Asymptotic Behaviour of Solutions to Some Fluid and Plasma Equations

一類流體和等離子體方程解的漸近行為

Student thesis: Doctoral Thesis

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Award date1 Jun 2020

Abstract

In this thesis some asymptotic behaviour of both gas dynamics in thermal nonequilibrium and plasma equations in isothermal or nonthermal state have been investigated. The gas dynamics in thermal nonequilibriun is characterized by a relaxation model of Euler equations, which is a less dissipative model since the Shizuta-Kawashima condition does not hold here. While the motion of ions in inviscid or viscous plasma is presented by Euler-Poisson equations or Navier-Stokes-Poisson equations, respectively, which also have different forms in isothermal and nonthermal plasma.

On one hand, the large time behaviour of the traveling wave solutions with shock profile for one-dimensional relaxation model has been studied when the initial disturbance are small and integral zero and the shock strength is also small, using relative entropy method and antiderivative method. Moreover, the relaxation limit of the outer pressure problem of gas dynamics with several nonequilibrium modes to equilibrium flow or frozen flow in one dimension has been verified in the Sobolev space by compactness argument when the initial data are well prepared and the relaxation scale are same. Especially, the relaxation limit to equilibrium flow is much singular, and the outer pressure problem is actually a free boundary problem in Eulerian coordinates.

Moreover, on the half space, for example, Rd+, d ≥ 2, we have also verified the multi-scale singular limits for gas dynamics in thermal nonequilibrium with several nonequilibrium modes with physical boundaries in analytic settings. In this case, a cancellation mechanism is utilized to deal with the nonlinear singular terms which cause the increase of both time and space derivatives in energy estimates. The rates of the relaxations corresponding to different non-equilibrium modes tending to zero discussed in this thesis can be arbitrarily different.

On the other hand, we have considered the incompressible limit behaviour of Euler-Poisson equations in nonthermal plasma with the same order for the square of Debye length and the ratio of temperature between ions and electrons, for which the advantage of the estimate for acoustic-electric waves is invalid. In this thesis, the singular limit from compressible Euler-Poisson equations to incompressible Euler equations has been obtained with convergence rate for an ill-prepared initial data, using matched initial layer expansion.

As for the Navier-Stokes-Poisson equations in isothermal plasma, we have expanded the linear asymptotic result for the system with Navier-type slip boundary condition to that with nonslip boundary condition, under the monotonicity assumption on the initial data. The nonslip boundary condition will lead to compressible Prandtl boundary layer near the boundary when the viscosity goes to zero.