Global Wellposedness Of The Compressible Navier-stokes System And Related Problems

This thesis is mainly concerned with the global xwellposedness and the precise de-scription of the large time behaviors of solutions to two types of nonlinear partial dif-ferential equations arising from the gas dynamics and the kinetic theory of dilute gases.The main contents include the existence and large-time behaviors of the global solutions to the one-dimensional and symmetric cases of multidimensional compressible Navier-Stokes equations,and the construction of the global solution of the Boltzmann equation with frictional force near a given global Maxwellian for soft potentials.The thesis is divided into the following two parts.In the first part,we are interested in the construction of global solutions to the one-dimensional and symmetric multidimensional compressible Navier-Stokes equations with large initial data.Firstly for the one-dimensional case,we investigate the motions of viscous heat-conducting fluids in half space and shear flows in unbounded domains,respectively.In Chapter 2,we study the one-dimensional compressible Navier-Stokes equations with constant transport coefficients,and consider the time asymptotical nonlinear stability of certain nonlinear wave patterns for the outflow problem.For such a problem,Qin[Nonlinearity 24(2011),no.5,1369-1394]obtained a Nishida-Smoller type nonlinear stability result for the rarefaction wave and its superposition with a non-degenerate sta-tionary solution(i.e.under the assumption that the adiabatic exponent is sufficiently close to 1).For the case with general adiabatic exponent,we obtained the asymptotical stability of its stationary solution in[Nonlinearity 29(2016),no.4,1329-1354].And in Chapter 2,we are interested in the nonlinear stability of the rarefaction wave and its superposition with a non-degenerate stationary solution,by applying the argument de-veloped by Kanel',the maximum principle and a well-designed continuation argument,we get the desired uniform lower and upper positive bounds on the density and tem-perature.Then we deduce the global solution with large initial data and it large-time behavior.Chapter 3 deals with the large-time behavior of a viscous heat-conducting flow with shear viscosity in unbounded domains.We obtain an entropy-type energy estimate involving the dissipative effects of viscosity and heat diffusion,which yields the uniform lower and upper bounds of the specific volume.Then one can deduce a positive lower bound for the temperature by employing the standard maximum principle.We deduce the uniform upper bound of the temperature and all required a priori estimates by using the entropy-type energy estimate and the nonlinear structure of the system itself.The global existence of solutions will be proved by extending the local solution to the global one based on the a priori estimates.Moreover,it follows that the solution converges to the constant steady state uniformly as time goes to infinity.Next we consider the symmetric solutions of multidimensional compressible Navier-Stokes equations.Chapter 4 is devoted to the cylindrically symmetric motion of a vis-cous and nonbarotropic fluid in the unbounded domain exterior to a three-dimensional ball.To deduce the uniform bounds of temperature,by making use of the special struc-ture of ideal polytropic gases,one can deduce a localized representation formula for the specific volume,which is essential for establishing the uniform-in-time bounds of the specific volume.Compared with one-dimensional and spherically symmetric compress-ible Navier-Stokes equations where the angular velocity ? vanishes,in our case,some new difficult term appears.Our idea to overcome this difficulty is to impose that the initial data of the angular velocity ?o satisfies ?o?o?L2(?)to control the new term.Then we can get the upper bound of specific volume,which combined with the uniform bounds of temperature ?(t,x)can finally deduce the global solvability and asymptotic behavior of cylindrically symmetric solutions.Chapter 5 is concerned with the multidimensional compressible Navier-Stokes equations in a bounded annular domain.We consider the case when the viscosity coefficients ?,A and thermal conductivity coefficient ? are smooth functions of temper-ature.We establish the global existence result of spherically or cylindrically symmetric solution.These results are of Xishida-Smoller type(i.e.the conductivity coefficient? is assumed to be sufficiently close to 1)and converge exponentially to the constant state as time tends to infinity.Our result extends the work[SIAM J.Math.Anal.46(2014),2185-2228]restricted to the one-dimensional flows.The second part is devoted to the Cauchy problem of Boltzmann equation with fric-tional force for both cutoff and non-cutoff soft potentials in the whole space.Compared with that of the Boltzmann equation,the main difficulty for the Boltzmann equation with frictional force for soft potentials lies in how to control the possible growth of the solutions induced by the frictional force.Based on the nonlinear energy method developed independently by Liu-Yang-Yu and Guo,we introduce a new time-velocity weight to control the nonlinear terms contain the frictional force,and finally establish the global solvability results near a given global Maxwellian and deduce the temporal convergence rates of such global solutions toward the global Maxwellian when initial perturbation is sufficiently small.