Akitaka Matsumura

Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids

The equations of motion of compressible viscous and heat-conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region. The global solution in time is proved to exist uniquely and approach the stationary state ast→∞, provided the prescribed initial data and the external force are sufficiently small.

Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion

The asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics. 1) The solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet−γ (for someγ>0) ast→∞, provided that the initial disturbance is small and of integral zero, and in addition decays at an algebraic rate for |x|→∞. 2) The traveling wave solution with Nishida and Smollers condition of the system of a viscous heat-conductive ideal gas is asymptotically stable, provided the initial disturbance is small and of integral zero. 3) The traveling wave solution with weak shock profile of the Broadwell model system of the Boltzmann equation is asymptotically stable, provided the initial disturbance is small and its hydrodynamical moments are of integral zero. Each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one. The property of integral zero of the initial disturbance plays a crucial role in this procedure.

Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas

This paper is concerned with the asymptotic behavior toward the rarefaction waves of the solution of a one-dimensional model system associated with compressible viscous gas. If the initial data are suitably close to a constant state and their asymptotic values atx=±∞ are chosen so that the Riemann problem for the corresponding hyperbolic system admits the weak rarefaction waves, then the solution is proved to tend toward the rarefaction waves ast→+∞. The proof is given by an elementaryL2 energy method.

On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas

Travelling wave solutions with shock profile for a one-dimensional model system associated with compressible viscous gas are investigated in terms of asymptotic stability. The travelling wave solution is proved to be asymptotically stable, provided the initial disturbance is suitably small and of zero constant component. The proof is given by the elementalL2 energy method.

Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas

This paper is concerned with the asymptotic behavior toward the rarefaction wave of the solution of a one-dimensional barotropic model system for compressible viscous gas. We assume that the initial data tend to constant states atx=±∞, respectively, and the Riemann problem for the corresponding hyperbolic system admits a weak continuous rarefaction wave. If the adiabatic constant γ satisfies 1≦γ≦2, then the solution is proved to tend to the rarefaction wave ast→∞ under no smallness conditions of both the difference of asymptotic values atx=±∞ and the initial data. The proof is given by an elementaryL2-energy method.

Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity

The asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsut+f(u)x=μuxx with the initial datau0 which tend to the constant statesu± asx→±∞. Stability theorems are obtained in the absence of the convexity off and in the allowance ofs (shock speed)=f′(u±). Moreover, the rate of asymptotics in time is investigated. For the casef′(u+)<s<f′(u−), if the integral of the initial disturbance over (−∞,x) is small and decays at the algebraic rate as |x|→∞, then the solution approaches the traveling wave at the corresponding rate ast→∞. This rate seems to be almost optimal compared with the rate in the casef=u2/2 for which an explicit form of the solution exists. The rate is also obtained in the casef′(u±=s under some additional conditions. Proofs are given by applying an elementary weighted energy method to the integrated equation of the original one. The selection of the weight plays a crucial role in those procedures.

On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation

The compressible and heat-conductive Navier-Stokes equation obtained as the second approximation of the formal Chapman-Enskog expansion is investigated on its relations to the original nonlinear Boltzmann equation and also to the incompressible Navier-Stokes equation. The solutions of the Boltzmann equation and the incompressible Navier-Stokes equation for small initial data are proved to be asymptotically equivalent (mod decay ratet−5/4) ast→+∞ to that of the compressible Navier-Stokes equation for the corresponding initial data.

Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas

We consider asymptotic behaviors in time of solutions to the initial boundary value problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. In particular, we focus our attention on inflow(or outflow) problems where the velocity on the boundary is given as a constant inward (or outward) flow, and try to classify all asymptotic behaviors of the solutions. It turns out that depending on the data both on the boundary and at far field (especially depending on whether the state is subsonic, transonic, or supersonic), the asymptotic state variously consists of rarefaction waves, viscous shock waves, and also stationary boundary layer. Moreover, we give a survey of our recent results on some particular cases which justify our classification.

Uniform boundedness of the solutions for a one-dimensional isentropic model system of compressible viscous gas

This paper studies an initial boundary value problem for a one-dimensional isentropic model system of compressible viscous gas with large external forces, represented by vt−ux=0,ut+(av−γ)x=μ(ux/v)x+f(∫0xvdx,t), with (v(x, 0),u(x, 0))= (v0(x),u0(x)),u(0,t)=u(1,t)=0. Especially, the uniform boundedness of the solution in time is investigated. It is proved that for arbitrary large initial data and external forces, the problem uniquely has an uniformly bounded, global-in-time solution with also uniformly positive mass density, provided the adiabatic constant γ(>1) is suitably close to 1. The proof is based on L2-energy estimates and a technique used in [9].

The fixed boundary value problems for the equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall condition

The equations of ideal Magneto-Hydrodynamics are investigated concerning initial boundary value problems with a perfectly conducting wall condition. The local in time solution is proved to exist uniquely, provided that the normal component of the initial magnetic field vanishes everywhere or nowhere on the boundary.