Contraction for large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model

TL;DR

This paper studies the stability of traveling shock waves in a hyperbolic-parabolic chemotaxis model, introducing a relative entropy method that demonstrates contraction properties even for large initial disturbances when shock strength is small.

Contribution

It develops a relative entropy framework for analyzing shock stability in a chemotaxis system, showing contraction properties independent of diffusion strength for small shocks.

Findings

Relative entropy captures shock proximity in $L^2$-sense.

Non-increasing entropy for large perturbations with small shocks.

Contraction property holds regardless of diffusion strength.

Abstract

We consider a hyperbolic-parabolic system arising from a chemotaxis model in angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost $L^2$-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.

Equations

$$\begin{array}{rl}{\partial }_{t}n-{\partial }_x(nq)& =\nu {\partial }_{xx}n,\\ {\partial }_{t}q-{\partial }_{x}n& =0\text{mbox}forx\in \mathbb{R}\text{mbox}andfort>0\end{array}$$

$$\begin{array}{rl}{\partial }_{t}n-\nu \Delta n& =-\nabla \cdot (n\chi (c)\nabla c),\\ {\partial }_{t}c-ϵ\Delta c& =-c^{m}n\text{mbox}forx\in {\mathbb{R}}^N\text{mbox}andfort>0\end{array}$$

$$\begin{array}{rl}{\partial }_{t}n-\nu {\partial }_{xx}n& =-{\partial }_x\left(n\frac{{\partial }_{x}c}{c}\right),\\ {\partial }_{t}c& =-cn.\end{array}$$

$$q:=-{\partial }_x[\ln c]=-\frac{{\partial }_{x}c}{c}.$$

$$\tilde{n}(-\infty )=n_{-}>0,\tilde{n}(+\infty )=n_{+}>0,\tilde{q}(-\infty )=q_{-},\tilde{q}(+\infty )=q_{+}$$

$$\displaystyle\begin{aligned} &\exists~{}\sigma\in\mathbb{R}\mbox{ such that }~{}\left\{\begin{array}[]{ll}-\sigma(n_{+}-n_{-})-(n_{+}q_{+}-n_{-}q_{-})=0,\\ -\sigma(q_{+}-q_{-})-(n_{+}-n_{-})=0,\end{array}\right.\\ &\mbox{and either $n_{-}>n_{+}$ and $q_{-}<q_{+}$ or $n_{-}<n_{+}$ and $q_{-}<q_{+}$ holds.}\end{aligned}$$

$$\sigma =\frac{-q_{-}\pm \sqrt{q_{-}^2+4n_{+}}}{2}.$$

$$\exists x_0\in \mathbb{R}\text{mbox}suchthat{\int }_{\mathbb{R}}\left(\begin{array}{c}{n}_0(x)-\tilde{n}(x-x_0)\\ q_0(x)-\tilde{q}(x-x_0)\end{array}\right)dx=\left(\begin{array}{c}0\\ 0\end{array}\right).$$

$$\eta (U_1|U_2):=\frac{|q_1-q_2{|}^2}{2}+\Pi (n_1|n_2),$$

$$\Pi (n_1|n_2):=\Pi (n_1)-\Pi (n_2)-\nabla \Pi (n_2)(n_1-n_2),\Pi (n):=n\log n-n.$$

$$\begin{array}{r}{\mathcal{X}}_T:=\{(n,q)\in L^{\infty }((0,T)\times \mathbb{R}{)}^2|n>0,n^{-1}\in L^{\infty }((0,T)\times \mathbb{R}),{\partial }_{x}n\in L^2((0,T)\times \mathbb{R})\}\end{array}$$

$${\int }_{-\infty }^{\infty }\eta (U_0|\tilde{U})dx<\infty .$$

$$\displaystyle\begin{aligned} &\int_{-\infty}^{\infty}a(x-\sigma t)\eta\big{(}U(t,x-X(t))|{\tilde{U}}(x-\sigma t)\big{)}dx\\ &\qquad+\delta_{0}\int_{0}^{t}\int_{-\infty}^{\infty}a(x-\sigma\tau)n\big{(}\tau,x-X(\tau)\big{)}\Big{|}\partial_{x}\Big{(}\log\frac{n(\tau,x-X(\tau))}{{\tilde{n}}(x-\sigma\tau)}\Big{)}\Big{|}^{2}dxd\tau\\ &\quad\leq\int_{-\infty}^{\infty}a(x)\eta\big{(}U_{0}(x)|{\tilde{U}}(x)\big{)}dx,\end{aligned}$$

$$\begin{split}&|\dot{X}(t)-\sigma|\leq\frac{1}{\varepsilon^{2}}\Big{(}f(t)+C\int_{-\infty}^{\infty}\eta(U_{0}|{\tilde{U}})dx+1\Big{)}\quad\mbox{ for {a.e.} }t\in[0,T]\\ &\mbox{where $f$ is some positive function satisfying}\quad\|f\|_{L^{1}(0,T)}\leq C\frac{\lambda}{\varepsilon}\int_{-\infty}^{\infty}\eta(U_{0}|{\tilde{U}})dx.\end{split}$$

$$0<\sigma =\frac{-q_{-}+\sqrt{q_{-}^2+4n_{+}}}{2}.$$

$$\int_{-\infty}^{\infty}\eta\big{(}U(t,x-X(t))|{\tilde{U}}(x-\sigma t)\big{)}dx\leq 4\int_{-\infty}^{\infty}\eta\big{(}U_{0}(x)|{\tilde{U}}(x)\big{)}dx.$$

$$\displaystyle\begin{aligned} &\int a^{\nu}(x-\sigma t)\eta\big{(}U^{\nu}(t,x-X^{\nu}(t))|{\tilde{U}}^{\nu}(x-\sigma t)\big{)}dx\\ &\qquad\quad=\nu\int a(x-\sigma t/\nu)\eta\big{(}U(t/\nu,x-X(t/\nu))|{\tilde{U}}(x-\sigma t/\nu)\big{)}dx,\end{aligned}$$

$$\int a^{\nu}(x-\sigma t)\eta\big{(}U^{\nu}(t,x-X^{\nu}(t))|{\tilde{U}}^{\nu}(x-\sigma t)\big{)}dx\leq\int a^{\nu}(x)\eta\big{(}U_{0}(x)|{\tilde{U}}^{\nu}(x)\big{)}dx.$$

$$\displaystyle\frac{d}{dt}\int_{-\infty}^{\infty}a(\xi)\eta\big{(}U(t,\xi+X(t))|\tilde{U}(\xi)\big{)}d\xi$$

$$=\dot{X}(t)\mathcal{Y}(U(t,\cdot +X(t)))+{\mathcal{I}}^{bad}(U(t,\cdot +X(t)))-{\mathcal{I}}^{good}(U(t,\cdot +X(t))).$$

$$-\frac{1}{{\epsilon }^4}{\mathcal{Y}}^2(U)+{\mathcal{I}}^{bad}(U)-{\mathcal{I}}^{good}(U)\le 0.$$

$$\begin{array}{rl}{\partial }_{t}n-{\partial }_x(nq)& ={\partial }_{xx}n,\\ {\partial }_{t}q-{\partial }_{x}n& =0.\end{array}$$

$$\begin{array}{rl}{\partial }_{t}n-\sigma {\partial }_{\xi }n-{\partial }_{\xi }(nq)& ={\partial }_{\xi \xi }n,\\ {\partial }_{t}q-\sigma {\partial }_{\xi }q-{\partial }_{\xi }n& =0.\end{array}$$

$$\begin{array}{rl}-\sigma {\partial }_{\xi }\tilde{n}-{\partial }_{\xi }(\tilde{n}\tilde{q})& ={\partial }_{\xi \xi }\tilde{n},\\ -\sigma {\partial }_{\xi }\tilde{q}-{\partial }_{\xi }\tilde{n}& =0.\end{array}$$

$$\sigma =\frac{-q_{-}+\sqrt{q_{-}^2+4n_{+}}}{2}>0.$$

$$\begin{array}{rl}& {\tilde{n}}^{\prime }<0,{\tilde{q}}^{\prime }=-\frac{{\tilde{n}}^{\prime }}{\sigma }>0,\text{mbox}and\\ & {\tilde{n}}^{\prime }=\frac{(\tilde{n}-n_{-})(\tilde{n}-n_{+})}{\sigma }.\end{array}$$

$$\begin{array}{r}-\frac{{\epsilon }^2}{{\sigma }_{-}}{e}^{-\frac{\epsilon |\xi |}{{\sigma }_{-}}}\le {\tilde{n}}^{\prime }(\xi )\le -\frac{{\epsilon }^2}{4{\sigma }_{-}}{e}^{-\frac{\epsilon |\xi |}{{\sigma }_{-}}},\end{array}$$

$${\sigma }_{-}:=\frac{-q_{-}+\sqrt{q_{-}^2+4n_{-}}}{2}.$$

$$0<\frac{{\sigma }_{-}}{2}\le ({\sigma }_{-}-C\epsilon )\le \sigma <{\sigma }_{-},$$

$$|{\tilde{n}}^{\prime \prime }(\xi )|\le C\epsilon |{\tilde{n}}^{\prime }(\xi )|.$$

$${\tilde{n}}^{\prime \prime }=-\sigma {\tilde{n}}^{\prime }-(\tilde{n}\tilde{q}{)}^{\prime },$$

Full text

Acknowledgement. The work of KC was partially supported by NRF-2018R1D1A1B07043065 and by the POSCO Science Fellowship of POSCO TJ Park Foundation. The work of MK was partially supported by NRF-2017R1C1B5076510. The work of YK was supported by Basic Science Research Program through the Na- tional Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A1B03030249). The work of AV was partially supported by the NSF grant: DMS 1614918.

