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Nonlinear Instability of the Two-Dimensional Striation Model About Smooth Steady States
Besse, C.,Degond, P.,Hwang, HJ.,Poncet, R. Marcel Dekker, Inc 2007 Communications in partial differential equations Vol.32 No.7
<P> The two-dimensional striation model consists of a nonlinear system of PDE's which arises in the modeling of the ionospheric plasma. The local-in-time existence of strong solutions is first proved using Banach's fixed point theorem. Then, under physically relevant assumptions, the system is shown to be nonlinearly unstable as soon as it is linearly unstable. Moreover, the instability occurs before the possible blow-up time of the solution. The proof relies on an earlier work of Hwang and Guo (2003). The first step of the proof is to investigate under which conditions the linearized system is unstable and to prove that its spectrum is bounded, by means of a variational formulation. The second one consists in constructing a family of solutions depending on the parameter &dgr; measuring the smallness of the perturbation to the steady-state. Thanks to the boundedness of the linearized spectrum, this family of solutions is shown to be unstable by means of a power series expansion in &dgr;.</P>
Well-posedness for Hall-magnetohydrodynamics
Chae, D.,Degond, P.,Liu, J.G. Gauthier-Villars 2014 Annales de l'Institut Henri Poincaré. Analyse non Vol.31 No.3
We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resistive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions.