Tetu Makino

Free Boundary Problem for the Equation of One-Dimensional Motion of Compressible Gas with Density-Dependent Viscosity.

SuntoSi considera un problema di frontiera libera per l’equazione del moto unidimensionale isoentropico con viscosità dipendente dalla densità secondo la legge μ =bϱβ, doveb e β sono costanti positive. Si dimostra che esiste un’unica soluzione debole globale nel tempo, purché β<1/3.AbstractWe consider a free boundary problem for the equation of the one-dimensional isentropic motion with density-dependent viscosity μ =bϱβ, whereb and β are positive constants. We prove that there exists an unique weak solution globally in time, provided that β<1/3.

Free boundary problem for the equation of spherically symmetric motion of viscous gas (III)

We study the spherically symmetric motion of viscous barotropic gas surrounding a solid ball. We are interested in the density distribution which contacts the vacuum at a finite radius. The equilibrium is asymptotically stable with respect to small perturbation, provided that γ > 4/3 anda is sufficiently small, when the equation of state isp =aργ,p being the pressure and π the density.

Spherically symmetric solutions to the compressible Euler equation with an asymptotic γ-law

This work investigates the spherically symmetric solutions of compressible Euler equation with an asymptotic γ-law. We generalize the method of [6, 7, 8] to show the existence of weak solution of the equation with initial data containing the vacuum state.

On spherically symmetric solutions of the Einstein-Euler equations

We construct spherically symmetric solutions to the Einstein-Euler equations, which give models of gaseous stars in the framework of the general theory of relativity. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein-Euler equations are given by the Tolman-Oppenheimer-Volkoff equations, and time periodic solutions around the equilibrium of the linearized equations can be considered. Our aim is to find true solutions near these time-periodic approximations. Solutions satisfying so called physical boundary condition at the free boundary with the vacuum will be constructed using the Nash-Moser theorem. This work also can be considered as a touchstone in order to estimate the universality of the method which was originally developed for the non-relativistic Euler-Poisson equations.

An application of the Nash–Moser theorem to the vacuum boundary problem of gaseous stars

Abstract We have been studying spherically symmetric motions of gaseous stars with physical vacuum boundary governed either by the Euler–Poisson equations in the non-relativistic theory or by the Einstein–Euler equations in the relativistic theory. The problems are to construct solutions whose first approximations are small time-periodic solutions to the linearized problem at an equilibrium and to construct solutions to the Cauchy problem near an equilibrium. These problems can be solved when 1 / ( γ − 1 ) is an integer, where γ is the adiabatic exponent of the gas near the vacuum, by the formulation by R. Hamilton of the Nash–Moser theorem. We discuss on an application of the formulation by J.T. Schwartz of the Nash–Moser theorem to the case in which 1 / ( γ − 1 ) is not an integer but sufficiently large.

On Slowly Rotating Axisymmetric Solutions of the Euler–Poisson Equations

We construct stationary axisymmetric solutions of the Euler–Poisson equations, which govern the internal structure of polytropic gaseous stars, with small constant angular velocity when the adiabatic exponent

The Mathematician K. Ogura and the “Greater East Asia War”

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On rotating axisymmetric solutions of the Euler–Poisson equations

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