1. Introduction and main theorem

We consider the following one dimensional hyperbolic-parabolic system:

[TABLE]

where $\nu>0$ is a positive constant. We are interested in stability of viscous shocks (so-called traveling waves) of the above system.

1.1. Model from Chemotaxis

The system (1.1) is related to the following Keller-Segel system [19]:

[TABLE]

with $m>0$ and $\epsilon\geq 0$. In chemotaxis, the unknown $n(\mathbb{x},t)>0$ represents the bacterial density while the unknown $c(\mathbb{x},t)>0$ means the concentration of chemical nutrient consumed by bacteria at position $\mathbb{x},$ and time $t$. We assume that the given sensitivity function $\chi(\cdot):\mathbb{R}^{+}\to\mathbb{R}^{+}$ is decreasing since the chemosensitivity gets usually lower as the concentration of the chemical gets higher. The positive constant $m$ indicates the consumption rate of nutrient $c$, and the non-negative constant $\epsilon\geq 0$ means the chemical diffusion rate for $c$.

Such a Keller-Segel system can play a role of a simplified model of angiogenesis on the formation of new blood vessels from pre-existing vessels, which is considered to be the mechanism for tumor progression and metastasis (see [7, 8, 21, 26, 27, 31], and references therein). In this interpretation, $n$ denotes the density of endothelial cells while $c$ does the concentration of the protein known as the vascular endothelial growth factor(VEGF). In biological implications, we usually consider $\epsilon$ small (or negligible) (e.g. see [21]).

To derive our system (1.1), we just take $\chi(c)=c^{-1}$ and $m=1$, $N=1$, and $\epsilon=0$ into (1.2) to get

[TABLE]

To have a traveling wave of (1.3), the chemosensitivity function $\chi(c)$ needs to be singular near $c=0$ (e.g. see [19]). In particular, $\chi(c)=c^{-1}$ was assumed in [19]. Thanks to the restriction $m=1$, we can treat the singularity in $c$ of the sensitivity by the Cole-Hopf transformation

[TABLE]

After the transform, we have (1.1) as in [36].

1.2. Traveling waves of (1.1)

We notice that if $n\geq 0$, which is biologically relevant by the derivation from chemotaxis, then the principal part (i.e. when $\nu=0$) of the system (1.1) is hyperbolic. By [36] (also see [24]), it has been known that for any $\nu>0$, (1.1) admits a smooth traveling wave $\begin{pmatrix}{{\tilde{n}}}\\ {{\tilde{q}}}\end{pmatrix}(x-\sigma t)$ connecting two end-states $(n_{-},q_{-})$ and $(n_{+},q_{+})$, i.e.,

[TABLE]

(we denote $\displaystyle\lim_{x\to\pm\infty}{f}(x)$ by ${f}(\pm\infty)$ in short), provided the two end-states satisfy the Rankine-Hugoniot condition and the Lax entropy condition:

[TABLE]

Here, the velocity $\sigma$ is given by

[TABLE]

More precisely, if $n_{-}>n_{+}>0$, then $\sigma=\frac{-q_{-}+\sqrt{q_{-}^{2}+4n_{+}}}{2}>0$, whereas if $0<n_{-}<n_{+}$, then $\sigma=\frac{-q_{-}-\sqrt{q_{-}^{2}+4n_{+}}}{2}<0$ (See Subsection 2.2 for more details). For this topic, we also refer to the survey paper [35] by Wang.

In this parabolic conservation laws, it is an interesting topic to discuss how stable these viscous shocks are. By [24], it has been known that these waves are stable if the anti-derivative of a perturbation $(n-{\tilde{n}},q-{\tilde{q}})$ is small in the Sobolev space $[H^{2}(\mathbb{R})]^{2}$. Thus the perturbation needs at least to have the mean-zero condition:

[TABLE]

This condition is quite common in studying stability of viscous shocks since [9] and [18].

In this paper, we introduce a relative entropy functional of the system, which plays a similar role of $L^{2}$-distance between a solution $(n,q)$ and a given shock profile $({\tilde{n}},{\tilde{q}})$. Then we show that the functional is non-increasing in time for any large initial perturbation. Therefore, we prove that the contraction property holds independently of the size of the perturbation or the strength of the viscosity $\nu$. It is remarkable that our result do not ask a perturbation to have either the mean-zero condition or the smallness in a Sobolev space. However, we need that the shock strength $|n_{-}-n_{+}|$ is small enough while this smallness on the wave amplitude was not required in [24].

For the Cauchy problem of (1.1), we refer to [10, 23, 25] for global well-posedness. For multi-dimentional cases, see [22] and references therein. For stability of planar shocks under the mean-zero condition, we refer to [1, 2].

1.3. Main result

For $U_{i}=\begin{pmatrix}n_{i}\\ q_{i}\end{pmatrix}$ with $n_{i}>0$ for $i=1,2$, we consider the relative entropy

[TABLE]

where

[TABLE]

Since $\Pi(n)$ is strictly convex in $n$, its relative functional ${\Pi}(\cdot|\cdot)$ above is positive definite, and so is $\eta(\cdot|\cdot)$. That is, $\eta(U_{1}|U_{2})\geq 0$ for any $U_{1}$ and $U_{2}$, and $\eta(U_{1}|U_{2})=0$ if and only if $U_{1}=U_{2}$.

Global existence and uniqueness of weak solutions to (1.1) belonging to the space

[TABLE]

for each $T>0$, is studied in [3].

Here is the main result. We first state it for a fixed viscosity $\nu=1$. Then, in Remark 1.5, we illustrate that the main result still holds for any $\nu>0$.

Theorem 1.1**.**

*Let $\nu=1$. For a given constant state $(n_{-},q_{-})\in\mathbb{R}^{+}\times\mathbb{R}$, there exist constants $\delta_{0}\in(0,1/2)$ and $C>0$ such that the following is true:

For any $\varepsilon,\lambda>0$ with $\varepsilon\in(0,n_{-})$ and $\delta_{0}^{-1}\varepsilon<\lambda<\delta_{0}$, and for any $(n_{+},q_{+})\in\mathbb{R}^{+}\times\mathbb{R}$ satisfying (LABEL:end-con) with $|n_{-}-n_{+}|=\varepsilon$, there exists a smooth monotone function $a:\mathbb{R}\to\mathbb{R}^{+}$ with $\lim_{x\to\pm\infty}a(x)=1+a_{\pm}$ for some constants $a_{-},a_{+}$ with $|a_{+}-a_{-}|=\lambda$ such that the following holds:

Let ${\tilde{U}}:=\begin{pmatrix}{\tilde{n}}\\ {\tilde{q}}\end{pmatrix}$ be a traveling wave of (1.1) with the boundary condition (1.4) and with the speed $\sigma$ from (1.6). For a given $T>0$, let $U(t,x):=\begin{pmatrix}n(t,x)\\ q(t,x)\end{pmatrix}$ be a solution to (1.1) belonging to $\mathcal{X}_{T}$ with initial data $U_{0}(x):=\begin{pmatrix}n_{0}(x)\\ q_{0}(x)\end{pmatrix}$ satisfying*

[TABLE]

Then there exists an absolutely continuous shift function $X:[0,T]\rightarrow\mathbb{R}$ with $X\in W^{1,1}_{loc}$ and $X(0)=0$ such that

[TABLE]

and

[TABLE]

Remark 1.2*.*

The result can be considered to be an a-priori estimate for solutions of (2.1). The existence issue of solutions in the class $\mathcal{X}_{T}$ for any $T>0$ with the initial condition (1.7) will be covered in the forthcoming paper [3]. The estimate on the dissipation in (LABEL:ineq_contraction), will be crucially used for the proof of the global existence of weak solutions to (1.1) in [3].

Remark 1.3*.*

Notice that it is enough to prove Theorem 1.1 in the case of $n_{-}>n_{+}>0$. Indeed, the result for $n_{+}>n_{-}>0$ is obtained by the change of variables $x\mapsto-x$ with $\sigma\mapsto-\sigma$. Therefore, from now on, we assume $n_{-}>n_{+}>0$ and thus

[TABLE]

Remark 1.4*.*

Since the weight function $a$ satisfies that $|a(x)-1|\leq\lambda<\delta_{0}<1/2$ for all $x\in\mathbb{R}$, the contraction estimate (LABEL:ineq_contraction) yields

[TABLE]

Remark 1.5*.*

In fact, such a contraction property (LABEL:ineq_contraction) holds for any $\nu>0$, by scaling as follows. This scaling argument makes sense because of no condition on the strength of the initial perturbation. Let $U^{\nu}$ and ${\tilde{U}}^{\nu}$ be a solution and traveling wave to (1.1) with initial data $U_{0}$, respectively. Then, $U(t,x):=U^{\nu}(\nu t,\nu x)$ (resp. ${\tilde{U}}(x):={\tilde{U}}^{\nu}(\nu x)$) is a solution (resp. traveling wave) to (1.1) with $\nu=1$. Therefore, using (LABEL:ineq_contraction) together with the fact that

[TABLE]

where $a^{\nu}(x):=a(x/\nu)$ and $X^{\nu}(t):=\nu X(t/\nu)$, we get

[TABLE]

**Notations ** Throughout the paper, $C$ denotes a positive constant which may change from line to line, but which is independent of $\varepsilon$ (the strength of the shock) and $\lambda$ (the total variation of the function $a$). The paper will consider two smallness conditions, one on $\varepsilon$, and the other on $\varepsilon/\lambda$. In the argument, $\varepsilon$ will be far smaller than $\varepsilon/\lambda$ .

1.4. Ideas of Proof

We basically take advantage of the new method introduced by Kang-Vasseur in [13], which is also used in the recent works [11, 14]. The main scenario of the method is briefly explained as follows.

For a given viscous traveling wave ${\tilde{U}}$ with small amplitude $|n_{-}-n_{+}|=\varepsilon$, the weight function $a$ is defined by ${\tilde{U}}$ (see (2.12)). We employ the weighted relative entropy with the weight $a$, to get the contraction of any large perturbation $U$ from ${\tilde{U}}$, up to a time-dependent shift $X(t)$. The shift function $X$ is constructed after the relative entropy computation in Lemma 2.3, which gives

[TABLE]

Because of the relative entropy structure, the bad terms $\mathcal{I}^{bad}$ and the good terms $\mathcal{I}^{good}$ (i.e. $\mathcal{I}^{good}\geq 0$) are quadratic when the perturbation is small. However, we have no uniform control on the size of the large perturbation $U(t,\cdot)$, therefore we should carefully estimate what happens for large values of $U(t,x)$.

The key idea of the technique is to exploit the degree of freedom of the shift $X(t)$ in the first term $\dot{X}(t)\mathcal{Y}(U(t,\cdot+X(t)))$. First of all, when $\mathcal{Y}(U(t,\cdot))$ is not too small, we can construct the shift $X(t)$ such that the term $\dot{X}(t)\mathcal{Y}(U(t,\cdot+X(t)))$ absorbs all the bad terms $\mathcal{I}^{bad}$ (see (3.2)). Specifically, we ensure algebraically that the contraction holds as long as $|\mathcal{Y}(U(t))|\geq\varepsilon^{2}$. Thus, the rest of the method is to show that the contraction still holds when $|\mathcal{Y}(U(t))|\leq\varepsilon^{2}$.

In the argument, for the values of $t$ such that $|\mathcal{Y}(U(t))|\leq\varepsilon^{2}$, we construct the shift as a solution to the ODE: $\dot{X}(t)=-\mathcal{Y}(U(t,\cdot+X(t)))/\varepsilon^{4}$. From this point, we forget that $U$ is a solution to the system and $X(t)$ is the shift. That is, we leave out $X(t)$ and the $t$-variable of $U$. Therefore, it remains to show that for any function $U$ satisfying $\mathcal{Y}(U)\leq\varepsilon^{2}$,

[TABLE]

This is proved by Proposition 3.1 together with Lemma 2.6. This completes the proof of Theorem 1.1. Proposition 3.1 is obtained thanks to a generic non-linear Poincaré type inequality (see Lemma 4.2), which is first introduced in [13]. It was first discovered for the scalar case in [16]. The general method then follows [13] by performing a careful expansion on the strength of the shock. Note that the parabolic system (1.1) is degenerate (that is, there is no diffusion in terms of $q$). Therefore, following [13], we first maximize the bad terms with respect to $q$ for $n$ fixed (see Lemma 2.6). The expansion is then performed only on $n$. A new feature compared to [13] is that the maximization can be performed only locally for $|n-{\tilde{n}}|\ll 1$.

The remaining parts of the paper are organized as follows. In Section 2, we introduce background materials including some properties of traveling waves, the definition of the weight function $a(\cdot)$, and the main inequality (Lemma 2.3) from the relative entropy. Then in Section 3, we give the definition of our shift $X$ and present the main proposition (Proposition 3.1), which implies our main result (Theorem 1.1). The proof of Proposition 3.1 is presented in Sections 4 and 5. In Section 4, we get sharp estimates when $|n-{\tilde{n}}|$ is small enough while in Section 5, we control all bad terms when $|n-{\tilde{n}}|$ is not small.

2. Background

2.1. Moving frame

From now on, we fix $\nu=1$ so our system (1.1) becomes

[TABLE]

For simplification of our analysis, we rewrite (2.1) into the following system, based on the change of variables $(t,x)\mapsto(t,\xi=x-\sigma t)$, where $\sigma=\frac{-q_{-}+\sqrt{q_{-}^{2}+4n_{+}}}{2}$:

[TABLE]

We are interested in a traveling wave solution ${\tilde{U}}=\begin{pmatrix}{\tilde{n}}\\ {\tilde{q}}\end{pmatrix}$ of (2.1) as a solution of

[TABLE]

2.2. Existence and properties of traveling wave solutions

In the sequel, without loss of generality, we consider the traveling wave $(\tilde{n},\tilde{q})$ satisfying $\tilde{n}(0)=\frac{n_{-}+n_{+}}{2}$.

Lemma 2.1**.**

(1) For any $n_{\pm},q_{\pm}$ with $n_{-}>n_{+}>0$ satisfying (LABEL:end-con), the system (2.1) admits a smooth traveling wave $\begin{pmatrix}{{\tilde{n}}}\\ {{\tilde{q}}}\end{pmatrix}(x-\sigma t)$ connecting the two end-states $(n_{-},q_{-})$ and $(n_{+},q_{+})$ as (1.4) with velocity

[TABLE]

Moreover,

[TABLE]

*(2) For any $(n_{-},q_{-})\in\mathbb{R}^{+}\times\mathbb{R}$, there exist positive constants $\varepsilon_{1}$ and $C$ such that for any $0<\varepsilon<\varepsilon_{1}$ and any $(n_{+},q_{+})\in\mathbb{R}^{+}\times\mathbb{R}$ satisfying (LABEL:end-con) with $n_{+}=n_{-}-\varepsilon$, the following is true:

Let $\begin{pmatrix}{{\tilde{n}}}\\ {{\tilde{q}}}\end{pmatrix}(x-\sigma t)$ be the traveling wave connecting the two end states $(n_{-},q_{-})$ and $(n_{+},q_{+})$ such that ${{\tilde{n}}}(0)=(n_{-}+n_{+})/2$.

Then,*

[TABLE]

where

[TABLE]

Moreover, we have

[TABLE]

and

[TABLE]

Proof.

$\bullet$ proof of (1) : The proof can be found in [24] and [36]. Here we sketch its proof for completeness. Since

[TABLE]

we have

[TABLE]

which can be written as

[TABLE]

But, since ${\tilde{q}}-q_{-}=-\frac{1}{\sigma}({\tilde{n}}-n_{-})$ from ${\tilde{q}}^{\prime}=-\frac{1}{\sigma}{\tilde{n}}^{\prime}$, we have

[TABLE]

Since it follow from (2.4) that

[TABLE]

we have

[TABLE]

That is,

[TABLE]

This ODE has a smooth solution ${\tilde{n}}$ connecting $n_{-}$ to $n_{+}$, and ${\tilde{n}}^{\prime}<0$. By ${\tilde{q}}^{\prime}=-\frac{1}{\sigma}{\tilde{n}}^{\prime}$ and (LABEL:end-con), we have ${\tilde{q}}$.

$\bullet$ proof of (2) : First of all, since it follows from (2.4) and $n_{+}=n_{-}-\varepsilon$ that

[TABLE]

taking $\varepsilon_{1}$ small enough such that

[TABLE]

which gives (2.8).

To show (2.6), we first observe that (2.9) yields

[TABLE]

Since ${\tilde{n}}^{\prime}<0$ and ${{\tilde{n}}}(0)=(n_{-}+n_{+})/2$ imply

[TABLE]

it follows from (2.10) and $n_{-}-n_{+}=\varepsilon$ that

[TABLE]

These together with ${{\tilde{n}}}(0)=(n_{-}+n_{+})/2$ imply

[TABLE]

Applying the above estimates to (2.10) together with (LABEL:od), we have

[TABLE]

Finally, using (2.8), we have the desired estimates in (2.6).

