The Study Of Well-posedness For Hydro Dynamics Equations In Multi-physics Coupling Processes

This dissertation is mainly devoted to the study of the well-posedness of three kinds of hydro dynamics equations in multi-physics coupling processes.These models are obtained from gas dynamic equations coupled with the equations of the self-consistent electromagnetic field,they have a wide range of application to describe the dynamics of charged particles in semiconductors or plasmas.Firstly,we study the well-posedness of the bipolar Navier-Stokes-Poisson equation which can be used to describe the dynamics of two species fluid.Secondly,we investigate the non-isentropic Navier-Stokes-Maxwell system which also takes the influence of the magnetic field into consideration.Lastly,we consider the Euler-Maxwell equation with a nonconstant background density,the main difference between this model and the first two models is that it neglect the viscosity of the fluid but take the relaxation effect into consideration.We are mainly concerned with the well-posedness and the large time behavior of the solution.Chapter 1 is the introduction part,we will give some backgrounds of the models and the research progress in this field.We also introduce the main results and some useful lemmas.In Chapter 2,we will study the well-posedness and the long time behavior of solution for the bipolar Navier-Stokes-Poisson system.We mainly use the Green func-tion method combined with high-low frequency decomposition and energy estimate to investigate this problem.Firstly,the Green function of the linearized system is es-tablished by direct calculation,then the Fourier analysis is used to derive the time decay estimate of the Green function.Then,some a priori estimate is needed to solve the nonlinear problem.To this end,we make a high-low frequency decomposition of solutions and then treat them in different ways respectively.The low frequency part is handled by the Green function method,while the high frequency part is estimated by energy method.The decay rates we obtained are optimal,and this is closely related to the anti-derivative conditions on the initial data.Moreover,we also construct the optimal decay of the high-order derivatives of solution and the Lp decay rate.In Chapter 3,we investigate the decay properties of the solution for the non-isentropic Navier-Stokes-Maxwell equation.Due to the complexity of the system,di-rect calculation of the Green function is hard.To overcome this difficulty,we need to use the Holmholtz decomposition to decompose the solution into the potential part and solenoidal part.Then the Fourier analysis method is used to derive the decay estimate of the Green function.One of the key point is that we need to treat two character-istic equations.The Navier-Stokes-Maxwell equation is of the regularity-loss type,it violates the Shizuta-Kawashima stability condition.We must use the energy estimate to overcome the difficulty of loss derivative.Actually,for the L2 decay estimate,the usual weighted energy estimate is sufficient.While for the general Lp decay estimate,the high order weighted energy estimate is dispensable.In Chapter 4,we are interested in the global existence and large time behavior of solution for the Euler-Maxwell system with nonconstant background density.Different from the Euler-Maxwell system with constant background,the system we studied here no longer admits a constant stationary solution,and this will cause difficulties in our proofs.Firstly,due to the regularity-loss property,the space-time integrability of the electric and magnetic field is lower than the others.So when we want to give the control of the nonlinear terms,we need to carefully handle the terms containing electromagnetic field.This can be overcome by elaborate integration by parts.Secondly,to get the decay rate of the solution,we have to use the decay of the linearized part.Then we need to reformulate the original system into a homogeneous system with constant coefficient and the reminders will be handled as sources.As a result,source terms will contain linear terms whose decay is worse than quadratic nonlinear terms.That will cause difficulties in closing the a priori estimate.The main idea is that we should make a clever use of the source terms.Actually,to recover the decay of the magnetic field,we have to assume that the decay of density,temperature,velocity and high order derivative of solution is strictly faster than(1+t)-1.Then we should appeal to weighted estimate for the energy and high-order energy simultaneously.Lastly,the Green function method is used to estimate the low-order terms,thus we can close the a priori assumption.