arXiv subjects

Tetu Makino

Publications and source records attributed to Tetu Makino.

At least 19 recordsLinked to original sources

arXiv2025

On the essential spectrum of adiabatic stellar oscillations

The generator $\mathbf{L}$ of the linearized evolution equation of adiabatic oscillations of a gaseous star, ELASO, is a second order integro-differential operator and is realized as a self-adjoint operator in the Hilbert space of square integrable unknown functions with weight, which is the density distribution of the compactly supported background. Eigenvalues and eigenfunctions of the operator $\mathbf{L}$ have been investigated in practical point of view of eigenmode expansion of oscillations. But it should be examined whether continuous spectra are absent in the spectrum of $\mathbf{L}$ or not. In order to discuss this question, the existence of essential spectra in a closely related evolution problem is established.

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arXiv2025

Linearized Analysis of Adiabatic Oscillations of Rotating Gaseous Stars

We study adiabatic oscillations of rotating self-gravitating gaseous stars in mathematically rigorous manner. The internal motion of the star is supposed to be governed by the Euler-Poisson equations with rotation of constant angular velocity under the equation of state of the ideal gas. The motion is supposed to be adiabatic, but not to be barotropic in general. This causes a free boundary problem to gas-vacuum interface. Existence of solutions to the linearized equation in the Lagrange coordinates of the perturbations around a fixed stationary solution, the eigenvalue problem with concept of quadratic pencil of operators, and the stability problem with a new concept of stability introduced in this article are discussed.

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arXiv2023

On a mathematical model of the rotating atmosphere of the Earth

In meteorology the analysis of motions of the atmosphere on the Earth has been done using various mathematical models and using various approximations. In this article as the simplest model the compressible Euler equations with barotropic equation of state of the ideal gas is analyzed under the co-ordinate system which rotates with constant angular velocity. Mathematically rigorous inquiry is tried. Although problems remain to be open, some fundamental results are exhibited.

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arXiv2021

Note on the descriptions of the Euler-Poisson equations in various co-ordinate systems

In this note we derive the descriptions of the system of Euler-Poisson equations which governs the hydrodynamic evolution of gaseous stars in various co-ordinate systems. This note does not contain essentially new results for astrophysicists, but mathematically rigorous derivations cannot be found in the literatures written by physicists so that it will be useful to prepare details of rather stupidly honest derivations of the equations in various flames when we are going to push forward the mathematical research of the problem.

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arXiv2021

On Linear Adiabatic Perturbations of Spherically Symmetric Gaseous Stars Governed by the Euler-Poisson Equations

The linearized operator for non-radial oscillations of spherically symmetric self-gravitating gaseous stars is analyzed in view of the functional analysis. The evolution of the star is supposed to be governed by the Euler-Poisson equations under the equation of state of the ideal gas, and the motion is supposed to be adiabatic. We consider the case of not necessarily isentropic, that is, not barotropic motions. Basic theory of self-adjoint realization of the linearized operator is established. Some problems in the investigation of the concrete properties of the spectrum of the linearized operator are proposed. The existence of eigenvalues which accumulate to 0 is proved in a mathematically rigorous fashion.The absence of continuous spectra and the completeness of eigenfunctions for the operators reduced by spherical harmonics is discussed.

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arXiv2021

On the axisymmetric metric generated by a rotating perfect fluid with the vacuum boundary

We consider the equations for the coefficients of stationary rotating axisymmetric metrics governed by the Einstein-Euler equations, that is, the Einstein equations together with the energy-momentum tensor of a barotropic perfect fluid. Although the reduced system of equations for the potentials in the co-rotating co-ordinate system is known, we derive the system of equations for potentials in the so called zero angular momentum observer co-ordinate system. We newly give a proof of the equivalence between the reduced system and the full system of Einstein equations. It is done under the assumption that the angular velocity is constant on the support of the density. Also the consistency of the equations of the system is analyzed. On this basic theory we construct on the whole space the stationary asymptotically flat metric generated by a slowly rotating compactly supported perfect fluid with vacuum boundary.

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arXiv2020

A Note on the Axisymmetric Stationary Metric in the General Theory of Relativity

We consider the equations for the coefficients of stationary rotating axisymmetric metrics governed by the Einstein-Euler equations, that is, the Einstein equations together with the energy-momentum tensor of a barotropic perfect fluid. Although the derived equations are not already known except for the case of the constant angular velocity described in the corotating coordinate system, the main content of this article is not to derive the equations, but to prove the equivalence of the derived equations with the full set of the Einstein equations, and to prove the consistency of the derived equations. These affairs have not yet been discussed except for the vacuum case, and are far from being self-evident, requiring tedious careful calculations and some tricks. The proof is done under the assumption that the angular velocity is constant on a neighborhood of the support of the density. The conclusions seem to be doubtful if this assumption does not hold.