Moreover, we differentiate (2.9) to get

[TABLE]

Since

[TABLE]

we have

[TABLE]

∎

2.3. Definition of the weight $a$

For a given stationary solution ${{\tilde{U}}}$ of (2.2)(i.e. a solution of (2.3)), we define $a(\cdot)$ by

[TABLE]

Note

[TABLE]

2.4. Relative entropy method

As mentioned in Subsection 1.4, we employ the new analysis in [13], which is based on the relative entropy method. The method is purely nonlinear, and allows to handle rough and large perturbations. The relative entropy method was first introduced by Dafermos [5] and Diperna [6] to prove the $L^{2}$ stability and uniqueness of Lipschitz solutions to the hyperbolic conservation laws endowed with a convex entropy.

Recently, the relative entropy method has been extensively used in studying on the contraction (or stability) of large perturbations of viscous (or inviscid) shock waves (see [4, 11, 12, 13, 15, 16, 17, 20, 28, 29, 30, 33, 34]).

To use the relative entropy method, we rewrite (2.2) into the following general system of viscous conservation laws:

[TABLE]

where

[TABLE]

Indeed, since

[TABLE]

we see that (2.2) is equivalent to (2.14).

Notice that $\eta$ is a strictly convex entropy of the system (2.14), since

[TABLE]

is the entropy flux of $\eta$ such that $\partial_{i}G(U)=\sum_{k=1}^{2}\partial_{k}\eta(U)\partial_{i}A_{k}(U),\quad 1\leq i\leq 2$.

In general, for a given function $f$, we define its relative function $f(\cdot|\cdot)$ of two variables by

[TABLE]

Then for $U_{i}=\begin{pmatrix}n_{i}\\ q_{i}\end{pmatrix},i=1,2$,

[TABLE]

and

[TABLE]

where

[TABLE]

Since ${\Pi}(n)=n\log n-n$, we find that

[TABLE]

We define the corresponding flux $G(\cdot;\cdot)$ for our relative entropy $\eta(\cdot|\cdot)$ by

[TABLE]

In what follows, we use a simple notation: for any function $f:\mathbb{R}_{\geq 0}\times\mathbb{R}\rightarrow\mathbb{R}$ and any shift $X:[0,\infty)\rightarrow\mathbb{R}$,

[TABLE]

We also introduce the function space

[TABLE]

Remark 2.2*.*

As mentioned before, we consider the solution $U$ to (1.1) belonging to $\mathcal{X}_{T}$. Then, since $\partial_{\xi}n\in L^{2}((0,T)\times\mathbb{R})$ and $n^{-1}\in L^{\infty}((0,T)\times\mathbb{R})$, using $\tilde{n}\in L^{\infty}(\mathbb{R})$ and $\tilde{n}^{\prime}\in L^{2}(\mathbb{R})$, we find

[TABLE]

which implies $U(t)\in\mathcal{H}$ for a.e. $t\in[0,T]$.

Lemma 2.3**.**

Let $\tilde{U}:={\tilde{n}\choose\tilde{q}}$ be the traveling wave in (2.3), and $a:\mathbb{R}\to\mathbb{R}^{+}$ be the weight function by (2.12). For any solution $U={n\choose q}\in\mathcal{X}_{T}$ of (2.2) for some $T>0$ and for any absolutely continuous shift $X:[0,T]\rightarrow\mathbb{R}$, we have, for a.e. $t\in[0,T]$,

[TABLE]

where

[TABLE]

Remark 2.4*.*

By Remark 2.2, we know $U(t)\in\mathcal{H}$ for a.e. $t\in[0,T]$. It makes the above functionals $\mathcal{Y},\mathcal{I}^{bad},\mathcal{I}^{good}$ in (LABEL:badgood) well-defined for a.e. $t\in[0,T]$.

Proof.

To derive the desired structure, we use here a change of variables $\xi\mapsto\xi-X(t)$ as

[TABLE]

Then, by a straightforward computation together with [32, Lemma 4] and the identity $G(U;V)=G(U|V)-\nabla\eta(V)A(U|V)$ (see also [13]), we have

[TABLE]

where

[TABLE]

We first use (2.17) and (2.19) to have

[TABLE]

For the parabolic part $I_{3}$, we rewrite it into

[TABLE]

Substituting the explicit quantities in (LABEL:def-flux), we have

[TABLE]

Since

[TABLE]

we use (2.18) to have

[TABLE]

Therefore, we have

[TABLE]

Again, we use a change of variable $\xi\mapsto\xi+X(t)$ to have

[TABLE]

∎

Remark 2.5*.*

Notice that since $\sigma>0$ and $a^{\prime}>0$, the three terms of $\mathcal{I}^{good}$ in (LABEL:badgood) are non-negative. Therefore, $-\mathcal{I}^{good}$ consists of good terms, while $\mathcal{I}^{bad}$ consists of bad terms.

2.5. Maximization in terms of $q-{\tilde{q}}$

In order to estimate the right-hand side of (2.21), we will use Proposition 4.1 on a sharp estimate with respect to $n-{\tilde{n}}$ when $|n-{\tilde{n}}|\ll 1$, for which we will first rewrite the functional $\mathcal{I}^{bad}$ in the right-hand side of (2.21) into the maximized representation in terms of $q-{\tilde{q}}$. More precisely, we use the first good term of $\mathcal{I}^{good}$ in (LABEL:badgood):

[TABLE]

to separate $q-{\tilde{q}}$ from the factors related to $n$ in the first term of $\mathcal{I}^{bad}$ in (LABEL:badgood). However, we will keep $\mathcal{I}^{bad}$ for remaining cases as follows.

Lemma 2.6**.**

Let $\tilde{U}:={\tilde{n}\choose\tilde{q}}$ be the traveling wave in (2.3), and $a:\mathbb{R}\to\mathbb{R}^{+}$ be the weight function by (2.12). Let $\delta$ be any positive constant. Then, for any $U={n\choose q}\in\mathcal{H}$, we have

[TABLE]

where

[TABLE]

Remark 2.7*.*

The bad term $\mathcal{B}_{\delta}(U)$ does not ask any information on $q$ when $|(n/{\tilde{n}})-1|\leq\delta$ .

Proof.

First of all, using $a^{\prime}=-\frac{\lambda}{\varepsilon}{\tilde{n}}^{\prime}$ and ${{\tilde{q}}}^{\prime}=-\frac{{{\tilde{n}}}^{\prime}}{\sigma}$, we have from (LABEL:badgood) that

[TABLE]

By using a simple identity $\alpha x^{2}+\beta x=\alpha(x+\frac{\beta}{2\alpha})^{2}-\frac{\beta^{2}}{4\alpha}$ with putting $x:=q-{\tilde{q}}$, we have

[TABLE]

Therefore, we have the desired relation. ∎

2.6. Global and local estimates on the relative quantity ${\Pi}(\cdot|\cdot)$

2.6.1. Global estimates on the relative quantity ${\Pi}(\cdot|\cdot)$

Lemma 2.8**.**

*For given constants $\delta\in(0,\frac{1}{2}]$ and $n_{-}>0$, there exist positive constants $C_{1}=C_{1}(n_{-}),C_{2}=C_{2}(n_{-},\delta)$ and $C_{3}=C_{3}(n_{-},\delta)$ such that the following inequalities hold:

1. For any $n_{1}>0$ and any $n_{2}>0$ with $\frac{n_{-}}{2}<n_{2}<n_{-}$,*

[TABLE]

[TABLE]

[TABLE]

*where $\log^{+}(y)$ is the positive part of $\log(y)$.

2) For any $n_{1},n_{2},m>0$ satisfying $m\leq n_{2}\leq n_{1}$ or $n_{1}\leq n_{2}\leq m$,

[TABLE]

Proof.

$\bullet$ proof of (2.26) : We use the fact that the definition of the relative functional implies

[TABLE]

Notice that since $\Pi^{\prime\prime}(n)=1/n$,

[TABLE]

Since $|\frac{n_{1}}{n_{2}}-1|\leq\delta\leq\frac{1}{2}$ and $\frac{n_{-}}{2}<n_{2}<n_{-}$, we have

[TABLE]

Thus for any $0\leq s,t\leq 1$,

[TABLE]

Hence

[TABLE]

where the constant $c_{1},c_{2}$ only depends on $n_{-}$ as

[TABLE]

$\bullet$ proof of (2.27) : First of all, we observe from (2.18) that

[TABLE]

Notice that $\hat{{\Pi}}$ is smooth and non-negative on $(0,\infty)$, and $\hat{\Pi}(y)=0$ if and only if $y=1$, since $\hat{{\Pi}}$ is strictly convex, and $y=1$ is the only critical point.

We will first estimate $\hat{{\Pi}}(y)$ as follows:

For any fixed $\delta\in(0,1/2]$, since $\hat{{\Pi}}^{\prime}(y)=\log y<0$ for $0<y\leq 1-\delta$, we have

[TABLE]

On the other hand, using

[TABLE]

we have a small constant $\kappa>0$ such that

[TABLE]

Moreover, since

[TABLE]

this together with (2.31) and (2.32) implies that there exists $C=C(\delta)>0$ such that

[TABLE]

Hence, this together with (2.30) and $\frac{n_{-}}{2}<n_{2}<n_{-}$ implies (2.27).

