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Luc Molinet

Publications and source records attributed to Luc Molinet.

At least 19 recordsLinked to original sources
arXiv2026

$L^2({\mathbb R}) $-Unconditional well-posedness for low dispersion fractional KdV equations

We show that the $ L^2({\mathbb R}) $-unconditional well-posedness, that is well-known for the KdV equation, is shared by KdV type equations with weaker dispersion. This is despite the difference in the nature of these equations, which are quasilinear while KdV is semilinear. More precisely we prove that the low dispersion fractional KdV equation $$ \partial_t u -D_x^\alpha \partial_x u +\partial_x(u^2)=0 $$ is unconditionally globally well-posed in $L^2({\mathbb R}) $ for $\alpha \in ]\frac{55}{38},2] $. Our method of proof combined refined bilinear estimates with the energy method enhanced with Bourgain's type estimates developed in Molinet-Vento (2015).

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arXiv2025

Improved refined bilinear estimates and well-posedness for generalized KdV type equations on $\mathbb{R}$

We study the Cauchy problem for one-dimensional dispersive equations posed on $\mathbb{R} $, under the hypotheses that the dispersive operator behaves, for high frequencies, as a Fourier multiplier by $ i |\xi|^\alpha \xi $ with $ 1 \le \alpha\le 2 $, and that the nonlinear term is of the form $ \partial_x f(u) $ where $f $ is a real analytic function satisfying certain conditions. We prove the unconditional local well-posedness of the Cauchy problem in $H^s(\mathbb{R}) $ for $ s\ge \frac{5-2\alpha}{4} $ whenever $ 1\le \alpha<\frac{3}{2} $, and for $ s>\frac{1}{2} $ whenever $\alpha\in [\frac{3}{2},2] $. This result is optimal in the case $\alpha\ge \frac{3}{2}$ in view of the restriction $ s>\frac{1}{2} $ required for the continuous embedding $ H^s(\mathbb{R}) \hookrightarrow L^\infty(\mathbb{R}) $. The main novelty of this work, compared to our previous studies, is an improvement of the refined linear and bilinear estimates on $\mathbb{R} $. Our local well-posedness results enable us to derive global existence of solutions for $ \alpha \in [\frac{5}{4},2] $.

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arXiv2025

Low regularity well-posedness of nonlocal dispersive perturbations of Burgers' equation

We consider the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations, which contains the low dispersion Benjamin-Ono equation, (also known as low dispersion fractional KdV equation), $$ \partial_tu-D_x^{\alpha}\partial_xu=\partial_x(u^2) \, ,$$ and prove that it is locally well-posed in $H^s(\mathbb K)$, $\mathbb K=\mathbb R$ or $\mathbb T$, for $s>s_{\alpha}$, where \begin{equation*} s_\alpha=\begin{cases} 1-\frac{3\alpha}4 & \text{for} \quad \frac23 \le \alpha \le 1; \frac 32(1-\alpha) & \text{for} \quad \frac13 \le \alpha \le \frac23; \frac 32-\frac{\alpha}{1-\alpha} & \text{for} \quad 0 < \alpha \le \frac13 . \end{cases} \end{equation*} The uniqueness is unconditional in $H^s(\mathbb K)$ for $s>\max\{\frac12,s_{\alpha}\}$. Moreover, we obtain \emph{a priori} estimates for the solutions at the lower regularity threshold $s>\widetilde{s}_\alpha$ where \begin{equation*} \widetilde{s}_\alpha=\begin{cases} \frac 12-\frac \alpha 4 & \text{for} \quad \frac23 \le \alpha \le 1; 1-\alpha & \text{for} \quad \frac12 \le \alpha \le \frac23; \frac 32-\frac{\alpha}{1-\alpha} & \text{for} \quad 0 < \alpha \le \frac12 . \end{cases} \end{equation*} As a consequence of these results and of the Hamiltonian structure of the equation, we deduce global well-posedness in $H^s(\mathbb K)$ for $s>s_{\alpha}$ when $\alpha>\frac23$, and in the energy space $H^{\frac{\alpha}2}(\mathbb K)$ when $\alpha>\frac45$.

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arXiv2025

Local well-posedness for the derivative nonlinear Schr\"odinger equation with nonvanishing boundary conditions

We consider the derivative nonlinear Schr\"odinger equation on the real line, with a background function $\psi(t,x)\in L^\infty(\mathbb{R}^2)$ that satisfies suitable conditions. Such a function may, for example, be a non-decaying solution of the equation, such as a dark soliton. By developing the energy method with correction terms, we prove that the Cauchy problem for perturbations around such an $L^\infty$ function is unconditionally locally well-posed in $ H^s(\mathbb{R}) $ for $ s>3/4 $. As a byproduct, we also establish local well-posedness in the Zhidkov space.

