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Luc Molinet

Publications and source records attributed to Luc Molinet.

At least 37 records · Page 2Linked to original sources
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].

math.AP↗
arXiv2018

A liouville property with application to asymptotic stability for the camassa-holm equation

We prove a Liouville property for uniformly almost localized (up to translations) H 1-global solutions of the Camassa-Holm equation with a momentum density that is a non negative finite measure. More precisely, we show that such solution has to be a peakon. As a consequence, we prove that peakons are asymptotically stable in the class of H 1-functions with a momentum density that belongs to M + (R). Finally, we also get an asymptotic stability result for train of peakons.

math.AP↗
arXiv2018

On well-posedness for some dispersive perturbations of Burgers' equation

We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation $\partial$\_t u -- D^$α$\_x $\partial$\_x u = $\partial$\_x(u^2), 0 < $α$ $\le$ 1, is locally well-posed in H^s (R) when s > 3 /2 -- 5$α$ /4. As a consequence, we obtain global well-posedness in the energy space H^{$α$/2} (R) as soon as $α$ > 6/7 .

math.AP↗
arXiv2017

Unconditional uniqueness for the modified Korteweg-de Vries equation on the line

We prove that the modified Korteweg- de Vries equation (mKdV) equation is unconditionally well-posed in $H^s(\mathbb R)$ for $s> \frac 13$. Our method of proof combines the improvement of the energy method introduced recently by the first and third authors with the construction of a modified energy. Our approach also yields \textit{a priori} estimates for the solutions of mKdV in $H^s(\mathbb R)$, for $s>0$, and enables us to construct weak solutions at this level of regularity.

math.AP↗
arXiv2016

On the stability of the solitary waves to the (generalized) kawahara equation

In this paper we investigate the orbital stability of solitary waves to the (generalized) Kawahara equation (gKW) which is a fifth order dispersive equation. For some values of the power of the nonlinearity, we prove the orbital stability in the energy space H 2 (R) of two branches of even solitary waves of gKW by combining the well-known spectral method introduced by Benjamin [3] with continuity arguments. We construct the first family of even solitons by applying the implicit function theorem in the neighborhood of the explicit solitons of gKW found by Dey et al. [8]. The second family consists of even travelling waves with low speeds. They are solutions of a constraint minimization problem on the line and rescaling of perturbations of the soliton of gKdV with speed 1.

math.AP↗
arXiv2015

Optimal transportation between hypersurfaces bounding some strictly convex domains

Let $M,N$ be two smooth compact hypersurfaces of $\mathbb{R}^n$ which bound strictly convex domains equipped with two absolutely continuous measures $μ$ and $ν$ (with respect to the volume measures of $M$ and $N$). We consider the optimal transportation from $μ$ to $ν$ for the quadratic cost. Let $(ϕ:m \to \mathbb{R},ψ:N \to \mathbb{R})$ be some functions which achieve the supremum in the Kantorovich formulation of the problem and which satisfy $$ ψ(y) = \inf_{z\in M} \Bigl( \frac{1}{2}|y-z|^2 -φ(z)\Bigr); φ(x)=\inf_{z\in N} \Bigl( \frac{1}{2}|x-z|^2 -ψ(z)\Bigr).$$ Define for $y \in N$, $$φ^\Box(y) = \sup_{z\in M} \Bigl( \frac{1}{2}|y-z|^2 -φ(z)\Bigr).$$ In this short paper, we exhibit a relationship between the regularity of $φ^\Box$ and the existence of a solution to the Monge problem.

math.DG↗
arXiv2014

Improvement of the energy method for strongly non resonant dispersive equations and applications

In this paper we propose a new approach to prove the local well-posedness of the Cauchy problem associated with strongly non resonant dispersive equations. As an example we obtain unconditional well-posedness of the Cauchy problem below $ H^1 $ for a large class of one-dimensional dispersive equations with a dispersion that is greater or equal to the one of the Benjamin-Ono equation. Since this is done without using a gauge transform, this enables us to prove strong convergence results for solutions of viscous versions of these equations towards the purely dispersive solutions.

math.AP↗
arXiv2013

Remarks on the Cauchy problem for the one-dimensional quadratic (fractional) heat equation

