantC dependent ofN(t). So, the proof of A priori estimate is done.

Let check the Stability of travelling wave.

Theorem 4.0.2. (Stability) Supposeψ0∈H². Then there exists a positive constant2 andC2

such that if ||ψ₀||₂< ₂, the problem has a unique global solution ψ∈X(0,∞),

||ψ(t)||²₂+ Z t

0

||ψ_ξ(τ)||²₂dτ ≤C₂²(||ψ₀||²₂). (4.0.6)

for anyt≥0. Also, ψ_ξ tends to 0 in the L^∞ norm ast→ ∞, supξ∈R|ψ_ξ(ξ, t)| →0 , as t→ ∞.

Proof. To prove this theorem, we have to use Continuation argument.

Suppose2= min{₃/2, 3/2C3} ,C2=C3.

By the local existence, we put||ψ₀||²₂ ≤²₂≤²₃/4. Because ofT₀ is dependent of₀ ,T₀ has the valueT0(3) such that the solution exists on [0, T0(3)]. It means

||ψ(t)||²₂ ≤4(||ψ₀||²₂)≤²₃, fort∈[0, T₀].

By A priori estimate with T =T0(3), we know

||ψ(t)||²₂+Rt

0||ψ_ξ(τ)||²₂dτ ≤C₃²(||ψ₀||²₂), for 0≤t≤T. So we get

||ψ(t)||²₂≤C₃²(||ψ₀||²₂)≤C₃²²₂≤ ₄²³. It means

||ψ(T₀)||²₂ ≤C₃²²₂ ≤ ₄²³.

Using above results, we knowψ(ξ, T₀)∈H² and ||ψ(T₀)||²₂ ≤ ||ψ(T₀)||²₂ ≤ ₄²³. So, we can apply Local existence again at the start timeT0. Then,

||ψ(t)||²₂ ≤4(||ψ₀||²₂)≤²₃ fort∈[T₀,2T₀].

It means Local existence holds for t∈[0,2T0].

Also, by A priori estimate, we get

||ψ(t)||²₂+Rt

0||ψ_ξ(τ)||²₂dτ ≤C₃²(||ψ₀||²₂),

⇒ ||ψ(t)||²₂ ≤C₃²(||ψ₀||²₂)≤C₃²²₂ ≤ ₄²³.

fort∈[0,2T₀].

Using the same methods, Local existence and A priori estimate hold fort∈[0, nT0], n∈N. It means a global solution ψ∈X(0,∞) exist.

The remain thing is to showψ_ξ tends to 0 in the L^∞ norm ast→ ∞,

supξ∈R|ψ_ξ(ξ, t)| →0 , as t→ ∞. (4.0.7) Using the Gagliardo-Nirenburg interpolation inequality, we get

sup

ξ∈R

|ψ_ξ(t)| ≤ ||ψ_ξξ(t)||^1/2||ψ_ξ||^1/2. (4.0.8)

Since

||ψ(t)||²+Rt

0 µ||ψ_ξ(τ)||²dτ ≤C||ψ₀||²

||ψ_ξ(t)||²+Rt

0µ||ψ_ξξ(τ)||²dτ ≤C||ψ_ξ(0)||²+cRt

0||ψ_ξ(τ)||²dτ ≤C||ψ₀||²₁

||ψ_ξξ(t)||²+Rt

0µ||ψ_ξξξ(τ)||²dτ ≤C||ψ_ξξ(0)||²+cRt

0||ψ_ξ(τ)||²+||ψ_ξξ(τ)||²dτ ≤C||ψ₀||²₂ are done, we know ||ψ_ξ(t)|| and||ψ_ξξ(t)||are uniformly bounded for t≥0.

Also, since ||ψ_ξ(t)||² is integrable over tand _dt^d||ψ_ξ(t)||² is also integrable overtand satisfies Rt

0(_dt^d||ψ_ξ(τ)||²)dτ =||ψ_ξ(t)||²− ||ψ_ξ(0)||² ≤C||ψ_ξ(0)||²,

we know||ψ_ξ(t)||²is lipschitz continuous. Therefore||ψ_ξ(t)||² is uniformly continuous. It means

||ψ_ξ(t)|| →0 , as t→ ∞. (4.0.9)

So we get,

sup

ξ∈R

|ψ_ξ(t)| ≤ ||ψ_ξξ(t)||^1/2||ψ_ξ||^1/2 →0, ast→ ∞ (4.0.10) Therefore, Theorem 4.0.2 is done.