$\bullet$ proof of (2.28) : Likewise, since there exists a constant $C=C(\delta)>0$ such that

[TABLE]

we have (2.28).

$\bullet$ proof of (2.29) : Since $z\mapsto\Pi(z|y)$ is convex in $z>0$ and zero at $z=y$, $z\mapsto\Pi(z|y)$ is increasing in $|z-y|$, which implies (2.29).

∎

Remark 2.9*.*

$C_{1}$ is independent of $\delta\in(0,1/2]$ while $C_{2}$ blows up as $\delta$ goes to zero.

2.6.2. Local inequalities on the relative quantity ${\Pi}(\cdot|\cdot)$

We present now some local estimates on ${\Pi}(n_{1}|n_{2})$ for $|n_{1}-n_{2}|\ll 1$, based on Taylor expansions. The specific coefficients of the estimates will be crucially used in our local analysis.

Lemma 2.10**.**

*For a given constant $n_{-}>0$, there exist positive constants $C$ and $\delta_{*}$ such that for any $0<\delta<\delta_{*}$, the following is true.

For any $(n_{1},n_{2})\in\mathbb{R}_{+}^{2}$ satisfying \Big{|}\frac{n_{1}}{n_{2}}-1\Big{|}<\delta and $\frac{n_{-}}{2}<n_{2}<2n_{-}$,*

[TABLE]

[TABLE]

Proof.

Since the function $\hat{\Pi}(y):=y\log y-(y-1)$ is smooth for $y>0$, we apply Taylor theorem to the function $\hat{\Pi}$. That is, using

[TABLE]

for any $0<\delta<1$ and any $y\in[1-\delta,1+\delta]$, there exists $y_{*}$ between $1$ and $y$ such that

[TABLE]

Then we take $\delta_{*}$ small enough such that for any $0<\delta<\delta_{*}$ and $y\in[1-\delta,1+\delta]$, we have

[TABLE]

Since $\Pi(n_{1}|n_{2})=n_{2}\hat{\Pi}(\frac{n_{1}}{n_{2}})$, for any $(n_{1},n_{2})\in\mathbb{R}_{+}^{2}$ satisfying \Big{|}\frac{n_{1}}{n_{2}}-1\Big{|}<\delta,

[TABLE]

which completes the proof. ∎

3. Proof of Theorem 1.1

Let $n_{-}>0$ and $q_{-}\in\mathbb{R}$. Consider $\lambda>0$ and $\varepsilon\in(0,n_{-})$. Define $n_{+}>0$ by $\varepsilon=(n_{-}-n_{+})$. Let ${\tilde{U}}:=\begin{pmatrix}{\tilde{n}}\\ {\tilde{q}}\end{pmatrix}$ be a traveling wave of (2.2) with the boundary condition (1.4) and with the speed $\sigma>0$ from (1.6). We define $a:\mathbb{R}\to\mathbb{R}_{>0}$ by (2.12).

3.1. Construction of the shift $X$

For any fixed $\varepsilon>0$, we consider a continuous function $\Phi_{\varepsilon}$ defined by

[TABLE]

For a given solution $U\in\mathcal{X}_{T}$, we define a shift function $X(t)$ as the solution of the nonlinear ODE:

[TABLE]

where the functionals $\mathcal{Y}$ and $\mathcal{I}^{bad}$ are as in (LABEL:badgood).

Then, for any solution $U\in\mathcal{X}_{T}$ for some $T>0$, an absolutely continuous shift $X$ satisfying (3.2) exists on $[0,T]$ and is unique. Indeed, if we call the right-hand side of the ODE by $F(t,X)$, then it can be shown that there exist functions $a,b\in L^{2}(0,T)$ such that

[TABLE]

by using the information from $U\in\mathcal{X}_{T}$ together with the change of variables $\xi\mapsto\xi-X(t)$ as in (2.23). Then we obtain the existence of a local solution by Picard’s iteration argument, and it is extended up to time $T$ thanks to the estimate $a,b\in L^{2}(0,T)$. Uniqueness also follows (see Appendix A for the detail).

The following is the main proposition as a corner stone of proof of Theorem 1.1.

Proposition 3.1**.**

There exist $\delta_{0}\in(0,1/2)$ and $\delta_{1}\in(0,1/2)$ such that if positive constants $\varepsilon$ and $\lambda$ satisfy $\delta_{0}^{-1}\varepsilon<\lambda<\delta_{0}$, then for any traveling wave $\tilde{U}:={\tilde{n}\choose\tilde{q}}$ in (2.3) and for any $U\in\mathcal{H}$ satisfying $|\mathcal{Y}(U)|\leq\varepsilon^{2}$, we have

[TABLE]

where the functional $\mathcal{Y}$ is as in (LABEL:badgood), $\mathcal{B}_{\delta_{1}}$ and $\mathcal{G}_{\delta_{1}}$ are as in (LABEL:bg-max), and $\mathcal{D}$ is defined by

[TABLE]

We will first show how this proposition implies Theorem 1.1.

3.2. Proof of Theorem 1.1 from Proposition 3.1

In order to prove the contraction (LABEL:ineq_contraction) in Theorem 1.1, by (2.21) and (3.2), it is enough to show that for almost every $t\in[0,T]$,

[TABLE]

For every $U\in\mathcal{H}$ we define

[TABLE]

Since it follows from (3.1) that

[TABLE]

we first find that for all $U\in\mathcal{H}$ satisfying $|\mathcal{Y}(U)|\geq\varepsilon^{2}$,

[TABLE]

On the other hand, using (2.24), we find that for any $\delta>0$ and any $U\in\mathcal{H}$ satisfying $|\mathcal{Y}(U)|\leq\varepsilon^{2}$,

[TABLE]

Then, Proposition 3.1 implies that for any $U\in\mathcal{H}$ satisfying $|\mathcal{Y}(U)|\leq\varepsilon^{2}$,

[TABLE]

Therefore, using the above estimates with $U=U^{X}$ and $\delta_{0}<\frac{1}{2}$, we find that for a.e. $t\in[0,T]$,

[TABLE]

which together with the initial condition $\int_{\mathbb{R}}\eta(U_{0}|{\tilde{U}})d\xi<\infty$ yields that

[TABLE]

To conclude (LABEL:ineq_contraction), we recover $x$ variable from $\xi$ variable (see Subsection 2.1).

Hence we have (LABEL:ineq_contraction) by redefining $X(t)$ by $(\sigma t-X(t))$.

Next, to estimate $|\dot{X}|$, we first observe that it follows from (3.1) and (3.2) that

[TABLE]

Since (LABEL:intemp) yields

[TABLE]

we have (using $\|a\|_{L^{\infty}}\leq 2$ by $\lambda<\delta_{0}<\frac{1}{2}$)

[TABLE]

Notice that (2.24) together with the definitions of $\mathcal{I}^{good}$ and $\mathcal{G}_{\delta_{1}}$ yields

[TABLE]

Since (2.26) implies that

[TABLE]

we use (2.26), $a^{\prime}\leq C\delta_{0}$, $\delta_{0}\leq\frac{1}{2}\leq a$ and by (2.6) and (2.13), to have

[TABLE]

Therefore, it follows from (3.6), (3.7) and (3.8) that

[TABLE]

where

[TABLE]

Hence we have (LABEL:est_shift) by redefining $X(t)$ by $(\sigma t-X(t))$ as mentioned above.

The remaining part is dedicated to prove Proposition 3.1. In Section 4, we study behaviour of a scalar function in a certain class near a given traveling wave ${\tilde{n}}$. Then, in Section 5, we construct a truncation $\bar{V}=\begin{pmatrix}\bar{m}\\ q\end{pmatrix}$ for $V\in\mathcal{H}$ with $|\mathcal{Y}(V)|\leq\varepsilon^{2}$ so that the truncated function $\bar{m}$ lies on the class covered in Proposition 4.1 while the error between $\bar{V}$ and $V$ in our functionals can be estimated in a proper way. It will give us Proposition 3.1.

4. Estimates near the traveling wave

4.1. Expansion in the size of the traveling wave.

We define the following functions:

[TABLE]

Proposition 4.1**.**

*For any $K>0$, there exist $\delta_{1}\in(0,\frac{1}{2})$ such that for any $\delta_{1}^{-1}\varepsilon<\lambda<\delta_{1}$ and for any $\delta\in(0,\delta_{1})$, the following is true:

For any function $n:\mathbb{R}\to\mathbb{R}^{+}$ such that if*

[TABLE]

then

[TABLE]

To prove this proposition, we will use the nonlinear Poincaré type inequality in [13]:

Lemma 4.2**.**

*[Proposition 3.3. in [13]] For any given $M>0$, there exists $\delta^{*}=\delta^{*}(M)>0$, such that, for any $\delta\in(0,\delta^{*})$, the following is true:

For any $W\in L^{2}(0,1)$ with $\sqrt{y(1-y)}\partial_{y}W\in L^{2}(0,1)$, if $\int_{0}^{1}|W(y)|^{2}\,dy\leq M$, then*

[TABLE]

where

[TABLE]

4.1.1. Proof of Proposition 4.1

We first consider $\delta_{1}\in(0,\frac{1}{2})$ such that $\delta_{1}$ is smaller than

[TABLE]

where $\varepsilon_{1}$ is as in Lemma 2.1, and $\delta_{*}$ is as in Lemma 2.10. Then it follows from Lemma 2.1 that

[TABLE]

where $\sigma_{-}$ denotes the constant in (2.7).