We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat equation: $u_t=D_x^{2α} u \mp u^2,\; t\in (0,T),\; x\in \R$ or $ \T $, with $ 0<α\le 1 $ is well-posed in $ H^s $ for $ s\ge \max(-α,1/2-2α) $ except in the case $ α=1/2 $ where it is shown to be well-posed for $ s>-1/2 $ and ill-posed for $ s=-1/2 $. As a by-product we improve the known well-posedness results for the heat equation ($α=1$) by reaching the end-point Sobolev index $ s=-1 $. Finally, in the case $ 1/2<α\le 1 $, we also prove optimal results in the Besov spaces $B^{s,q}_2.$

math.AP↗
arXiv2013

Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications

This article is concerned with the Zakharov-Kuznetsov equation {equation} \label{ZK0} \partial_tu+\partial_xΔu+u\partial_xu=0 . {equation} We prove that the associated initial value problem is locally well-posed in $H^s(\mathbb R^2)$ for $s>\frac12$ and globally well-posed in $H^1(\mathbb R\times \mathbb T)$ and in $H^s(\R^3) $ for $ s>1$. Our main new ingredient is a bilinear Strichartz estimate in the context of Bourgain's spaces which allows to control the high-low frequency interactions appearing in the nonlinearity of \eqref{ZK0}. In the $\mathbb R^2$ case, we also need to use a recent result by Carbery, Kenig and Ziesler on sharp Strichartz estimates for homogeneous dispersive operators. Finally, to prove the global well-posedness result in $ \R^3 $, we need to use the atomic spaces introduced by Koch and Tataru.

math.AP↗
arXiv2012

Dispersive limit from the Kawahara to the KdV equation

We investigate the limit behavior of the solutions to the Kawahara equation $$ u_t +u_{3x} + \varepsilon u_{5x} + u u_x =0, $$ as $ 0<\varepsilon \to 0 $. In this equation, the terms $ u_{3x} $ and $ \varepsilon u_{5x} $ do compete together and do cancel each other at frequencies of order $ 1/\sqrt{\varepsilon} $. This prohibits the use of a standard dispersive approach for this problem. Nervertheless, by combining different dispersive approaches according to the range of spaces frequencies, we succeed in proving that the solutions to this equation converges in $ C([0,T];H^1(\R)) $ towards the solutions of the KdV equation for any fixed $ T>0$.

math.AP↗
arXiv2011

Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation

In this paper, we prove that the Cauchy problem associated to the following higher-order Benjamin-Ono equation $$ \partial_tv-b\mathcal{H}\partial^2_xv- aε\partial_x^3v=cv\partial_xv-dε\partial_x(v\mathcal{H}\partial_xv+\mathcal{H}(v\partial_xv)), $$ is globally well-posed in the energy space $H^1(\mathbb R)$. Moreover, we study the limit behavior when the small positive parameter $ε$ tends to zero and show that, under a condition on the coefficients $a$, $b$, $c$ and $d$, the solution $v_ε$ to this equation converges to the corresponding solution of the Benjamin-Ono equation.

math.AP↗
arXiv2011

Sharp ill-posedness results for the KdV and mKdV equations on the torus

We establish a new a priori bound for $ L^2 $-bounded sequences of solutions to the mKdV equations on the torus. This first enable us to construct weak solutions in $ L^2$ for this equation and to check that the "solutions" constructed by Kappeler and Topalov in the defocusing case satisfy the equation in some weak sense. In a second time, we prove that the solution-map associated with the mKdV and the KdV equation are discontinuous for the $ H^s(\T) $ topology for respectively $ s<0$ and $ s<-1$. These last results are sharp.

math.AP↗
arXiv2011

Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equations

Sufficient and necessary conditions for the generalized Gagliardo-Nirenberg (GN) inequality in Besov spaces and Triebel-Lizorkin spaces are obtained. Applying the GN inequality, we show that the finite-time blowup solutions have concentration phenomena in critical Lebesgue space L^3. Moreover, we consider the minimizer for a class of variational problem by applying the fractional GN inequality.

math.FA↗
arXiv2010

Global well-posedness for the KP-II equation on the background of a non localized solution

Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations : perturbations that are square integrable in $ \R\times \T $ and perturbations that are square integrable in $ \R^2 $. In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.

math.AP↗