5

Stability of travelling wave for non convex flux case

In this section, we check A priori estimate and Stability of travelling waves for non convex flux using continuation argument.

The key point of this section is to use the energy method with weighted functionw(U).

Theorem 5.0.1. (A priori estimate) Letψbe a solution of the(3.0.5) and satisfying inX(0, T) for a positive constant T. Then there exists a positive constant3 such that ifsup_0≤t≤T||ψ(t)||₂<

₃, then ψ(t) satisfies

||ψ(t)||²₂+|ψ(t)|²_w(U)+Rt

0||ψ_ξ(τ)||²₂+|ψ_ξ(τ)|²_w(U)dτ ≤C₂²(||ψ₀||²₂+|ψ₀|²_w(U)), for any0≤t≤T.

Lemma 5.0.1. Let ψ∈X(0, T) be a solution of the reformulation form. Then, it satisfies

1

2|ψ(t)|²_w(U)+Rt 0 ||p

−U_ξψ(τ)||²+µ|ψ_ξ(τ)|²_w(U)dτ ≤ ¹₂|ψ₀|²_w(U)+Rt 0

R

Rw(U)ψF dxdτ, Proof. Multiplying the (3.0.5) by 2w(U)ψ we get

2wψψt+ 2wψψξh⁰(U)−2wµψψξξ= 2wψF. (5.0.1) Using the product rule, we get

2wψψ_t+ 2wψψ_ξh⁰(U) + 2w⁰ψψ_ξh(U)−2w⁰ψψ_ξh(U)−2wµψψ_ξξ= 2wψF, (5.0.2)

⇒(wψ²)_t+ 2(wh)⁰ψψ_ξ−2w⁰ψψ_ξh(U)−2wµψψ_ξξ = 2wψF, (5.0.3)

⇒(wψ²)_t+ ((wh)⁰ψ²)_ξ−(wh)⁰⁰U_ξψ²−2w⁰ψψ_ξh(U)−2wµψψ_ξξ= 2wψF, (5.0.4)

In the chapter 2, we knowh(U) =µU_ξ such that we get

(wψ²)t+ ((wh)⁰ψ²)ξ−(wh)⁰⁰Uξψ²−2µw⁰Uξψψξ−2wµψψξξ= 2wψF. (5.0.5) Using the product rule, we get

(wψ²)t+ ((wh)⁰ψ²)_ξ−(wh)⁰⁰U_ξψ²−2(µwψψ_ξ)_ξ+ 2µwψ_ξ² = 2wψF. (5.0.6) It means

(¹₂w(U)ψ²)_t+ (¹₂(w(U)h(U))⁰ψ²−µw(U)ψψ_ξ)_ξ +µw(U)ψ_ξ²−¹₂(w(U)h(U))⁰⁰U_ξψ² =w(U)ψF. Integrating the result over (0, t)×R we get

1

2|ψ(t)|²_w(U)+Rt 0

R

Rµw(U)ψ_ξ²−¹₂(w(U)h(U))⁰⁰U_ξψ²dξdτ = ¹₂|ψ₀|²_w(U)+Rt 0

R

Rw(U)ψF dξdτ. Since we selected the weight w = w(U) = |^(U−u+_h(U)^)(U−u−)|, we know w(U)h(U) is convex function of U such that ¹₂(wh)⁰⁰= 1.

1

2|ψ(t)|²_w(U)+Rt

0µ|ψ_ξ(τ)|²_w(U)+|| −p

U_ξψ(τ)||²dτ = ¹₂|ψ₀|²_w(U)+Rt 0

R

Rw(U)ψF dξdτ

Let

N(t) = sup_0≤τ≤t||ψ(τ)||₂ and assumeN(t)≤0.

By the taylor extension, |F|=|f(U +ψξ)−f(U)−f⁰(U)ψξ|=O(ψ²_ξ). It satisfies |F| →0 as ξ→ ±∞.

