Large Time Behavior to the Solution of Nonequilibrium Thermodynamic System and Navier–Stokes–Poisson Equations

熱動力學非平衡系統及納維–斯托克斯–泊松方程的解的大時間行為

Student thesis: Doctoral Thesis

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Award date1 Jun 2020

Abstract

Some mathematical modes are concerned in this thesis. The phenomenon is well-known that gases are composed of molecules and atoms whose overall behavior is described by kinetic theory and whose individual properties are governed by the laws of quantum mechanics. Gases can adjust instantaneously to any change in its environment, which means all molecular processes take place within the gas infinitely rapidly, then the gases come back to equilibrium flow. In one-dimensional space, gas flows with any finite number of thermal nonequilibrium modes can be governed by a hyperbolic system with several relaxation equations, which is the object of mathematical research in the second and third chapters. The last chapter is devoted to the global existence of the compressible Navier-Stokes-Poisson system, the Navier-Stokes equations coupled with the self-consistent Poisson equation, which can be used to simulate the transport of charged particles under the electric field of electrostatic potential force, such as in semiconductor devices.

In the first part, some background knowledge of mathematical modes we considered is presented.

In the second part of this thesis, a simplest case of thermal nonequilibrium modes: vibrational nonequilibrium mode has been taken into account, which only has one nonequilibrium mode. For the one-dimensional gas flow in vibrational nonequilibrium, the linearized asymptotic stability of rarefaction waves is obtained in this part with convergence rate, and the life-span of the solution in terms of the rarefaction wave strength is also given when the initial data are perturbations of a smooth rarefaction wave of the equilibrium of the compressible Euler equations. The main feature of the problems is that the L2-norm of the perturbations may grow in time.

In third part, for gas dynamics with several thermal nonequilibrium modes, the global existence of smooth solutions to an initial boundary value problem is established. The key difference compared with the initial value problem studied in  is the boundary effects, e.g. boundary layers. Moreover, the exponential decay of solutions to an initial boundary value problem of the linearized system with a vibrational nonequilibrium mode is proved via the Fourier analysis, which illustrates a key distinction from that for the initial value problem without a boundary for which the decay is only algebraic.

The last part is contributed to analyze the compressible Navier-Stokes-Poisson equations in the domain exterior to a ball in Rn (n≥1). We shall prove the global existence of spherically symmetric smooth solutions for the large initial data with symmetry assumption. Moreover, such solution approaches the steady state as time tends to infinity in three-dimensional space, provided that the pertubations between the initial data and steady states is small enough.