Note also that

[TABLE]

We define

[TABLE]

Since ${\tilde{n}}^{\prime}(\xi)<0$, we will use a change of variables $\xi\in\mathbb{R}\mapsto y\in[0,1]$ to rewrite the functionals $\mathcal{Y}_{g},\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{3},\mathcal{G}_{2},\mathcal{D}$ in (LABEL:badgood-n).

Notice that it follow from (2.12) that $a=1+\lambda y$ and

[TABLE]

In what follows, for simplification, we use the notation

[TABLE]

$\bullet$ Change of variables for $\mathcal{Y}_{g}$: We first set

[TABLE]

We use the change of variables with $|a-1|\leq\delta_{1}$ to have

[TABLE]

Since

[TABLE]

it follows from (2.33) and (2.34) in Lemma 2.10 together with $\frac{\varepsilon}{\lambda}a<2\delta_{1}$ that for any $n$ satisfying $\|\frac{n}{{\tilde{n}}}-1\|_{L^{\infty}(\mathbb{R})}\leq\delta_{1}$,

[TABLE]

Then we use the change of variables to have

[TABLE]

Thus, using (4.4) with (4.5), we have

[TABLE]

Likewise, since it follows from (2.33) and (2.34) that

[TABLE]

we have

[TABLE]

Therefore, combining (4.8), (LABEL:y14est), (4.10) with the notation (4.7), we have

[TABLE]

Setting $\beta:=2\left(\frac{1}{n_{-}}+\frac{1}{\sigma_{-}^{2}}\right)^{-1}$, we have

[TABLE]

$\bullet$ Change of variables for $\mathcal{I}_{1},\mathcal{I}_{2}$: We first use (2.33) and (2.34) to find that for any $n$ satisfying $\|\frac{n}{{\tilde{n}}}-1\|_{L^{\infty}(\mathbb{R})}\leq\delta_{1}$,

[TABLE]

Then using (4.4), we have

[TABLE]

Since

[TABLE]

using (2.34), we have

[TABLE]

Thus,

[TABLE]

$\bullet$ Change of variables for $\mathcal{G}_{2}$: We use (2.33) and (4.4) to find that

[TABLE]

$\bullet$ Estimates on $\mathcal{I}_{1}+\mathcal{I}_{2}-\mathcal{G}_{2}$: We combine (4.12), (4.13) and (4.14) to have

[TABLE]

Since the constant $\sigma_{-}=\frac{-q_{-}+\sqrt{q_{-}^{2}+4n_{-}}}{2}$ solves the quadratic equation $\sigma_{-}^{2}+q_{-}\sigma_{-}-n_{-}=0$, the above coefficients $J_{1},J_{2}$ become $J_{1}=0$, $J_{2}=\frac{4}{3}n_{-}$.

Therefore, we have

[TABLE]

which can be rewritten as (by normalizing the right-hand side above)

[TABLE]

As in (4.12), (4.13) and (4.14), we can estimate

[TABLE]

which yields

[TABLE]

Therefore, we have

[TABLE]

$\bullet$ Change of variables for $\mathcal{D}$: Since

[TABLE]

we have

[TABLE]

To compute $\frac{dy}{d\xi}$, using (4.6) with $n_{-}-n_{+}=\varepsilon$, and (2.5), we have

[TABLE]

which implies

[TABLE]

Since

[TABLE]

using (4.4), we have

[TABLE]

Therefore,

[TABLE]

Hence

[TABLE]

$\bullet$ A uniform bound of ${\int_{0}^{1}}W^{2}dy$: Using (4.2) and (4.11), we have

[TABLE]

where $K$ is the constant in the assumption (4.2).

Using

[TABLE]

we have

[TABLE]

by taking $\delta_{1}$ small enough. Therefore there exists a positive constant $M$ depending on $K$ such that

[TABLE]

$\bullet$ Control on $-|\mathcal{Y}_{g}|^{2}$ : As in [13], we here use the following inequality: For any $a,b\in\mathbb{R}$,

[TABLE]

Using this inequality with

[TABLE]

we find

[TABLE]

Then by (4.11), we have

[TABLE]

Using (4.18), we have

[TABLE]

Therefore, we have

[TABLE]

$\bullet$ Conclusion: Since $\mathcal{G}_{2}\geq 0$, we see that for any $\delta<\delta_{1}$,

[TABLE]

Multiplying (4.17) by $(1-\delta_{1})$, and summing it with (LABEL:sum-bg), (4.16) and (4.19) with putting $C_{*}:=\frac{2\beta^{2}}{n_{-}\sigma_{-}}$, we find

[TABLE]

Let $\delta^{*}$ be the constant in Lemma 4.2 corresponding to the constant $M$ of (4.18).

Taking $\delta_{1}$ small enough such that $\max(C_{*},C)\delta_{1}\leq\delta^{*}$, therefore we have

[TABLE]

Then we have $R_{\delta_{*}}(W)\leq 0$ by Lemma 4.2. Therefore $\mathcal{R}_{\varepsilon,\delta}(n)\leq 0$.

5. Proof of Proposition 3.1

5.1. Truncation of the big values of $|(n/{\tilde{n}})-1|$

In order to use Proposition 4.1, we need to show that the values for $n$ such that $|(n/{\tilde{n}})-1|\geq\delta_{1}$ have a small effect. However, the value of $\delta_{1}$ is itself conditioned to the constant $K$ in Proposition 4.1. Therefore, we need first to find a uniform bound on $\mathcal{Y}_{g}$ which is not yet conditioned on the level of truncation $\delta_{1}$.

We define a truncation on $|(n/{\tilde{n}})-1|$ with any constant $\theta\in(0,1/2)$ as follows:

[TABLE]

Notice that

[TABLE]

Lemma 5.1**.**

There exist constants $\delta_{0}\in(0,1/2)$, $C,K>0$ such that for any $\varepsilon,\lambda>0$ with $\delta_{0}^{-1}\varepsilon<\lambda<\delta_{0}$, the following holds for $U\in\mathcal{H}$ whenever $|\mathcal{Y}(U)|\leq\varepsilon^{2}$:

[TABLE]

and

[TABLE]

Proof.

$\bullet$ proof of (5.3) : We consider $\delta_{0}$ small enough such that it is smaller than (4.3), and therefore there exists $C>0$ such that $\sigma,{\tilde{n}}\in(C^{-1},C)$.

First of all, using (2.16) together with $a^{\prime}=-\frac{\lambda}{\varepsilon}{\tilde{n}}^{\prime}$ and ${{\tilde{q}}}^{\prime}=-\frac{{{\tilde{n}}}^{\prime}}{\sigma}$, we rewrite $\mathcal{Y}(U)$ in (LABEL:badgood) as

[TABLE]

Then we have

[TABLE]

Thus we use (2.26) and (2.28) to have

[TABLE]

by taking $\delta_{0}$ small enough. Hence we have

[TABLE]

which implies (5.3).

$\bullet$ proof of (5.4) : Let $\theta\in(0,1/2)$. Recall the functional $\mathcal{Y}_{g}$ and $\varphi$ in (LABEL:badgood-n). Since

[TABLE]

we have

[TABLE]

Since it follows from (2.26) with Remark 2.9 and (5.2) that

[TABLE]

we have

[TABLE]

Likewise, using (2.26), we have

[TABLE]

and

[TABLE]

Since (2.29) and (5.1) imply

[TABLE]

we use (5.3) to find that there exists $K>0$ such that

[TABLE]

∎

From now until the end, we take and fix the constant $\delta_{1}$ from Proposition 4.1 associated to the constant $K$ of Lemma 5.1. In what follows, we use the simple notation: (without confusion)

[TABLE]

Note that from Lemma 5.1, we have

[TABLE]

In what follows, we will set $\Omega:=\{\xi~{}|~{}|\frac{n}{{\tilde{n}}}-1|\leq\delta_{1}\}$.

We decompose $\mathcal{G}=\mathcal{G}_{1}^{I}+\mathcal{G}_{1}^{O}+\mathcal{G}_{2}+\mathcal{D}$ where

[TABLE]

Notice that the functionals $\mathcal{G}_{2},\mathcal{D}$ are as in (LABEL:badgood-n) and they do not depend on $q$.

