Mathematical Analysis Of Hydrodynamics Equations In Multi-physics Processes

The thesis is aimed to study the well-posedness of solutions,singular per-turbation problems and blow-up criteria of hydrodynamics equations in multi-physics processes.First,we consider the Cauchy problem of the incompressible MHD equations,of which we establish the existence,uniqueness and regular-ity theory of mild solutions provided that the initial data belong to Morrey space.Then,we study two kinds of singular perturbation problems including the combined non-relativistic and quasi-neutral limit of the two-fluid Euler-Maxwell equations and the zero-electron-mass limit of isothermal Euler-Poisson equations with variable ion density.We prove that the limiting system of the two-fluid Euler-Maxwell equations is the system of compressible Euler equations.While,we show that Euler-Poisson equations' limiting system is the system of incom-pressible Euler equations,at the same time,we give a rigorous proof of the Boltzmann relation.Finally,we study two kinds of blow-up criteria for the strong solutions to the initial boundary value problem of the three-dimensional compressible isentropic MHD equations with zero magnetic diffusion and initial vacuum and to the Cauchy problems of three-dimensional compressible radiation fluids with vacuum.Chapter 1 is the introduction,dedicated to several models of hydrodynamics equations in multi-physics processes and related results.We also state our main results.Some basic inequalities are given for later use.In Chapter 2,we establish the existence,uniqueness and regularity theory of mild solutions to the MHD equations with the initial data within Morrey space.This Chapter is inspired by Giga,Miyakawa[58]and Kato[86]'s work in incompressible Navier-Stokes equations.To solve the system when the initial data is very "rough",we will study the time evolution of the vorticity and the current density.The main difficulty is the coupling terms.Because of the unboundedness of Calderon-Zygmund singular operator in L1 space,we can not use the vorticity to control the gradient of velocity appearing in the coupling terms.To this end,we should use more equations to get more estimates of the solutions in order to close the estimates.We use several inequalities about heat kernel,Biot-Savart kernel and Riesz potentials in Morrey space to get the estimates.Then,we use the Banach fixed point Theorem to prove the main results.In Chapter 3,we consider two-fluid Euler-Maxwell equations for magnetized plasmas composed of electrons and ions.By using the method of asymptotic expansions,we analyze the combined non-relativistic and quasi-neutral limit for periodic problems with well-prepared initial data.It is shown that the small parameter problems have a unique solution existing in a finite time interval where the corresponding limit problems(compressible Euler equations)have smooth solutions.The proof is based on energy estimates for symmetrizable hyperbolic equations and on the exploration of the coupling between the Euler equations and the Maxwell equations.The convergence from two-fluid Euler-Maxwell equations to two-fluid Euler-Poisson equations under non-relativistic limit has been studied in[162]and the quasi-neutral limit from two-fluid Euler-Poisson equations to compressible Euler equations has been studied in[102].The formal derivation of the combined limit has been given in[123].The main work of Chapter 3 is to prove rigorously that it makes no difference taking the limit either simultaneously or successively,while at the same time verifying the results in[123].In Chapter 4,by means of asymptotic expansions and energy estimates,we justify the Boltzmann relation as the zero-electron-mass limit of an isothermal Euler-Poisson system with variable ion density.The Boltzmann relation for elec-trons is well-known in plasma physics.It can be formally derived from fluid equations when the electron mass is neglected.The result is obtained for peri-odic smooth solutions of the system on a uniform time interval with respect to the electron-mass.Comparing to the problem when the ion density is constant,variable ion density case leads to additional difficulties in higher order energy estimates,which occur in particular in crossing terms.Our treatment for these terms is successful only for the isothermal fluid.It should be noted that the condition of isothermal fluid is only used in the estimate of crossing terms in the higher order energy estimates.Other estimates are all valid for any strictly increasing p.In Chapter 5,we prove a blow-up criterion in terms of the magnetic field H and the mass density p for the strong solutions to the three-dimensional compress?ible isentropic MHD equations with zero magnetic diffusion and initial vacuum.Due to the vanishing of magnetic diffusion,there are two significant difficulties in our proof:the strong coupling between the velocity u and H and the lack of regularity of H.Under the assumption of the viscosity coefficients(3A<29?),we show that the L?norms of the magnetic field H and the mass density p control the possible blow-up for strong solutions,which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time,then the formation of singularity must be caused by losing the upper bound of H or p as the critical time approaches.In Chapter 6,we consider the blow-up criteria for the Cauchy problems of three-dimensional compressible radiation fluids with vacuum.It is shown to own the same BKM-type criterion as the compressible Navier-Stokes equations[71],while the LP(p ?[2,3])norm of the gradient of density should be involved for the Serrin-type criterion.