Yoshiyuki Kagei

The Eckhaus criterion for convection roll solutions of the Oberbeck-Boussinesq equations

Abstract This paper is concerned with the stability problem of the convection roll solutions of the Oberbeck-Boussinesq equations. We consider the stability of the roll solutions with respect to the two-dimensional disturbances. The Eckhaus criterion for the stability and instability of the roll solutions is derived in a mathematically rigorous way.

LOCAL SOLVABILITY OF AN INITIAL BOUNDARY VALUE PROBLEM FOR A QUASILINEAR HYPERBOLIC-PARABOLIC SYSTEM

This paper investigates the solvability of initial boundary value problem for a quasilinear hyperbolic-parabolic system which consists of a transport equation and strongly parabolic system. The characteristics of the transport equation are assumed to be outward on the boundary of the domain. The unique local (in time) existence of solutions is shown in the class of continuous functions with values in Hs, where s is an integer satisfying s ≥ [n/2]+1.

Asymptotic behavior of solutions to the compressible Navier-Stokes equation in a cylindrical domain

Asymptotic behavior of solutions to the compressible Navier-Stokes equation around a given constant state is investigated on a cylindrical domain in R, under the no slip boundary condition for the velocity field. The L decay estimate is established for the perturbation from the constant state. It is also shown that the time-asymptotic leading part of the perturbation is given by a function satisfying a 1 dimensional heat equation. The proof is based on an energy method and asymptotic analysis for the associated linearized semigroup.

Decay properties of solutions to the linearized compressible Navier-Stokes equation around time-periodic parallel flow

Decay estimates on solutions to the linearized compressible Navier–Stokes equation around time-periodic parallel flow are established. It is shown that if the Reynolds and Mach numbers are sufficiently small, solutions of the linearized problem decay in L2 norm as an (n - 1)-dimensional heat kernel. Furthermore, it is proved that the asymptotic leading part of solutions is given by solutions of an (n - 1)-dimensional linear heat equation with a convective term multiplied by time-periodic function.

Spectral Properties of the Linearized Semigroup of the Compressible Navier–Stokes Equation on a Periodic Layer

The linearized problem for the compressible Navier–Stokes equation around a given constant state is considered in a periodic layer of R with n ≥ 2, and spectral properties of the linearized semigroup are investigated. It is shown that the linearized operator generates a C0-semigroup in L 2 over the periodic layer and the time-asymptotic leading part of the semigroup is given by a C0-semigroup generated by an n − 1-dimensional elliptic operator with constant coefficients that are determined by solutions of a Stokes system over the basic period domain. 2010 Mathematics Subject Classification: 35Q30, 35Q35, 76N15.

Asymptotic stability of higher order norms in terms of asymptotic energy stability for viscous incompressible fluid flows heated from below

We consider a steady viscous incompressible fluid flow in an infinite layer heated from below. The steady flow is assumed to be periodic with respect to the plane variables. If this flow turns out to be asymptotically energy-stable with respect to a particular disturbance then it is also asymptotically stable in higher order norms with respect to the same perturbation. No smallness of the initial values is needed.

Large time behavior of solutions to the compressible Navier-Stokes equations in an infinite layer under slip boundary condition

This paper is concerned with large time behavior of solutions to the compressible Navier–Stokes equations in an infinite layer of ℝ2 under slip boundary condition. It is shown that if the initial data is sufficiently small, the global solution uniquely exists and the large time behavior of the solution is described by a superposition of one-dimensional diffusion waves.

The Oberbeck--Boussinesq Approximation as a Constitutive Limit

We derive the usual Oberbeck–Boussinesq approximation as a constitutive limit of the full system describing the motion of an compressible linearly viscous fluid. To this end, the starting system is written, using the Gibbs free energy, in the variables v, θ and p. The Oberbeck–Boussinesq system is then obtained as the thermal expansion coefficient α and the isothermal compressibility coefficient β tend to zero.