The Existence And Stability Of Euler-Poisson Equations And Stochastic Magnetohydrodynamic Systems

This paper mainly focuses on the Euler-Poisson equations and the stochastic incompressible MHD systems.The compressible Euler-Poisson equations describe the dynamical behavior of the compressible gas stars.The stochastic incompressible MHD systems describe the dynamical behavior of plasma that mainly governed by the stochastic force and the internal electromagnetic fields produced by the particles themselves.This thesis mainly studied the existence of steady-state solutions to the Euler-Poisson equations,linear stability/instability for rotating stars,the existence of a martingale solution to the stochastic MHD systems.In Chapter 3,the existence of steady-state solutions to the steady compressible EulerPoisson equations parametrized by the rotation speed κω(r)r and center density μ is proven by implicit function theorem.These solutions continuously depend on μ and κ.In Chapter 4,for Rayleigh stable case,the linearized Euler-Poisson equations can be written as separable Hamiltonian PDEs.By using a general framework of separable Hamiltonian PDEs,the stability/instability problem can be reduced to find the number of negative direction of an operator restricted on a generalized angular momentum conserve space which has infinite constraint conditions.After we reduce these infinite constraint conditions to the mass constraint condition by finding the minimum of a quadric form,we can get a criterion of stability/instability for rotating stars.If for Rayleigh unstable case,by constructing a family of Weyl functions in the admissible perturbation space of linear dynamics,we prove that,the linearized operator has negative continuous spectrum.In Chapter 5,We consider the existence of a martingale solution to the stochastic incompressible MHD systems with Levy noises in a bounded domain.First,we use the time splitting method to construct the approximate solutions to the problems.Meanwhile,we can get the energy estimates by using the Ito formula and the BDG inequality.By showing that approximate solutions satisfies the Aldous condition,we can get the existence of martingale solution to the stochastic incompressible MHD systems.