Mathematical Theory Research On Some Equations Of Hydrodynamics And Population Dynamics

The paper falls into two parts.One part is composed of the second chapter,the third chapter and the fourth chapter,which is concerned with some equations of hydrodynamics,including the compressible Euler equations with damping,the compressible magnetohy-drodynamic equations,the steady equations of compressible nematic liquid crystals flows.This part mainly studied the well-posedness of solutions for the three partial differential equations.The other part is the fifth chapter,which mainly discussed the well-posedness of strong solutions to an approximation and instability behaviour of equilibrium state for the fitness gradient system of population dynamics.In Chapter 2,we considered the existence and uniqueness of a time periodic solution to the three-dimensional compressible damped Euler equations in a periodic domain.By adapting a regularized approximation scheme and applying the topological degree theory,we establish the existence of the time periodic solution under some smallness and structure assumptions imposed on a time periodic force.And based on energy estimates,the uniqueness of the periodic solution is proved.In Chapter 3,we investigated the Cauchy problem for the three-dimensional com-pressible magnetohydrodynamic equations without heat conductivity.The global solution is obtained by combining the local existence and a priori estimates under the smallness assumption on the initial perturbation in Hl(l>3).But we don’t need the bound of L1 norm.This is different from the work[21].Our proof is based on pure estimates to get the time decay estimates on the pressure,velocity and magnet field.In particular,we use a fast decay of velocity gradient to get the uniform boundedness of the non-dissipative entropy,which is sufficient to close the priori estimates.In addition,we study the optimal convergence rates of the global solution.In Chapter 4,we considered the Dirichlet problem of three-dimensional equations for the steady compressible flow of nematic liquid crystals are considered for the adiabatic exponent γ>1.We establish the existence and uniform boundedness of a weak solution by three-level approximation,weighted estimates and weak convergence.In Chapter 5,we studied a fitness gradient system for two populations interacting via a symmetric game.The population dynamics are governed by a conservation law,with a spatial migration flux determined by the fitness.By applying the Galerkin method,we establish the existence,regularity and uniqueness of global solutions to an approximate system,which retains most of the interesting mathematical properties of the original fitness gradient system.Furthermore,we show that a Turing instability occurs for equilibrium states of the fitness gradient system,and its approximations.