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Seminários do IMPA

Análise / Equações Diferenciais Parciais

Short Wave-Long Wave Interactions in Planar Magnetohydrodynamics
Daniel Rodriguez Marroquin

Quinta-feira, 22 de junho 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 through particle paths. The model may be regarded as the evolution of a quantum particle in a magnetohydrodynamic flow. 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 focus on a one dimensional version of the model for which we show the existence and uniqueness of smooth solutions for large initial data. We also show convergence of the sequence of solutions as the bulk viscosity of the fluid, the magnetic permeability of the fluid, and the interaction coefficient of the coupling terms tend to zero, showing that the limit functions satisfy the limit problem, which consists of a decoupled system involving the Euler Equations of compressible fluid dynamics and the nonlinear Schrödinger equation.