Then, for some ₃ satisfying N(t)< ₃< ₀,

|ψ(t)|²_w(U)+Rt

0|ψ_ξ(τ)|²_w(U)dτ ≤C|ψ₀|²_w(U)

Moreover, we need to check the similarly case when we apply ∂_ξ and ∂_ξξ to (3.0.5). Then, we can get a new lemma

Lemma 5.0.2. There is a positive constant 3 where is smaller than 0 such that if N(t) ≤ ₃< ₀,

||ψ_ξ(t)||²₁+Rt

0||ψ_ξξ(τ)||²₁dτ ≤C(||(ψ₀)_ξ||²₁+|ψ₀|²_w(U))

Proof. For the estimates ofψ_ξ andψ_ξξ, we apply∂_ξ and∂_ξξ to (3.0.5) and multiplyψand ψ_ξξ. Applying ∂_ξ to (3.0.5), and multiplying 2ψ_ξ to the result, we get

2ψ_ξψ_ξt+ 2(h⁰(U)ψ_ξ)_ξψ_ξ−2µψ_ξξξψ_ξ = 2F_ξψ_ξ. (5.0.7) Using the product rule, we get

(ψ²_ξ)t+ 2h⁰⁰(U)Uξψ²_ξ+{(h)⁰ψ²_ξ}_ξ−(h)⁰⁰Uξψ²_ξ− {2µψ_ξψξξ}_ξ+ 2µψ_ξξ²

= 2F_ξψ_ξ Integrating over (0, t)×R, we get

||ψ_ξ(t)||²− ||ψ_ξ(0)||²+ Z t

0

Z

R

h⁰⁰Uξψ_ξ²dξdτ+ Z t

0

2µ||ψ_ξξ(τ)||²dτ = Z t

0

Z

R

2Fξψξdξdτ (5.0.8) Under the smallness assumption onN(T), we get

||ψ_ξ(t)||²+ Z t

0

Z

R

h⁰⁰U_ξψ_ξ²dξ+ 2µ||ψ_ξξ(τ)||²dτ ≤ ||ψ_ξ(0)||² (5.0.9) Since U is a smooth function and bothU_ξ and U_ξξ converge to 0 asξ → ±∞,|U_ξ|and|U_ξξ|are bounded, so is sup_ξ∈R|h⁰⁰(U)Uξ|<∞. It means

||ψ_ξ(t)||²+ Z t

0

2µ||ψ_ξξ(τ)||²dτ ≤ ||ψ_ξ(0)||²− Z t

0

Z

R

2h⁰⁰U_ξψ_ξ²dξdτ (5.0.10)

≤ ||ψ_ξ(0)||²+cRt 0

R

Rψ_ξ²dξdτ ≤ ||ψ_ξ(0)||²+cRt 0

R

Rwψ_ξ²dξdτ

≤ ||ψ_ξ(0)||²+C|ψ(0)|²_w(U) wherec= 2 sup_ξ∈R|h⁰⁰(U)Uξ|

So, the estimates ofψ_ξ follows

||ψ_ξ(t)||²+ Z t

0

µ|ψ_ξξ(τ)|²_w(U)dτ ≤C(||ψ_ξ(0)||²+|ψ(0)|²_w(U)) (5.0.11)

Applying ∂_ξξ to (3.0.5), and multiplying 2ψ_ξξ to the result, we get

2ψξξψξξt+ 2(h⁰(U)ψξ)ξξψξξ−2µψξξξξψξξ = 2Fξξψξξ. (5.0.12) Using the product rule, we get

2ψ_ξξψ_ξξt+ 2(h⁰(U)ψ_ξ)_ξξψ_ξξ− {2µψ_ξξξψ_ξξ}_ξ+ 2wµψ²_ξξξ= 2wF_ξξψ_ξξ.

The second term, 2(h⁰(U)ψ_ξ)_ξξψ_ξξ is calculated as follows

2(h⁰(U)ψ_ξ)_ξξψ_ξξ={(2h⁰ψ_ξ)_ξψ_ξξ}_ξ−(2h⁰ψ_ξ)_ξψ_ξξξ

={2h⁰ψ²_ξξ+ 2h⁰⁰U_ξψ_ξψ_ξξ}_ξ−2h⁰⁰U_ξψ_ξψ_ξξξ−2h⁰ψ_ξξψ_ξξξ

={2h⁰ψ²_ξξ+ 2h⁰⁰U_ξψ_ξψ_ξξ}_ξ+{−(2h⁰⁰U_ξψ_ξψ_ξξ)_ξ+ (2h⁰⁰U_ξ)_ξψ_ξψ_ξξ+ 2h⁰⁰U_ξψ_ξξ} +{−(h⁰ψ_ξξ² )_ξ+h⁰⁰U_ξψ_ξξ² }