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arXiv2024

Asymptotic stability of fast solitary waves to the Benjamin Equation

We prove the asymptotic stability of the high speed solitary waves to the Benjamin equation. This is done by establishing a Liouville property for the nonlinear evolution of the Benjamin equation around these solitary waves. To do this, inspired by Kenig-Martel-Robbiano 2011, we make use of the KdV limit of the Benjamin equation together with known rigidity property of the KdV flow. The main difficulties are linked to the presence of the Hilbert transform, that is a non-local operator, as well as the non-positivity of the quadratic part of the energy in the case $ \gamma<0$ which is the physical case.

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arXiv2024

Self-similar solutions for the generalized fractional Korteweg-de Vries equation

We consider the Cauchy problem for the generalized fractional Korteweg-de Vries equation $$ u_t+D^\alpha u_x + u^p u_x= 0, \quad 1<\alpha\le 2, \quad p\in {\mathbb N}\setminus\{0\}, $$ with homogeneous initial data $\Phi$. We show that, under smallness assumption on $\Phi$, and for a wide range of $(\alpha, p)$, including $p=3$, we can construct a self-similar solution of this problem.

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arXiv2024

Global solutions for the one-dimensional Boussinesq-Peregrine system under small bottom variation

The Boussinesq-Peregrine system is derived from the water waves system in presence of topographic variation under the hypothesis of shallowness and small amplitude regime. The system becomes significantly simpler (at least in the mathematical sens) under the hypothesis of small topographic variation. In this work we study the long time and global well-posedness of the Boussinesq-Peregrine system. We start by showing the intermediate time well-posedness in the case of general topography (i.e. the amplitude of the bottom graph $\beta=O(1)$). The novelty resides in the functional setting, $H^s({\mathbb R}), \, s> \frac {1} {2}$. Then we show our main result establishing that the global existence result obtained in Molinet-Talhouk-Zaiter in the flat bottom case is still valid for the Boussinesq-Peregrine system under the hypothesis of small amplitude bottom variation (i.e. $\beta =O(\mu)$). More precisely we prove that this system is unconditionally globally well-posed in the Sobolev spaces of type $ H^s ({\mathbb R}), \, s> \frac {1} {2}$. Finally, we show the existence of a weak global solution in the Schonbek sense, i.e. existence of low regularity entropic solutions of the small bottom amplitude Boussinesq-Pelegrine equations emanating from $ u_0 \in H^1 $ and $ \zeta_0 $ in an Orlicz class as weak limits of regular solutions.

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arXiv2024

On the well-posedness of the KP-I equation

We revisit the local well-posedness for the KP-I equation. We obtain unconditional local well-posedness in $H^{s,0}({\mathbb R}^2)$ for $s>3/4$ and unconditional global well-posedness in the energy space. We also prove the global existence of perturbations with finite energy of non decaying smooth global solutions.

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arXiv2024

On the Uniqueness and Orbital Stability of Slow and Fast Solitary Wave Solutions of the Benjamin Equation

This paper is devoted to the study of existence and properties of solitary waves of the Benjamin equation. The studied equation includes a parameter $\gamma$ in front of the Benjamin-Ono term. We show the existence, uniqueness, decay and orbital stability of solitary wave solutions obtained as a solution to a certain minimization problem, associated either with high speeds without a sign condition on the parameter $\gamma$ or with low speeds for the appropriate sign.

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arXiv2022

Refined bilinear Strichartz estimates with application to the well-posedness of periodic generalized KdV type equations

We improve our previous result [L. Molinet and T. Tanaka, Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations, J. Funct. Anal. 283 (2022), 109490] on the Cauchy problem for one dimensional dispersive equations with a quite general nonlinearity in the periodic setting. Under the same hypotheses that the dispersive operator behaves for high frequencies as a Fourier multiplier by $ i |\xi|^\alpha \xi $ with $ 1 \le \alpha\le 2 $, and that the nonlinear term is of the form $ \partial_x f(u) $ where $f $ is a real analytic function whose Taylor series around the origin has an infinite radius of convergence, we prove the unconditional LWP of the Cauchy problem in $H^s(\mathbb{T}) $ for $ s\ge 1-\frac{\alpha}{4} $ with $ s>1/2 $. It is worth noting that this result is optimal in the case $\alpha=2$ (generalized KdV equation) in view of the restriction $ s>1/2 $ for the continuous injection of $ H^s(\mathbb{T}) $ into $ L^\infty(\mathbb{T}) $. Our main new ingredient is the replacement of refined Strichartz estimates with refined bilinear estimates in the treatment of the worst resonant interactions. Such refined bilinear estimates already appeared in the work of Hani in the context of Schr\"odinger equations on a compact manifold. Finally, the main theorem yields global existence results for $ \alpha \in [4/3,2] $.