We first notice that it follows from (5.6) that

[TABLE]

which together with (5.3) yields

[TABLE]

On the other hand, since $\bar{n}/{\tilde{n}}$ is constant for any $n$ satisfying either $(n/{\tilde{n}})>1+\delta_{1}$ or $(n/{\tilde{n}})<1-\delta_{1}$ by the definition of $\bar{n}$, we see

[TABLE]

Therefore we have

[TABLE]

Hence, since (5.7), (5.9) and (5.10) together with (2.20) imply that for any $(n,q)\in\mathcal{H}$, $\bar{n}$ satisfies the assumptions (4.2), Proposition 4.1 implies

[TABLE]

Before specifying the following proposition, we first recall (5.5) as

[TABLE]

We split $\mathcal{Y}$ into four parts $\mathcal{Y}_{g}$, $\mathcal{Y}_{b}$, $\mathcal{Y}_{l}$, $\mathcal{Y}_{s}$ as follows:

[TABLE]

where

[TABLE]

Notice that the functional $\mathcal{Y}_{g}$ is as in (LABEL:badgood-n). We also notice that $\mathcal{Y}_{g}$ consists of the terms related to $n$, while $\mathcal{Y}_{b}$ and $\mathcal{Y}_{l}$ consist of terms related to $q$. While $\mathcal{Y}_{b}$ is quadratic, and $\mathcal{Y}_{l}$ is linear in $q$.

For the bad terms $\mathcal{B}$ in (LABEL:bg-max), we will use the following notations :

[TABLE]

where

[TABLE]

Notice that $\mathcal{B}_{1}(U)=\mathcal{I}_{1}({n})$ and $\mathcal{B}_{2}^{I}(U)=\mathcal{B}_{2}^{I}(\bar{U})\leq\mathcal{I}_{2}(\bar{n})$ in (LABEL:badgood-n).

We now state the following proposition.

Proposition 5.2**.**

There exist constants $\delta_{0}\in(0,1/2),C,C^{*}>0$ such that for any $\delta_{0}^{-1}\varepsilon<\lambda<\delta_{0}$, the following statements hold.

For any $U\in\mathcal{H}$ such that $|\mathcal{Y}(U)|\leq\varepsilon^{2}$,

[TABLE]

• 2.

For any $U\in\mathcal{H}$ such that $|\mathcal{Y}(U)|\leq\varepsilon^{2}$ and $\mathcal{D}(U)\leq\frac{C^{*}}{4}\frac{\varepsilon^{2}}{\lambda}$,

[TABLE]

5.2. Proof of Proposition 5.2

We will first derive a point-wise estimate on $|n-{\tilde{n}}|{\mathbf{1}}_{\{|\frac{n}{{\tilde{n}}}-1|>\delta_{1}\}}$ as follows:

Lemma 5.3**.**

For a sufficiently small $\delta_{0}>0$, there exists $C>0$ such that for any $\delta_{0}^{-1}\varepsilon<\lambda<\delta_{0}$ and any $U\in\mathcal{H}$ satisfying $|\mathcal{Y}(U)|\leq\varepsilon^{2}$, the following estimates hold:

[TABLE]

whenever $\xi\in\mathbb{R}$ satisfies

[TABLE]

Remark 5.4*.*

Recall that we assumed ${\tilde{n}}(0)=(n_{-}+n_{+})/2$.

Proof.

We set $\alpha:=\frac{1}{\lambda}\int_{-1/\varepsilon}^{1/\varepsilon}a^{\prime}\,d\xi$. Using $\frac{1}{\lambda}\int_{\mathbb{R}}a^{\prime}d\xi=1$ and $a^{\prime}=(\lambda/\varepsilon)|{\tilde{n}}^{\prime}|$ together with (2.6), we obtain

[TABLE]

Notice that $\frac{1}{2}(1-e^{-1/\sigma_{-}})$ is a positive constant.

Since (5.3) implies

[TABLE]

we have

[TABLE]

Since $\int_{-1/\varepsilon}^{1/\varepsilon}\frac{a^{\prime}}{\lambda\alpha}\,d\xi=1$, there exists a point $\xi_{0}\in[-\frac{1}{\varepsilon},\frac{1}{\varepsilon}]$ such that

[TABLE]

where $\tilde{C}$ is some constant. We take $\delta_{0}$ small enough to get

[TABLE]

where $C_{2}$ is the constant in (2.27) by plugging $\delta=\delta_{1}$.

We observe that (2.26) and (2.27) imply Then from (5.15), we get

[TABLE]

Thus, by taking $\delta_{0}$ small enough, we can assume that

[TABLE]

For the reference point $\xi_{0}$, since for any $\xi\in\mathbb{R}$,

[TABLE]

we have

[TABLE]

On the other hand, we claim that there exists $L=L(\delta_{1})>0$ such that if $y>0$ and $y_{0}>0$ with

[TABLE]

then

[TABLE]

Indeed, we can split it into two cases: $0<y\leq 1-\delta_{1}$ and $y\geq 1+\delta_{1}$.

Denote $\beta:=|\sqrt{y}-\sqrt{y_{0}}|>0$.

For the first case $0<y\leq 1-\delta_{1}$, since $y\leq 1-\delta_{1}<1-(\delta_{1}/2)\leq y_{0}\leq 1+(\delta_{1}/2)$, we have

[TABLE]

Thus we get $1\leq\frac{8\beta}{\delta_{1}}$. Therefore

[TABLE]

For the second case $y\geq 1+\delta_{1}$, since

[TABLE]

we have

[TABLE]

Thus we get

[TABLE]

which yields

[TABLE]

Let $\beta_{0}=\beta_{0}(\delta_{1})$ be the positive constant satisfying $4\beta_{0}(\beta_{0}+1)=\delta_{1}$.

Since $4\beta_{0}(\beta_{0}+1)\leq 4\beta(\beta+1),$ we have $\beta\geq\beta_{0}$ so $1\leq\frac{\beta}{\beta_{0}}$.

Therefore, using (5.19), we get

[TABLE]

It proves the above claim (5.18) by taking $L:=\frac{64}{(\delta_{1})^{2}}+4(1+\frac{1}{\beta_{0}})$.

By considering $y:=\frac{n}{{\tilde{n}}}(\xi)$ and $y_{0}:=\frac{n}{{\tilde{n}}}(\xi_{0})$ in the claim (5.18) together with (5.16) and (5.14), it follows from (5.17) that

[TABLE]

∎

Lemma 5.5**.**

Under the same assumption as in Lemma 5.3, we have

[TABLE]

[TABLE]

[TABLE]

Proof.

$\bullet$ proof of (5.20) : Since \log^{+}\frac{n}{{\tilde{n}}}\leq\frac{1}{\log(1+\delta_{1})}\Big{[}\log^{+}\frac{n}{{\tilde{n}}}\Big{]}^{2} whenever $|\frac{n}{{\tilde{n}}}-1|\geq\delta_{1}$, the desired result (5.20) follows from (5.21).

$\bullet$ proof of (5.21) : Since if $n$ satisfies $\frac{n}{{\tilde{n}}}-1\leq-\delta_{1}$ then

[TABLE]

and

[TABLE]

we find that there exists a constant $C>0$ (depending on $\delta_{1}$) such that

[TABLE]

Since if $n$ satisfies $\frac{n}{{\tilde{n}}}-1\geq\delta_{1}$ then (by (2.27))

[TABLE]

using the inequality:

[TABLE]

we find that there exists a constant $C>0$ such that

[TABLE]

Indeed for large $n$, the left-hand side is bounded above by $C(1+Cn^{7/6})$ while the right one is bounded below by $\frac{1}{C}(1+\frac{1}{C}n^{3/2})$.

By combining these two cases, we obtain

[TABLE]

Then, we have

[TABLE]

Since it follows from (2.28) that $\delta_{1}\leq C|n-{\tilde{n}}|\leq C\Pi(n|{\tilde{n}})$ whenever $|\frac{n}{{\tilde{n}}}-1|\geq\delta_{1}$, we use (5.13) and (5.3) to find that there exists a constant $C>0$ (depending on $\delta_{1}$) such that

[TABLE]

Using (5.13) and (5.3), we have

[TABLE]

Notice that

[TABLE]

Taking $\delta_{0}$ small enough such that for any $\varepsilon/\lambda\leq\delta_{0}$, $|\xi|^{2}\leq e^{(c/2)|\xi|}$ for $\xi\geq\sqrt{\lambda/\varepsilon}$ and

[TABLE]

we have

[TABLE]

which gives the desired estimate.

$\bullet$ proof of (5.22) : Following the same estimates together with (5.23) as above, and using \log^{+}\frac{n}{{\tilde{n}}}\leq\frac{1}{\log(1+\delta_{1})}\Big{[}\log^{+}\frac{n}{{\tilde{n}}}\Big{]}^{2}, we find that there exists a constant $C>0$ such that

[TABLE]

Indeed for large $n$, the right-hand side is bounded below by $\frac{1}{C}(1+\frac{1}{C}n^{5/4})$.

