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Seminários do IMPA

Análise / Equações Diferenciais Parciais

Short Wave-Long Wave Interactions in Magnetohydrodynamics
Daniel Rodriguez Marroquin

Quinta-feira, 10 de agosto de 2017, 15:30
Sala 232

We consider a Benney-type system consisting of a coupling between the equations of Magnetohydrodynamics (MHD) and a nonlinear Schrödinger equation. Such model was proposed and studied recently (in 2016) by Frid, Jia and Pan in the three dimensional context, showing existence, uniqueness and decay rates of smooth solutions for small initial data. A similar model involving the Navier-Stokes Equations instead of the MHD equations was proposed earlier by Dias and Frid in 2011, and was further studied by Frid, Pan and Zhang in 2014.

In this talk, we will discuss the existence of solutions with large initial data in the two-dimensional context. In this case, the main difficulty is the possible occurrence of vacuum. As the Lagrangian transformation becomes singular in the presence of vacuum an effective coupling of the MHD equations with the nonlinear Schrödinger equation cannot be made in a straightforward way. In order to overcome these difficulties, we define the interaction through a regularized system that provides a good definition for an approximate Lagrangian coordinate. Then, after showing existence of solutions, we show compactness of the sequence of solutions to the regularized system thus making sense of the desired SW-LW interaction in the limit process.