={h⁰ψ²_ξξ}_ξ+ 3h⁰⁰U_ξψ²_ξξ+ 2(h⁰)_ξξψ_ξψ_ξξ So, we get

2ψξξψξξt+{h⁰ψ_ξξ² }_ξ+ 3h⁰⁰Uξψ²_ξξ+ 2(h⁰)ξξψξψξξ

−{2µψ_ξξξψ_ξξ}_ξ+ 2µψ_ξξξ² = 2F_ξξψ_ξξ. Integrating over (0, t)×R, we get

||ψ_ξξ(t)||²− ||ψ_ξξ(0)||²+Rt 0

R

R3(h)⁰⁰U_ξψ²_ξξ+ 2(h⁰)_ξξψ_ξψ_ξξdξdτ +Rt

02µ||ψ_ξξξ(τ)||²dτ =Rt 0

R

R2Fξξψξξdξdτ. Under the smallness assumption onN(T), we get

||ψ_ξξ(t)||²+Rt

02µ||ψ_ξξξ(τ)||²dτ ≤ ||ψ_ξξ(0)||²−Rt 0

R

R3(h)⁰⁰U_ξψ²_ξξ+ 2(h⁰)_ξξψ_ξψ_ξξdξdτ. Since U is a smooth function and bothU_ξ and U_ξξ converge to 0 asξ → ±∞,|U_ξ|and|U_ξξ|are bounded, so is sup_ξ∈R|(h⁰(U))_ξ+ (h⁰(U))_ξξ|<∞. It means

||ψ_ξξ(t)||²+Rt

02µ||ψ_ξξξ(τ)||²dτ ≤ ||ψ_ξξ(0)||²−Rt 0

R

R3(h)⁰⁰U_ξψ_ξξ² + 2(h⁰)_ξξψ_ξψ_ξξdξdτ

≤ ||ψ_ξξ(0)||²+cRt

0||ψ_ξξ(τ)||+||ψ_ξ(τ)||dτ ≤ ||ψ_ξξ(0)||²+C(||ψ_ξ(0)||²+|ψ(0)|²_w(U)) wherec= 5 sup_ξ∈R|(h⁰(U))_ξ+ (h⁰(U))_ξξ|<∞.

So, the estimates ofψ_ξξ follows

||ψ_ξξ(t)||²+ Z t

0

2µ||ψ_ξξξ(τ)||²dτ ≤C(||ψ_ξξ(0)||²+||ψ_ξ(0)||²+|ψ(0)|²_w(U)) (5.0.13)

Combining (5.0.11) with (5.0.13) , we get

||ψ_ξ(t)||²₁+Rt

0||ψ_ξξ(τ)||²₁dτ ≤C(||ψ_ξ(0)||²₁+|ψ₀|²_w(U)).

So, Combineing Lemma 5.0.1 with Lemma 5.0.2, we get

||ψ(t)||²₂+|ψ(t)|²_w(U)+Rt

0||ψ_ξ(τ)||²₂+|ψ_ξ(τ)|²_w(U)dτ ≤C(||ψ₀||²₂+|ψ₀|²_w(U)).

where constantC dependent ofN(t). So, the proof of A priori estimate is done.

Let check the Stability of travelling wave.

Theorem 5.0.2. (Stability) Suppose ψ₀ ∈H²∩L²_w(U). Then there exists a positive constant₂ andC2 such that if||ψ₀||₂+|ψ₀|_w(U) < 2, the problem has a unique global solutionψ∈X(0,∞),

||ψ(t)||²₂+|ψ(t)|²_w(U)+ Z t

0

||ψ_ξ(τ)||²₂+|ψ_ξ(τ)|²_w(U)dτ ≤C₂²(||ψ₀||²₂+|ψ₀|²_w(U)). (5.0.14)

for anyt≥0. Also, ψ_ξ tends to 0 in the L^∞ norm ast→ ∞,

supξ∈R|ψ_ξ(ξ, t)| →0 , as t→ ∞. (5.0.15) Proof. To prove this theorem, we have to use continuation argument, again. The proof is similar to Theorem 4.0.2. The difference point is the weightedL²_w norm.

Suppose2= min{₃/2, 3/2C3} ,C2=C3.