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arXiv2021

On well-posedness for some Korteweg-De Vries type equations with variable coefficients

In this paper, KdV-type equations with time- and space-dependent coefficients are considered. Assuming that the dispersion coefficient in front of $u_{xxx}$ is positive and uniformly bounded away from the origin and that a primitive function of the ratio between the anti-dissipation and the dispersion coefficients is bounded from below, we prove the existence and uniqueness of a solution $u$ such that $h u$ belongs to a classical Sobolev space, where $h$ is a function related to this ratio. The LWP in $H^s(\mathbb{R})$, $s>1/2$, in the classical (Hadamard) sense is also proven under an assumption on the integrability of this ratio. Our approach combines a change of unknown with dispersive estimates. Note that previous results were restricted to $H^s(\mathbb{R})$, $s>3/2$, and only used the dispersion to compensate the anti-dissipation and not to lower the Sobolev index required for well-posedness.

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arXiv2021

Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations

We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity in the periodic setting. Our main hypotheses are both that the dispersive operator behaves for high frequencies as a Fourier multiplier by $ i |\xi|^\alpha \xi $, with $ 1\le \alpha \le 2 $, and that the nonlinear term is of the form $ \partial_x f(u) $ where $ f $ is the sum of an entire series with infinite radius of convergence. Under these conditions, we prove the unconditional local well-posedness of the Cauchy problem in $H^{s}(\mathbb{T})$ for $ s\ge 1-\frac{\alpha}{2(\alpha+1)}$. This leads to some global existence results above the energy space $ H^{\alpha/2}(\mathbb{T}) $, for $ \alpha \in [\sqrt{2},2]$.

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arXiv2020

The classical Boussinesq system revisited

In this work, we revisit the study by M. E. Schonbek [11] concerning the problem of existence of global entropic weak solutions for the classical Boussinesq system, as well as the study of the regularity of these solutions by C. J. Amick [1]. We propose to regularize by a "fractal" operator (i.e. a differential operator defined by a Fourier multiplier of type $\epsilon |\xi |^\lambda, \, (\epsilon,\lambda) \in\,\mathbb{R}_+\times ] 0,2]$). We first show that the regularized system is globally unconditionally well-posed in Sobolev spaces of type $H^s(\mathbb{R}),\,s > \frac {1}{2},$, uniformly in the regularizing parameters $(\epsilon,\lambda) \in\,\mathbb{R}_+\times ]0,2]$. As a consequence we obtain the global well-posedness of the classical Boussinesq system at this level of regularity as well as the convergence in the strong topology of the solution of the regularized system towards the solution of the classical Boussinesq equation as the parameter e goes to 0. In a second time, we prove the existence of low regularity entropic solutions of the Boussinesq equations emanating from $u_0 \in H^1$ and $\zeta_0$ in an Orlicz class as weak limits of regular solutions.

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arXiv2018

A rigidity result for the Holm-Staley b-family of equations with application to the asymptotic stability of the Degasperis-Procesi peakon

We prove that the peakons are asymptotically H 1-stable, under the flow of the Degasperis-Procesi equation, in the class of functions with a momentum density that belongs to M + (R). The key argument is a rigidity result for uniformly in time exponentially decaying global solutions that is shared by the Holm-Staley b-family of equations for b $\ge$ 1. This extends previous results obtained for the Camassa-Holm equation (b = 2).

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arXiv2018

Ill-posedness of the Camassa-Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B^{1+1/p}_{p,r}$ for $p\in [1,\infty], r\in (1,\infty]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works (\cite{Danchin2,Byers,HHK}).

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arXiv2018

Asymptotic stability for some non positive perturbations of the Camassa-Holm peakon with application to the antipeakon-peakon profile

We continue our investigation on the asymptotic stability of the peakon. In a first step we extend our asymptotic stability result [29] in the class of functions whose negative part of the momentum density is supported in ] -- $\infty$, x 0 ] and the positive part in [x 0 , +$\infty$[ for some x 0 $\in$ R. In a second step this enables us to prove the asymptotic stability of well-ordered train of antipeakons-peakons and, in particular, of the antipeakon-peakon profile. Finally, in the appendix we prove that in the case of a non negative momentum density the energy at the left of any given point decays to zero as time goes to +$\infty$,. This leads to an improvement of the asymptotic stability result stated in [29].

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