Leandro Domingues | Searcharxiv

arXiv2019

A Note on $C^2$ Ill-Posedness Results for the Zakharov System in Arbitrary Dimension

This work is concerned with the Cauchy problem for a Zakharov system with initial data in Sobolev spaces $H^k(\mathbb R^d)\!\times\!H^l(\mathbb R^d)\!\times\!H^{l-1}\!(\mathbb R^d)$. We recall the well-posedness and ill-posedness results known to date and establish new ill-posedness results. We prove $C^2$ ill-posedness for some new indices $(k,l)\in\mathbb R^2$. Moreover, our results are valid in arbitrary dimension. We believe that our detailed proofs are built on a methodical approach and can be adapted to obtain similar results for other systems and equations.

math.AP↗

arXiv2014

Sharp Well-Posedness Results for the Schrödinger-Benjamin-Ono System

This work is concerned with the Cauchy problem for a coupled Schrödinger-Benjamin-Ono system $$\left \{ \begin{array}{l} i\partial_tu+\partial_x^2u=αuv,\qquad t\!\in\![-T,T], \ x\!\in\!\mathbb R,\\ \partial_tv+ν\mathcal H\partial^2_xv=β\partial_x(|u|^2),\\ u(0,x)=ϕ, \ v(0,x)=ψ, \qquad (ϕ,ψ)\!\in\!H^{s}(\mathbb R)\!\times\!H^{s'}\!(\mathbb R). \end{array} \right. $$ In the non-resonant case $(|ν|\ne1)$, we prove local well-posedness for a large class of initial data. This improves the results obtained by Bekiranov, Ogawa and Ponce (1998). Moreover, we prove $C^2$-ill-posedness at low-regularity, and also when the difference of regularity between the initial data is large enough. As far as we know, this last ill-posedness result is the first of this kind for a nonlinear dispersive system. Finally, we also prove that the local well-posedness result obtained by Pecher (2006) in the resonant case $(|ν|=1)$ is sharp except for the end-point.

math.AP↗