By the local existence, we put||ψ₀||²₂+|ψ₀|²_w(U) ≤²₂≤²₃/4. Because of T₀ is dependent of₀ , T0 has the value T0(3) such that the solution exists on [0, T0(3)]. It means

||ψ(t)||²₂+|ψ(t)|²_w(U)≤4(||ψ₀||²₂+|ψ₀|²_w(U))≤²₃ fort∈[0, T₀].

By A priori estimate with T =T0(3), we know for 0≤t≤T

||ψ(t)||²₂+|ψ(t)|²_w(U)+R_t

0||ψ_ξ(τ)||²₂+|ψ_ξ(τ)|²_w(U)dτ ≤C₃²(||ψ₀||²₂+|ψ₀|²_w(U)).

So, we get

||ψ(t)||²₂+|ψ(t)|²_w(U)≤C₃²(||ψ₀||²₂+|ψ₀|²_w(U))≤C₃²²₂≤ ₄²³. It means

||ψ(T₀)||²₂+|ψ(T₀)|²_w(U)≤C₃²²₂ ≤ ₄²³.

Using above results, we know ψ(ξ, T₀)∈H²∩L²_w(U) and||ψ(T₀)||²₂ ≤ ||ψ(T₀)||²₂+|ψ(T₀)|²_w(U)≤

²₃

4. So, we can apply Local existence again at the start timeT0,

||ψ(t)||²₂+|ψ(t)|²_w(U) ≤4(||ψ₀||²₂+|ψ₀|²_w(U))≤²₃ fort∈[T₀,2T₀].

It means Local existence holds for t∈[0,2T0].

Also, by A priori estimate,

||ψ(t)||²₂+|ψ(t)|²_w(U)+Rt

0 ||ψ_ξ(τ)||²₂+|ψ_ξ(τ)|²_w(U)dτ ≤C₃²(||ψ₀||²₂+|ψ₀|²_w(U))

⇒ ||ψ(t)||²₂+|ψ(t)|²_w(U)≤C₃²(||ψ₀||²₂+|ψ₀|²_w(U))≤C₃²²₂ ≤ ₄²³, fort∈[0,2T₀].

Using the same method, Local existence and A priori estimate hold fort∈[0, nT0], n∈N. It

means a global solution ψ∈X(0,∞) exist.

The remain thing is to showψ_ξ tends to 0 in the L^∞ norm ast→ ∞,

supξ∈R|ψ_ξ(ξ, t)| →0 , as t→ ∞. (5.0.16) In fact, this is the same method that of Theorem 4.0.2. Using the Gagliardo-Nirenburg inter- polation inequality, we get

sup

ξ∈R

|ψ_ξ(t)| ≤ ||ψ_ξξ(t)||^1/2||ψ_ξ||^1/2. (5.0.17)

Since

|ψ(t)|²_w(U)+Rt

0µ|ψ_ξ(τ)|²_w(U)dτ ≤C|ψ₀|²_w(U)

||ψ_ξ(t)||²+Rt

0µ||ψ_ξξ(τ)||²dτ ≤C||ψ_ξ(0)||²+cRt

0||ψ_ξ(τ)||²dτ ≤C||ψ₀||²₁

||ψ_ξξ(t)||²+Rt

0µ||ψ_ξξξ(τ)||²dτ ≤C||ψ_ξξ(0)||²+cRt

0||ψ_ξ(τ)||²+||ψ_ξξ(τ)||²dτ ≤C||ψ₀||²₂ are done, we know ||ψ_ξξ(t)|| is uniformly bounded fort≥0.

Since ||ψ_ξ(t)||² is integrable overtand _dt^d||ψ_ξ(t)||² is also integrable overtand satisfies Rt

0(_dt^d||ψ_ξ(τ)||²)dτ =||ψ_ξ(t)||²− ||ψ_ξ(0)||² ≤C||ψ_ξ(0)||²,

we know||ψ_ξ(t)||²is lipschitz continuous. Therefore||ψ_ξ(t)||² is uniformly continuous. It means

||ψ_ξ(t)|| →0 , as t→ ∞. (5.0.18)

So we get,

sup

ξ∈R

|ψ_ξ(t)| ≤ ||ψ_ξξ(t)||^1/2||ψ_ξ||^1/2 →0, ast→ ∞ (5.0.19) Therefore, Theorem 5.0.2 is done.