On local well-posedness and ill-posedness results for a coupled system of mkdv type equations

Xavier Carvajal; Liliana Esquivel; Raphael Santos

doi:10.3934/dcds.2020382

Article Contents

2021, Volume 41, Issue 6: 2699-2723. Doi: 10.3934/dcds.2020382

This issue Previous Article Čech cohomology, homoclinic trajectories and robustness of non-saddle sets Next Article Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families

On local well-posedness and ill-posedness results for a coupled system of mkdv type equations

1.
Instituto de Matemática, Universidade Federal do Rio de Janeiro-UFRJ, Ilha do Fundão, 21945-970. Rio de Janeiro-RJ, Brazil
2.
Gran Sasso Science Institute, CP 67100, L' Aquila, Italia
3.
Universidade Federal do Rio de Janeiro, Campus Macaé/RJ, Brazil

Received: February 29, 2020

Revised: August 31, 2020

Early access: November 2020

Published: June 2021

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Abstract

We consider the initial value problem associated to a coupled system of modified Korteweg-de Vries type equations

$ \begin{equation*} \begin{cases} \partial_tv + \partial_x^3v + \partial_x(vw^2) = 0,&v(x,0) = \phi(x),\\ \partial_tw + \alpha\partial_x^3w + \partial_x(v^2w) = 0,& w(x,0) = \psi(x), \end{cases} \end{equation*} $

and prove the local well-posedness results for a given data in low regularity Sobolev spaces $ H^{s}( \rm{I}\! \rm{R})\times H^{k}( \rm{I}\! \rm{R}) $, $ s,k> -\frac12 $ and $ |s-k|\leq 1/2 $, for $ \alpha\neq 0,1 $. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be $ C^3 $ at the origin, when $ s<-1/2 $ or $ k<-1/2 $ or $ |s-k|>2 $; (II) the trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) $ s-2k>1 $ or $ k<-1/2 $ (b) $ k-2s>1 $ or $ s<-1/2 $; (c) $ s = k = -1/2 $;

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Mathematics Subject Classification: 35A01, 35M31, 35Q53.

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