Shinya Nishibata

Shock waves for a model system of the radiating gas

This paper is concerned with the existence and the asymptotic stability of traveling waves for a model system derived from approximating the one-dimensional system of the radiating gas. We show the existence of smooth or discontinuous traveling waves and also prove the uniqueness of these traveling waves under the entropy condition, in the class of piecewise smooth functions with the first kind discontinuities. Furthermore, we show that the C3 -smooth traveling waves areasymptotically stable and that the rate of convergence toward these waves is t-1/4 , which looks optimal. The proof of stability is given by applying the standard energy method to the integrated equation of the original one.

CAUCHY PROBLEM FOR A MODEL SYSTEM OF THE RADIATING GAS: WEAK SOLUTIONS WITH A JUMP AND CLASSICAL SOLUTIONS

This paper deals with the global existence and the time asymptotic state of solutions to the initial value problems for the system derived from approximating a one-dimensional model of a radiating gas. When the spatial derivative of the initial data is larger than a certain negative critical value, a unique solution exists globally in time. But if it is smaller than another negative critical value, the spatial derivative of the corresponding solution blows up in a finite time. Thus it is natural to think about weak solutions in a suitable sense. As a prototype of weak solutions, we consider the Cauchy problem with the Riemann initial data of which the left state is larger than the right state. This condition ensures the existence of the corresponding traveling wave, connecting the left state and the right state asymptotically. This Riemann problem admits a global weak solution, provided that the magnitude of the initial discontinuity is smaller than 1/2. Although the solution has a discontinuity, we have the uniqueness of a solution in weak sense by imposing the entropy condition. Furthermore, the magnitude of the discontinuity contained in the solution decays to zero with an exponential rate as the time t goes to infinity. Also, the solution approaches the corresponding traveling wave with the rate t-1/4 uniformly. The first result is obtained by the maximal principles. To show the second result, we have used an energy method with some estimates, which are also obtained through maximal principles.

Lp ENERGY METHOD FOR MULTI-DIMENSIONAL VISCOUS CONSERVATION LAWS AND APPLICATION TO THE STABILITY OF PLANAR WAVES

We introduce a new Lp energy method for multi-dimensional viscous conservation laws. Our energy method is useful enough to derive the optimal decay estimates of solutions in the W1,p space for the Cauchy problem. It is also applicable to the problem for the stability of planar waves in the whole space or in the half space, and gives the optimal convergence rate toward the planar waves as time goes to infinity. This energy method makes use of several special interpolation inequalities.

Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors

We study the existence and the asymptotic stability of a stat ion ry solution to the initial boundary value problem for a one-dimensional hy drodynamic model of semiconductors. This problem is considered, in the previou s researches [2] and [11], under the assumption that a doping profile is flat, which makes the stationary solution also flat. However, this assumption is too narrow to cover the doping profile in actual diode devices. Thus, the main purpose of the presen t paper is to prove the asymptotic stability of the stationary solution without th is assumption on the doping profile. Firstly, we prove the existence of the stationary so luti n. Secondly, the stability is shown by an elementary energy method, where the equation for an energy form plays an essential role.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A MODEL SYSTEM OF RADIATING GAS WITH DISCONTINUOUS INITIAL DATA

A one-dimensional model of radiating gas is obtained from approximating the system of radiating gas with thermo-nonequilibrium. The model system consists of a conservation law and a linear elliptic equation. In this paper, we study the global existence and the time asymptotic behavior of solutions to the model system with discontinuous initial data. Since the first equation is hyperbolic, the solutions contain discontinuities for any positive time. But, the uniqueness of solutions in weak sense holds by imposing the entropy condition. The main concern of this research is to investigate the behavior of the discontinuities contained in the solutions. It is proved that the set of discontinuous points consists of a certain C1-curve. This discrepancy of values at the discontinuities of the solutions is shown to decay to zero exponentially fast as time tends to infinity. This property is utilized in showing that the solutions approach the corresponding smooth traveling waves with the rate t-1/4 in the supremum norm.

Stationary waves to viscous heat-conductive gases in half-space: Existence, stability and convergence rate

The main concern of this paper is to study large-time behavior of solutions to an ideal polytropic model of compressible viscous gases in one-dimensional half-space. We consider an outflow problem and obtain a convergence rate of solutions toward a corresponding stationary solution. Here the existence of the stationary solution is proved under a smallness condition on the boundary data with the aid of center manifold theory. We also show the time asymptotic stability of the stationary solution under smallness assumptions on the boundary data and the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. The proof is based on deriving a priori estimates by using a time and space weighted energy method.

The asymptotic behavior of the hyperbolic conservation laws with relaxation on the quarter-plane

The hyperbolic conservation laws with relaxation appear in many physical models such as those for gas dynamics with thermo-nonequilibrium, elasticity with memory, flood flow with friction, and traffic flow. The main concern of this article is the long-time behavior of the interaction between the relaxations and the boundary conditions. In this article, we investigate this problem for a simple model of a 2

Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas

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Convergence Rate Toward Planar Stationary Waves for Compressible Viscous Fluid in Multidimensional Half Space

2 system. It is proven that the solution of the system asymptotically converges to a traveling wave moving away from the boundary under suitable conditions on the boundary.

Asymptotic Stability of Boundary Layers to the Euler–Poisson Equations Arising in Plasma Physics

We study the large-time behavior of solutions to an ideal polytropic model of compressible viscous gases in one-dimensional half space. We consider an inflow problem where the gas enter into the region through the boundary, and we show that a corresponding stationary solution is time-asymptotically stable in both the subsonic and transonic cases. The proof of asymptotic stability is based on a priori estimates of the perturbation from the stationary solution, which are derived by a standard energy method, provided the boundary strength and the initial perturbation in a certain Sobolev space are sufficiently small.