Blow-up for self-interacting fractional Ginzburg-Landau equation

TL;DR

This paper investigates the blow-up behavior of solutions to a fractional Ginzburg-Landau equation with non-positive nonlinearity, providing an ODE-based proof and lifespan estimates in one dimension.

Contribution

It introduces an ODE approach to prove blow-up and derives optimal lifespan estimates for initial data in the one-dimensional case.

Findings

Solutions blow up under certain conditions.

An ODE argument effectively demonstrates blow-up.

Optimal lifespan estimates are established for 1D cases.

Abstract

The blow-up of solutions for the Cauchy problem of fractional Ginzburg-Landau equation with non-positive nonlinearity is shown by an ODE argument. Moreover, in one dimensional case, the optimal lifespan estimate for size of initial data is obtained.

Equations

$${\partial }_t\psi =-(\alpha +\mathrm{i}\beta )\Delta \psi +F(\psi ,\overline{\psi }),$$

$$F(\psi ,\overline{\psi })=-\sum _{j=1}^K({\alpha }_j+\mathrm{i}{\gamma }_j)\psi |\psi {|}^{p_j-1},$$

$${\partial }_{t}U=A\Delta U=F(U),$$

$$U(t,x)=\left(\begin{array}[]{c}u_{1}(t,x)\\ u_{2}(t,x)\\ \end{array}\right),\quad A=\left(\begin{array}[]{cc}-\alpha&\beta\\ -\beta&-\alpha\\ \end{array}\right).$$

$${\partial }_t\psi =-\mathrm{i}\beta \Delta \psi -\sum _{j=1}^N\mathrm{i}{\gamma }_j\psi |\psi {|}^{p_j-1}.$$

$${\partial }_t{\psi }_k=-\mathrm{i}{H}_k{\psi }_k+F_k(\Psi ,\overline{\Psi }),k=1,\cdots ,N.$$

$$\mathrm{Im}\left(F_k(\Psi ,\overline{\Psi })\overline{{\psi }_k}\right)=0,k=1,\cdots ,N.$$

$$\mathcal{M}=\underset{N\text{mbox}times}{\underbrace{{\mathbb{S}}^1\times \cdots \times {\mathbb{S}}^1}},$$

$${\partial }_t\Psi =\mathrm{i}\mathcal{H}\Psi -(\alpha +\mathrm{i}\beta )\Delta \Psi -({\alpha }_1+\mathrm{i}{\beta }_1)\Psi |\Psi {|}^2,$$

$${\partial }_t\psi =-\mathrm{i}\beta \Delta \psi -{\alpha }_1\psi |\psi {|}^{p-1}.$$

$${\partial }_t\psi =-\mathrm{i}\sqrt{-\Delta }\psi \pm \psi |\psi {|}^{p-1}.$$

$${\partial }_t\psi =-\mathrm{i}|D|\psi -\psi |\psi {|}^{p-1},|D|=\sqrt{-\Delta },$$

$$\{\begin{array}{ll}{\partial }_{t}u=-\mathrm{i}|D|u+|u{|}^{p-1}u,& t\in [0,T),x\in {\mathbb{R}}^n,\\ u(0,x)=u_0(x),& x\in {\mathbb{R}}^n,\end{array}$$

$$\displaystyle\bigg{\|}\frac{1}{h(\cdot)}\int_{\mathbb{R}^{n}}\langle\cdot-y\rangle^{-n-1}h(y)f(y)dy\bigg{\|}_{2}\leq C\|f\|_{2}.$$

$$\Vert \frac{1}{h}{u}_0{\Vert }_2\ge \Vert \frac{1}{h}[D,h]{\Vert }^{\frac{1}{p-1}}\Vert \frac{1}{h}{\Vert }_2.$$

$$\Vert \frac{1}{h}u(t){\Vert }_2$$

$$\displaystyle\geq e^{-2\|\frac{1}{h}[D,h]\|t}\Big{(}\|\frac{1}{h}u_{0}\|_{2}^{-p+1}+\|\frac{1}{h}[D,h]\|^{-1}\|\frac{1}{h}\|_{2}^{-p+1}\big{\{}e^{-\|\frac{1}{h}[D,h]\|(p-1)t}-1\big{\}}\Big{)}^{-\frac{1}{p-1}}.$$

$$\displaystyle T\leq-\frac{2}{p-1}\|\frac{1}{h}[D,h]\|^{-1}\log\bigg{(}1-\|\frac{1}{h}[D,h]\|\|\frac{1}{h}\|_{2}^{p-1}\|\frac{1}{h}u_{0}\|_{2}^{-p+1}\bigg{)}.$$

$$Q_{-1}(t)={\int }_{{\mathbb{R}}^n}⟨x{⟩}^{-1}|u(t,x){|}^{2}dx.$$

$${\partial }_{t}u=-\mathrm{i}|D|u+|u{|}^{p-1}u$$

$$Q_{-1}^{\prime }(t)\ge C_0{\left(Q_{-1}(t)\right)}^{(p+1)/2}-C_1{Q}_{-1}(t),C_0,C_1>0.$$

$${\partial }_{t}u=\mathrm{i}\Delta u+\mathrm{i}|u{|}^{p-1}u,$$

$$Q_2(t)={\int }_{{\mathbb{R}}^n}|x{|}^2|u(t,x){|}^{2}dx,$$

$$Q_2^{\prime \prime }(t)\sim E(u)(t),$$

$$E(u)(t)\le -\delta ,\delta >0,$$

$$Q_2^{\prime \prime }(t)\le -\delta$$

$$f^{\prime}+C_{1}f=C_{2}f^{q}\quad\mbox{on $[0,T)$ for some $T>0$},$$

$$f(t)=e^{-C_{1}t}\bigg{(}f(0)^{-(q-1)}+C_{1}^{-1}C_{2}e^{-C_{1}(q-1)t}-C_{1}^{-1}C_{2}\bigg{)}^{-\frac{1}{q-1}}.$$

$$g^{\prime }=C_2{e}^{-C_1(q-1)t}{g}^q.$$

$$\frac{1}{1-q}\bigg{(}g^{1-q}(t)-g^{1-q}(0)\bigg{)}=\frac{C_{2}}{C_{1}(1-q)}(e^{-C_{1}(q-1)t}-1).$$

$$|D_x^{\beta }{D}_{\xi }^{\alpha }p(x,\xi )|\le C_{\alpha ,\beta }⟨\xi {⟩}^{1-|\alpha |}$$

Full text

The first author was partly supported by the Japan Society for the Promotion of Science, Grant-in-Aid for JSPS Fellows no 16J30008 and Top Global University Project of Waseda University.

The second author was supported in part by INDAM, GNAMPA - Gruppo Nazionale per l’Analisi Matematica, la Probabilita e le loro Applicazioni, by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and Top Global University Project, Waseda University.

The third author was supported by Grant-in-Aid for Scientific Research (A) Number 26247014.

††footnotetext: Key words: fractional Ginzburg-Landau equation, blow-up††footnotetext: AMS Subject Classifications: 35Q40, 35Q55

1. Introduction

The classical complex Ginzburg-Landau (CGL) equation takes the form

[TABLE]

where $\alpha,\beta$ are real parameters. The standard CGL equation has a self-interaction term $F$ of the form

[TABLE]

where $\alpha_{j},\beta_{j}$ are real parameters. We refer to [5] for a review on this subject. Using the representation $\psi(t,x)=u_{1}(t,x)+\mathrm{i}u_{2}(t,x)$, where $u_{1},u_{2}$ are real-valued functions, we see that the equation (1.1) can be rewritten in the form of a system of reaction diffusion equations

[TABLE]

where

[TABLE]

The limiting case $\alpha\to 0,\alpha_{j}\to 0$ leads to the nonlinear Schrödinger equation (NLS)

[TABLE]

The oscillation synchronization of phenomena modeled by Kuramoto equations (see [4]) lead to a system of ODE having a similar qualitative behavior

[TABLE]

The nonlinear terms $F_{k}$ in the system obey the property

[TABLE]

This system simulates the behavior of $N$ oscillators, so that $\Psi=(\psi_{1},\cdots,\psi_{N}),$ with $\psi_{j}$ being complex-valued functions. The nonlinearities in (1.3) are chosen so that the evolution flow associated to the Kuramoto system leaves the manifold

[TABLE]

invariant.

The derivation of the Kuramoto system in [4] is based on complex Landau-Ginzburg equation (see equation (2.4.15) in [4])

[TABLE]

where $\Psi=(\psi_{1},\cdots,\psi_{N})^{t}$, $\mathcal{H}$ is a diagonal matrix with real entries. If $\beta$ and $\beta_{1}$ become very large, then we have an equation very close to Schrödinger self-interacting system (1.2). As it was pointed out (p. 20, [4]), a chemical turbulence of a diffusion-induced type are possible only for regions intermediate between the two extreme cases, where $\beta$ and $\beta_{1}$ are very small or very large.

Turning back to CGL equation and comparing (1.1) with Kuramoto system, we see that it is natural to take $\alpha\to 0,$ $\beta_{j}\to 0$ so that we have the following simplified CGL equation

[TABLE]

A similar system was discussed in [1] with nonlinearity typical for the Kuramoto system.

