Atusi Tani

FREE BOUNDARY PROBLEM FOR AN INCOMPRESSIBLE IDEAL FLUID WITH SURFACE TENSION

We prove that a free boundary problem for an incompressible Euler equation with surface tension is uniquely solvable, locally in time, in a class of functions of finite smoothness. Moreover, it is shown that the solution of this problem converges to the solution of the problem without surface tension as the coefficient of the surface tension tends to zero.

Initial and Initial-Boundary Value Problems for a Vortex Filament with or without Axial Flow

The equations which describe the motion of a vortex filament with or without an axial flow inside its core are considered. The initial and the initial-boundary value problems are proved to have unique and smooth solutions globally in time. These results are obtained by adding vanishing parabolic terms which conserve the length of the filament.

CLASSICAL SOLVABILITY OF THE TWO-PHASE STEFAN PROBLEM IN A VISCOUS INCOMPRESSIBLE FLUID FLOW

In this paper we study the two-phase Stefan problem for a viscous incompressible fluid which is a model of melting a solid to a liquid or of soldificating a liquid to a solid with a liquid flowing. The unique solvability is established in Holder spaces locally in time. The method of the proof is rather standard, however the result obtained is completely new: the existence in the Holder spaces with the same Holder exponents as the data and the uniqueness also in the same Holder spaces.

On the classical solvability of the Stefan problem in a viscous incompressible fluid flow

This paper is devoted to the study of a solidification/melting process in the case where the fluid is flowing. Such a phenomenon is described by the Stefan problem with transport terms in the equation of the temperature distribution and the initial-boundary value problem for the incompressible Navier--Stokes equations. The existence of the classical solution is proved locally in time.

Solvability of the localized induction equation for vortex motion

The initial and the initial-boundary value problems for the localized induction equation which describes the motion of a vortex filament are considered. We prove the existence of solutions of both problems globally in time in the sense of distribution by the method of regularization.

TEMPORALLY GLOBAL SOLUTION TO THE EQUATIONS FOR A SPHERICALLY SYMMETRIC VISCOUS RADIATIVE AND REACTIVE GAS OVER THE RIGID CORE

In this paper, we consider a system of classical model equations describing a motion of gaseous star, spherically symmetric, with a free-surface and a rigid core in the center under the influence of gaseous self-gravitation and potential force of the core. In addition to it, we investigate the above model equations in the physically more realistic situation: the presence of radiation and reacting process. By introducing Lagrangian mass coordinate, this free-boundary problem is reduced to the one in a fixed domain. Based on the fundamental local existence result and a priori estimates, we can construct a unique global classical solution.

Overlapping domain problems in the crack theory with possible contact between crack faces

The paper is concerned with the analysis of a new class of overlapping domain problems for elastic bodies having cracks. Inequality type boundary conditions are imposed on the crack faces. We prove an existence of invariant integrals and analyze the asymptotic behavior of the solution. It is shown that the limit problem describes an equilibrium state for the elastic body with a thin inclusion.

A Boundary Value Problem for an Infinite Elastic Strip with a Semi-infinite Crack

In this paper we study a boundary value problem for an infinite elastic strip with a semi-infinite crack. By using the single and double layer potentials this problem is reduced to a singular integral equation, which is uniquely solved in the Hölder spaces by the Fredholm alternative.

On inverse problems for pseudoparabolic and parabolic equations of filtration

The inverse problems concerning the identification of the coefficient in the second order terms of linear pseudoparabolic equations of filtration in a fissured rock are investigated. The physical and mathematical justification of possible statements of the inverse problem for pseudoparabolic equations is given. New boundary conditions of overdetermination are discussed. Certain parabolic inverse problems relevant to pseudoparabolic ones are considered. The existence and uniqueness of a strong solution to one of the discussed inverse problems for the pseudoparabolic equation is proved in the case of one space variable. It is also established that the inverse problem for the pseudoparabolic equation approximates weakly some inverse problem for the parabolic one.

Existence of solution for Eguchi-Oki-Matsumura equation describing phase separation and order-disorder transition in binary alloys

Abstract In this paper we consider the Eguchi–Oki–Matsumura equation which consists of the fourth- and second-order coupled equations of parabolic type. It is shown that this system admits the unique global solution.