Then, we have

[TABLE]

Using the same argument as in $J_{1}$ above, we have

[TABLE]

Since $|q-{\tilde{q}}|{\Pi}(n|{\tilde{n}})^{1/4}\leq C\eta(U|\tilde{U})^{3/4}$, we have

[TABLE]

∎

5.2.1. Proof of (1.)

We first use (2.27) and (5.20) to have

[TABLE]

We use (2.28), (2.27) and (5.22) to have

[TABLE]

We use Young’s inequality to have

[TABLE]

We separate the remaining term $B_{4}(n)$ into

[TABLE]

Since there exists a constant $C>0$ such that

[TABLE]

we use (5.21) to have

[TABLE]

Since there exists a constant $C>0$ such that

[TABLE]

using $\bar{n}\leq(1+\delta_{1}){\tilde{n}}\leq C$ and (2.26), we have

[TABLE]

Using $\varepsilon<\delta_{0}(\varepsilon/\lambda)$, we have

[TABLE]

Therefore, by taking $\delta_{0}$ small enough, we get

[TABLE]

5.2.2. Proof of (1.)

Using (5.9) and (5.3), we have

[TABLE]

Since $|\bar{n}|\leq C$, using (2.26), (5.9) and (5.3), we have

[TABLE]

Hence, combining these estimates with (1.), using (5.3), and taking $\delta_{0}$ small enough, there exists $C^{*}>0$ such that

[TABLE]

5.2.3. Proof of (LABEL:m1)

We split the proof in two steps. Step 1: We use the good term $\mathcal{G}_{1}^{I}$ defined in (LABEL:ggd) and (5.25) to have

[TABLE]

In particular, since

[TABLE]

we use (5.3) to have

[TABLE]

We use the notations $Y_{1}^{s},Y_{2}^{s},Y_{3}^{s}$ and $Y_{4}^{s}$ for the terms of $\mathcal{Y}_{s}$ as follows:

[TABLE]

Using (5.24), we have

[TABLE]

Using (2.28), we have

[TABLE]

We use $\mathcal{G}_{1}^{O}$ defined in (LABEL:ggd) to control

[TABLE]

Therefore, we have

[TABLE]

Using (5.3) together with the assumption $|\mathcal{D}(U)|\leq C\frac{\varepsilon^{2}}{\lambda}$, we have

[TABLE]

Step 2: First of all, using Young’s inequality and (5.25), we estimate

[TABLE]

Then we have

[TABLE]

Therefore, this together with (5.27), (5.26) and (5.28) implies

[TABLE]

We use Hölder’s inequality to have

[TABLE]

Hence we have (LABEL:m1)

5.3. Conclusion

We are now ready to complete the proof of Proposition 3.1. We split the proof into two steps, depending on the strength of the dissipation term $\mathcal{D}(U)$. Step 1: We first consider the case of $\mathcal{D}(U)\geq 4C^{*}\frac{\varepsilon^{2}}{\lambda},$ where the constant $C^{*}$ is defined as in Proposition 5.2. Then using $\eqref{n2}$ and taking $\delta_{0}$ small enough, we have

[TABLE]

which gives the desired result.

Step 2: We now assume the other alternative, i.e., $\mathcal{D}(U)\leq 4C^{*}\frac{\varepsilon^{2}}{\lambda}.$

We will use Proposition 4.1 to get the desired result. First of all, we recall the constant $K$ satisfying (5.7) and the fixed small constant $\delta_{1}$ of Proposition 4.1 associated to the constant $K$.

Since $\mathcal{Y}_{g}(U)=\mathcal{Y}_{g}(\bar{U})$, we have

[TABLE]

Thus we have

[TABLE]

which can be written as

[TABLE]

Below, we will take $\delta_{0}$ small enough compared to the fixed constant $\delta_{1}$ (e.g. $\delta_{0}\leq C\delta_{1}^{9}$). Then, using the facts that $\mathcal{B}_{2}^{I}(U)=\mathcal{B}_{2}^{I}(\bar{U})$, $\mathcal{G}_{2}(\bar{U})\leq\mathcal{G}_{2}(U)$ and $\mathcal{D}(\bar{U})\leq\mathcal{D}(U)$, we find that for sufficiently small $\delta_{0}$ and for any $\delta_{0}^{-1}\varepsilon<\lambda<\delta_{0}$,

[TABLE]

We claim that $J_{1}+J_{2}+J_{3}\leq 0$ for sufficiently small $\delta_{0}>0$. Indeed, it follows from (1.) and (LABEL:m1) that for sufficiently small $\delta_{0}$ and for any $\varepsilon/\lambda<\delta_{0}$, we have

[TABLE]

and

[TABLE]

Therefore, if $\delta_{0}>0$ is small enough, then we have $J_{1}+J_{2}+J_{3}\leq 0$. Thus we get

[TABLE]

Since the above quantities $\mathcal{Y}_{g}(\bar{U}),\mathcal{B}_{1}(\bar{U}),\mathcal{B}_{2}^{I}(\bar{U}),\mathcal{G}_{2}(\bar{U})$ and $\mathcal{D}(\bar{U})$ depends only on $\bar{n}$ through $\bar{U}$ and we have $\mathcal{B}_{1}(\bar{U})=\mathcal{I}_{1}(\bar{n})$ and $0\leq\mathcal{B}_{2}^{I}(\bar{U})\leq\mathcal{I}_{2}(\bar{n})$, it follows from Proposition 4.1 that $\mathcal{R}(U)\leq 0$ (or see (5.11)).

Hence we complete the proof of Proposition 3.1.

Appendix A Existence of the shift

In this subsection, we present the existence of the shift satisfying (3.2) in Subsection 3.1. For a fixed $\varepsilon>0$ and for a given solution $U\in\mathcal{X}_{T}$, we define $F:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ by

[TABLE]

where $\Phi_{\varepsilon}$ is as in (3.1) and $\mathcal{Y}$ and $\mathcal{I}^{bad}$ are as in (LABEL:badgood).

We observe that $\Phi_{\varepsilon},a$, ${\tilde{n}}$ and $(1/{\tilde{n}})$ are bounded, $\tilde{n}^{\prime},\tilde{n}^{\prime\prime},\tilde{q}^{\prime},$ and $a^{\prime}$ are bounded and integrable. Together with the information from $U\in\mathcal{X}_{T}$, we get $\mathcal{D}(U)\in L^{1}(0,T)$ where $\mathcal{D}$ is defined in (LABEL:d_only_a). From these information, we can show

[TABLE]

Since we have $\sup_{x\in\mathbb{R}}\mathcal{D}(U^{x})(t)\leq C(\mathcal{D}(U)(t)+1)$ for each $t\in[0,T]$ and $\mathcal{D}(U)\in L^{1}(0,T)$, we can estimate

[TABLE]

for some $a\in L^{2}(0,T)$.

Similarly, we can prove

[TABLE]

for some function $b\in L^{2}(0,T)$. Indeed, we can use the same idea as in (2.23) in order to move the translation symbol $(\cdot)^{X}$ from $U$ into smooth functions such as $a,{\tilde{U}}$ and so on. It enables us to differentiate $F(t,x)$ with respect to $x$ without requiring any higher regularity of $U$. Then we can get a similar control for $|(D_{x}F)|$ as in (A.1).

Then we can use the following lemma which is a simple adaptation of the well-known Cauchy-Lipschitz theorem.

Lemma A.1**.**

Let $p>1$ and $T>0$. Suppose that a function $F:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ satisfies

[TABLE]

for some functions $a\in L^{1}(0,T)$ and $\,b\in L^{p}(0,T)$. Then for any $x_{0}\in\mathbb{R}$, there exists a unique absolutely continuous function $X:[0,T]\rightarrow\mathbb{R}$ satisfying

[TABLE]

Proof.

First we note that (A.4) is equivalent to

[TABLE]

Then, the proof follows the classical Picard’s iteration argument:

[TABLE]

Indeed, we observe that $a\in L^{1}$ makes the iteration possible. In particular, $x_{n}:[0,T]\rightarrow\mathbb{R}$ is continuous and it satisfies

[TABLE]

Thanks to $b\in L^{p}$ with $p>1$, we take $t_{*}>0$ such that $\|b\|_{L^{p}(0,T)}\cdot(t_{*})^{1-(1/p)}\leq\frac{1}{2}$ and $t_{*}\leq T$. Then we get, for each $n\geq 1$,

[TABLE]

Thus we obtain $\|x_{n+1}-x_{n}\|_{L^{\infty}(0,t_{*})}\leq 2^{-n}\|a\|_{L^{1}(0,T)}$ so that the uniform limiting function $X:[0,t_{*}]\rightarrow\mathbb{R}$ of the sequence $\{x_{n}:[0,t_{*}]\rightarrow\mathbb{R}\}_{n=1}^{\infty}$ exists and it satisfies (A.5) for every $t\in[0,t_{*}]$. If $t_{*}<T$, then we just do the process again with new data $X(t_{*})$ in order to obtain $X$ on $[t_{*},2t_{*}]$. Since we can repeat as many times as we want, we get $X$ up to the given time $T$. Similarly, uniqueness follows the assumption $p>1$.

∎