Search

The purpose of this paper is to investigate mathematically the fundamental mechanism of gas discharge. Townsend discovered that α- and γ-mechanisms are essential for a gas ionization process. Morrow, Degond, and Lucquin-Desreux derived a mathematical model taking these two mechanisms into account. This model consists of nonlinear hyperbolic, parabolic, and elliptic partial differential equations. In this paper, we establish a framework for analyzing this model rigorously, because no mathematical result has been announced. More precisely, we show the unique existence of a time-local solution to an initial-boundary value problem over an unbounded domain. The main difficulty of this problem lies in the fact that the direction of boundary characteristics of a hyperbolic equation degenerates at infinite distance. Generally speaking, the regularity of solutions to linearized hyperbolic equations with degenerate boundary characteristics may be lost near the boundary. This implies the possibility that loss on the derivatives arises at each step of the inductive scheme to solve the nonlinear problem. We first use the weighted Sobolev spaces in the construction of solutions to make clear the direction of characteristics. Furthermore, we reduce the initial-boundary value problem for the hyperbolic equation to an initial value problem by applying several extension operators. This reduction enables us to avoid the loss on the derivatives. [ABSTRACT FROM AUTHOR]

The pattern formation of a radially growing interface in a Hele-Shaw cell is studied. In contrast to the previous studies based on the Young-Laplace equation, a boundary condition is employed that included the effect of the wetting layer of the displacing fluid in the cell. Under this boundary condition, a weakly nonlinear analysis is carried out and the time evolution of the interface is numerically calculated. These results, compared to those obtained in the previous studies, suggest that our model reflects the nonlinear features of the viscous fingering phenomena more precisely. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with free boundary problem for compressible viscous isotropic Newtonian fluid. Our problem is to find the three-dimensional domain occupied by the fluid which is bounded below by the fixed bottom and above by the free surface together with the density, the velocity vector field and the absolute temperature of the fluid satisfying the system of Navier-Stokes equations and the initial-boundary conditions. The Navier-Stokes equations consist of the conservations of mass, momentum under the gravitational field in a downward direction and energy. The effect of the surface tension on the free surface is taken into account. The purpose of this paper is to establish two existence theorems to the problem mentioned above: the first concerns with the temporary local solvability in anisotropic Sobolev-Slobodetskii spaces and the second the global solvability near the equilibrium rest state. Here the equilibrium rest state (heat conductive state) means that the temperature distribution is a linear function with respect to a vertical direction and the density is determined by an ordinary differential equation which involves equation of state. For the proof, we rely on the methods due to Solonnikov in the case of incompressible fluid with some modifications, since our problem is hyperbolic-parabolic coupled system. [ABSTRACT FROM AUTHOR]