The fractional dynamics seems more adapted to synchronization models due to the considerations in [6], therefore we can consider the following fractional Ginzburg-Landau equations

[TABLE]

The study of the attractive case

[TABLE]

is initiated in [2], where the well-posedness is established for the cases $1\leq n\leq 3.$

In this article, we study the repulsive case

[TABLE]

where $n\geq 1$, and $p>1$. Our main goal is to obtain a blow-up result under the assumption that initial data are in $H^{s}(\mathbb{R}^{n})$ with $s>n/2$, where $H^{s}(\mathbb{R}^{n})$ is the usual Sobolev space defined by $(1-\Delta)^{-s/2}L^{2}(\mathbb{R}^{n})$.

We denote $\langle x\rangle=(1+|x|^{2})^{1/2}$. We abbreviate $L^{q}(\mathbb{R}^{n})$ to $L^{q}$ and $\|\cdot\|_{L^{q}(\mathbb{R}^{n})}$ to $\|\cdot\|_{q}$ for any $q$. We also denote by $\|T\|$ the operator norm of bounded operator $T:L^{2}\to L^{2}$.

The following statements are the main results of this article.

Proposition 1.1**.**

Let $h$ be a Lipschitz function satisfying $\frac{1}{h}\in L^{\infty}$ and

[TABLE]

Let $u_{0}\in hL^{2}$ satisfy

[TABLE]

If there is a solution $u\in C([0,T);hL^{2})$ for (1.4), then

[TABLE]

Therefore, the lifespan is estimated by

[TABLE]

We remark that $\langle\cdot\rangle$ is a typical example of weight functions $h$ for Proposition 1.1. Proposition 1.1 is a blow-up result for a kind of large data of $hL^{2}$. However, in a subcritical case where $p<p_{F}=1+2/n$, solutions blow up even for small $L^{2}$ initial data.

Corollary 1.2**.**

Let $u_{0}\in L^{2}(\mathbb{R}^{n})\backslash\{0\}$ and $1<p<p_{F}$. Then the corresponding solution in $C([0,T);L^{2})$ blows up at a finite positive time.

Remark 1.1**.**

If we choose $h(x)=\langle x\rangle$, the statement of our main result guarantees the blow-up of the momentum

[TABLE]

for the solution to the fractional CGL equation

[TABLE]

in (1.4). The blow-up mechanism is based on the differential inequality

[TABLE]

Comparing the fractional CGL equation with the classical NLS

[TABLE]

we see that introducing the momentum

[TABLE]

and using a Virial identity one can show that

[TABLE]

where $E(u)(t)=\|Du(t)\|^{2}_{L^{2}}-c\|u(t)\|^{p+1}_{L^{p+1}}$ and $c>0$ is an appropriate constant. Therefore, the blow-up mechanism for NLS is based on the estimate

[TABLE]

that implies differential inequality

[TABLE]

and the last inequality can not be satisfied for the whole interval $t\in(0,\infty)$ since $Q_{2}(t)$ is a positive quantity.

Moreover, for large $R$, if $u_{0}$ is given by $Rf$ with $f\in hL^{2}$ and $h$ satisfying (1.5), then (1.8) means $T\leq CR^{-p+1}$. In one dimensional case, this upper bound is shown to be sharp for $f\in(hL^{2}\cap H^{1})(\mathbb{R})$.

Proposition 1.3**.**

Let $u_{0}=Rf$ with $R>0$ sufficiently large and $f\in H^{1}(\mathbb{R})$. Then there exists an $H^{1}(\mathbb{R})$ solution for $u_{0}$ for which its lifespan is estimated by $T\geq CR^{-p+1}$ with some positive constant $C$.

2. Preliminary

In this section, we recall the blow-up solutions for an ODE which gives the mechanism of blow-up for weighted $L^{2}$ norm of solutions. We also study the condition for weight functions of Proposition 1.1.

2.1. Blow-up solutions for an ODE

Lemma 2.1**.**

Let $C_{1},C_{2}>0$ and $q>1$. If $f\in C^{1}([0,T);\mathbb{R})$ satisfies $f(0)>0$ and

[TABLE]

then

[TABLE]

Moreover, if $f(0)>C_{1}^{\frac{1}{q-1}}C_{2}^{-\frac{1}{q-1}}$, then $T<-\frac{1}{C_{1}(q-1)}\log(1-C_{1}C_{2}^{-1}f(0)^{-q+1})$.

Proof.

Let $f=e^{-C_{1}t}g$. Then

[TABLE]

Therefore,

[TABLE]

∎

2.2. Condition for weight function

Lemma 2.2** (Coiffman - Meyer).**

Let $p\in C^{\infty}(\mathbb{R}^{2n})$ satisfy the estimates

[TABLE]

for all multi-indices $\alpha$ and $\beta$. Then for any Lipschitz function $h$,

[TABLE]

Lemma 2.3**.**

Let $\phi\in C_{0}^{\infty}([0,\infty);\mathbb{R})$ satisfy

[TABLE]

Then

[TABLE]

Proof.