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arXiv2020

On Adiabatic Oscillations of a Stratified Atmosphere on the Flat Earth

We consider the oscillations of the atmosphere around a stratified back ground density and entropy distribution under the gravitation on the flat Earth. The atmosphere is supposed to be an ideal gas and the motion is supposed to be governed by the compressible Euler equations. The density distribution of the back ground equilibrium is supposed to touch the vacuum at the finite height of the stratosphere. Considering the linearized approximation for small perturbations, we show that time periodic oscillations with a sequence of time periods which accumulate to infinity, say, slow and slow oscillations, so called `g-modes', can appear when the square of the Brunt-Vaisala frequency is positive everywhere for the considered back ground equilibrium.

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arXiv2020

Linearized Analysis of Barotropic Perturbations around Spherically Symmetric Gaseous Stars Governed by the Euler-Poisson Equations

The hydrodynamic evolution of self-gravitating gaseous stars is governed by the Euler-Poisson equations. We study the structure of the linear approximation of barotropic perturbations around spherically symmetric equilibria based on functional analytic tools. In contrast to folklore, we show that the spectrum of the linearized operator for general perturbations is not of the Sturm-Liouville type unless the perturbations are restricted. In particular, we prove that it is of the Sturm-Liouville type for irrotational perturbations.

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arXiv2020

On Incompressible Vibrations of the Stratified Atmosphere on the Flat Earth

We consider the vibrations and waves in the atmosphere under the gravitation on the flat earth. The stratified density distribution of the back ground equilibrium is supposed to touch the vacuum at the finite height of the stratosphere. We show that incompressible motions are possible to create vibrations or progressive waves with countably many time frequencies of the vibrations or speeds of the wave propagation which accumulate to 0, say, of countably infinitely many slow and slow modes.

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arXiv2019

A Remark on the Matter-Vacuum Matching Problem for Axisymmetric Metrics Governed by the Einstein-Euler Equations

Axially symmetric stationary metrics governed by the Einstein-Euler equations for slowly rotating perfect fluids have been constructed in an arbitrarily large bounded domain containing the support of the mass density. However the problem of global prolongation of the metric is still open. On the other hand the so called matter-vacuum matching problem, particularly as the source problem for the Kerr metric, has been discussed by several authors. This can be regarded as the approach to the same open problem in the opposite direction. We give a remark on this open problem.

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arXiv2019

On spherically symmetric solutions of the Einstein-Euler-de Sitter equations

We construct spherically symmetric solutions to the Einstein-Euler equations, which contains a positive cosmological constant, say, the Einstein-Euler-de Sitter equations. We assume a realistic barotropic equation of state. Equilibria of the spherically symmetric Einstein-Euler-de Sitter equations are given by the Tolman-Oppenheimer-Volkoff-de Sitter equation. We can construct solutions near time periodic linearized solutions around the equilibrium. The Cauchy problem around the equilibrium can be solved. This work can be considered as a trial of the generalization of the previous work on the problem without cosmological constants.

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arXiv2018

On Rotating Axisymmetric Solutions of the Euler-Poisson Equations

We consider stationary axisymmetric solutions of the Euler-Poisson equations, which govern the internal structure of barotropic gaseous stars. We take the general form of the equation of states which cover polytropic gaseous stars indexed by $6/5<γ<2$ and also white dwarfs. A generic condition of the existence of stationary solutions with differential rotation is given, and the existence of slowly rotating configurations near spherically symmetric equilibria is shown. The problem is formulated as a nonlinear integral equation, and is solved by an application of the infinite dimensional implicit function theorem. Oblateness of star surface is shown and also relationship between the central density and the total mass is given.

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arXiv2018

On slowly rotating axisymmetric solutions of the Einstein-Euler equations

In recent works we have constructed axisymmetric solutions to the Euler-Poisson equations which give mathematical models of slowly uniformly rotating gaseous stars. We try to extend this result to the study of solutions of the Einstein-Euler equations in the framework of the general theory of relativity. Although many interesting studies have been done about axisymmetric metric in the general theory of relativity, they are restricted to the region of the vacuum. Mathematically rigorous existence theorem of the axisymmetric interior solutions of the stationary metric corresponding to the energy-momentum tensor of the perfect fluid with non-zero pressure may be not yet established until now except only one found in the pioneering work by U. Heilig done in 1993. In this article, along a different approach to that of Heilig's work, axisymmetric stationary solutions of the Einstein-Euler equations are constructed near those of the Euler-Poisson equations when the speed of light is sufficiently large in the considered system of units, or, when the gravitational field is sufficiently weak.

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arXiv2016

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 $γ$ belongs to $(\frac65,\frac32]$. The problem is formulated as a nonlinear integral equation, and is solved by iteration technique. By this method, not only we get the existence, but also we clarify properties of the solutions such as the physical vacuum condition and oblateness of the star surface.

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arXiv2016

An Application of the Nash-Moser Theorem to the Vacuum Boundary Problem of Gaseous Stars

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.

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arXiv2016

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.

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arXiv2016

On the Tolman-Oppenheimer-Volkoff-de Sitter equation

Spherically symmetric static solutions of the Einstein equations with a positive cosmological constant for the energy-momentum tensor of a barotropic perfect fluid are governed by the Tolman-Oppenheimer-Volkoff-de Sitter equation. Existence and regularity at the vacuum boundary of the solutions with finite radii are investigated. The interior metric of the solution is connected with the Schwarzschild-de Sitter metric on the vacuum region.

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