It suffices to consider the case where $|x|$ is sufficiently large. Let $\psi\in C_{0}^{\infty}([0,\infty);\mathbb{R})$ satisfy

[TABLE]

Let $e_{1}=(1,0,\cdots,0)$. Let $\xi_{1}=\xi\cdot e_{1}$ and $\xi^{\prime}=\xi-\xi_{1}e_{1}$. Assume $x=|x|e_{1}$. Then

[TABLE]

By integrating by parts $k$ times,

[TABLE]

where $R_{n+1}\in L^{1}(\mathbb{R}^{n})$. Here $\partial_{1}^{k}|\xi|$ is estimated by $C|\xi|^{1-k}$. Moreover,

[TABLE]

since $|e^{\mathrm{i}|x|\xi_{1}}-1|\leq|x||\xi|$. The first integral is estimated by

[TABLE]

By letting $\Psi(\rho)=\int_{0}^{\rho}|1-\psi(\rho^{\prime})|d\rho^{\prime}$ and integrating by parts once again, the second integral is estimated by

[TABLE]

This proves the lemma. ∎

Lemma 2.4**.**

Let $h$ be a Lipschitz function on $\mathbb{R}^{n}$ satisfying the estimate

[TABLE]

for any $f\in L^{2}$. Then $\frac{1}{h}[D,h]$ is a bounded operator from $L^{2}$ to $L^{2}$.

Proof.

Let $\phi$ be a smooth function on $[0,\infty)$ satisfying that $\phi(\xi)=1$ if $|\xi|\leq 1$ and $\phi(\xi)=0$ if $|\xi|\geq 2$. Let $\phi(D)f=\mathfrak{F}^{-1}\phi\hat{f}$. We divide the proof into the following two estimate: $\|\frac{1}{h}\phi(D)(Dhf)\|_{2}\leq\|f\|_{2}$ and $\|\frac{1}{h}[(1-\phi(D))D,h]f\|_{2}\leq\|f\|_{2}$.

At first,

[TABLE]

since

[TABLE]

Secondly, $(1-\phi(|\xi|))|\xi|$ satisfies the condition of Lemma 2.2. So the second estimate follows from Lemma 2.2. ∎

Remark 2.1**.**

$h(x)=\langle x\rangle$* satisfies the condition of Lemma 2.4. Actually $h$ is Lipshitz and by using triangle inequality,*

[TABLE]

where $n<q<\infty$.

Corollary 2.5**.**

Let $h$ satisfy the condition of Lemma 2.4 and let $h_{R}$ be $h_{R}=h(\cdot/R)$. Then

[TABLE]

Proof.

[TABLE]

This implies

[TABLE]

∎

3. Proof

3.1. Proof of Proposition 1.1

Let $u(t,x)=h(x)v(t,x)$. Then

[TABLE]

Multiplying both hand sides of (3.1) by $\overline{v}$, integrating over $\mathbb{R}^{n}$, and taking the imaginary part of the resulting integrals, we obtain

[TABLE]

where we used the following estimate:

[TABLE]

Then (1.7) follows from Lemma 2.1 with $q=(p+1)/2$.

3.2. Proof of Corollary 1.2

Let $h_{R}(x)=\langle x/R\rangle$ with $R>0$. Then $\frac{1}{h_{R}}u_{0}\to u_{0}$ in $L^{2}$ as $R\to\infty$. Moreover, $\|\frac{1}{h_{R}}[D,h_{R}]\|\sim R^{-1}$, and $\|\frac{1}{h_{R}}\|_{2}\sim R^{n/2}$. Therefore

[TABLE]

as $R\to\infty$ if $p<1+\frac{2}{n}$. It means that for any $u_{0}\in L^{2}(\mathbb{R}^{n})\backslash\{0\}$, there exists $R_{0}$ such that (1.6) is satisfied with $h(x)=\langle x/R_{0}\rangle$.

3.3. Proof of Proposition 1.3

The local well-posedness in $H^{1}(\mathbb{R})$ is easily obtained by the Sobolev embedding and standard contraction argument. By multiplying (1.4) by $\overline{u}$ and $(-\Delta)\overline{u}$, integrating over $\mathbb{R}$, we obtain

[TABLE]

where $\|f\|_{H^{1}(\mathbb{R})}^{2}=\|f\|_{2}^{2}+\|\nabla f\|_{2}^{2}$. By solving the following ordinary differential equality:

[TABLE]

we get

[TABLE]

This proves the Proposition 1.3.