I gt(P 2 c 2 + P 2)-F a a x"(P 2 c 2 ± P 2vk)=°k =1 3 2 0 c 2 ± P V a (PC 2 +p v v " + 0 , k) = 0... more I gt(P 2 c 2 + P 2)-F a a x"(P 2 c 2 ± P 2vk)=°k =1 3 2 0 c 2 ± P V a (PC 2 +p v v " + 0 , k) = 0 , i = 1 , 2 , 3. gt ' c 2-v 2 V ') a X k cv 2 3
In recent works we have constructed axisymmetric solutions to the Euler-Poisson equations which g... more 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.
It is known that spherically symmetric static solutions of the Einstein equations with a positive... more It is known that 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. Some sufficient conditions for the existence of monotone-short solutions (with finite radii) of the equation are given in this article. Then we show that the interior metric can extend to the exterior Schwarzschild-de Sitter metric on the exterior vacuum region with twice continuous differentiability. In addition, we investigate the analytic property of the solutions at the vacuum boundary. Our result (Theorem 1) can be considered as the de Sitter version of the result by Rendall and Schmidt [Classical Quantum Gravity 8, 985-1000 (1991)]. Furthermore, one can see that there are different properties of the solutions with those of the Tolman-Oppenheimer-Volkoff equation (with zero cosmological constant) in certain situation.
The linearized operator for non-radial oscillations of spherically symmetric self-gravitating gas... more 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.
Japan Journal of Industrial and Applied Mathematics, Jun 1, 1995
We study the spherically symmetric motion of viscous barotropic gas surrounding a solid ball. We ... more We study the spherically symmetric motion of viscous barotropic gas surrounding a solid ball. We are interested in the density distribution which contacts with the vacuum at a ¡ radius. This is a free boundary problem. We obtained the existence of a global weak solution with some regular properties. We can show that such a solution is unique.
Japan Journal of Industrial and Applied Mathematics, Jun 1, 1997
We study the spherically symmetric motion of viscous baxotropic gas surrounding a solid ball. We ... more We study the spherically symmetric motion of viscous baxotropic 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 "y > 4 anda is sufficiently small, when the equation of state is p = ap ~, p being the pressure and p the density.
We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball u... more We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball under the gravity of the core. Equilibria touch the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this timeperiodic solution plus the equilibrium is the first approximation. But this leads us to difficulty caused by singularities at the free boundary touching the vacuum. We solve this problem by the Nash-Moser theorem.
We construct spherically symmetric solutions to the Einstein-Euler equations, which give models o... more 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.
Japan Journal of Industrial and Applied Mathematics, Feb 1, 1994
We study the spherically symmetric motion of an ideal gas surrounding a solid ball. This is gover... more We study the spherically symmetric motion of an ideal gas surrounding a solid ball. This is governed by the compressible Euler equation of isentropic gas dynamics. The associated initial boundary value problem is solved by using the compensated compactness method for initial data containing the vacuum. The constructed weak solutions are temporally local but the class of initial data includes the stationaxy solutions.
In meteorology the analysis of motions of the atmosphere on the Earth has been done using various... more 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 coordinate system which rotates with constant angular velocity. Mathematically rigorous inquiry is tried. Although problems remain to be open, some fundamental results are exhibited.
We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball u... more We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball under the gravity of the core. Equilibria touch the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this timeperiodic solution plus the equilibrium is the first approximation. But this leads us to difficulty caused by singularities at the free boundary touching the vacuum. We solve this problem by the Nash-Moser theorem.
Abstract The equation of the hydrodynamical evolution of an adiabatic gaseous star, which consist... more Abstract The equation of the hydrodynamical evolution of an adiabatic gaseous star, which consists of the compressible Euler equation coupled with Poisson's equation, is waiting for mathematical treatises. Although many interesting studies from the numerical point of view supported by the development of big computers can be found among astrophysicists, rigorous mathematics have not caught up with them. The author has heard nothing about even the proof of the existence of solutions for the associated Cauchy problem. Now this article is devoted to establishing the existence of local solutions under a suitable condition on the initial data. The discussion is along a standard line except for the crucial difficulty of integrating the Euler equation for solutions with compact support, which does not appear in the hydrodynamics on the Earth. However the result is too restrictive so that the author hopes that he has presented a tentative treatise initiating the further development of mathematical theory on this interesting equation.
I gt(P 2 c 2 + P 2)-F a a x"(P 2 c 2 ± P 2vk)=°k =1 3 2 0 c 2 ± P V a (PC 2 +p v v " + 0 , k) = 0... more I gt(P 2 c 2 + P 2)-F a a x"(P 2 c 2 ± P 2vk)=°k =1 3 2 0 c 2 ± P V a (PC 2 +p v v " + 0 , k) = 0 , i = 1 , 2 , 3. gt ' c 2-v 2 V ') a X k cv 2 3
In recent works we have constructed axisymmetric solutions to the Euler-Poisson equations which g... more 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.
It is known that spherically symmetric static solutions of the Einstein equations with a positive... more It is known that 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. Some sufficient conditions for the existence of monotone-short solutions (with finite radii) of the equation are given in this article. Then we show that the interior metric can extend to the exterior Schwarzschild-de Sitter metric on the exterior vacuum region with twice continuous differentiability. In addition, we investigate the analytic property of the solutions at the vacuum boundary. Our result (Theorem 1) can be considered as the de Sitter version of the result by Rendall and Schmidt [Classical Quantum Gravity 8, 985-1000 (1991)]. Furthermore, one can see that there are different properties of the solutions with those of the Tolman-Oppenheimer-Volkoff equation (with zero cosmological constant) in certain situation.
The linearized operator for non-radial oscillations of spherically symmetric self-gravitating gas... more 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.
Japan Journal of Industrial and Applied Mathematics, Jun 1, 1995
We study the spherically symmetric motion of viscous barotropic gas surrounding a solid ball. We ... more We study the spherically symmetric motion of viscous barotropic gas surrounding a solid ball. We are interested in the density distribution which contacts with the vacuum at a ¡ radius. This is a free boundary problem. We obtained the existence of a global weak solution with some regular properties. We can show that such a solution is unique.
Japan Journal of Industrial and Applied Mathematics, Jun 1, 1997
We study the spherically symmetric motion of viscous baxotropic gas surrounding a solid ball. We ... more We study the spherically symmetric motion of viscous baxotropic 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 "y > 4 anda is sufficiently small, when the equation of state is p = ap ~, p being the pressure and p the density.
We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball u... more We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball under the gravity of the core. Equilibria touch the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this timeperiodic solution plus the equilibrium is the first approximation. But this leads us to difficulty caused by singularities at the free boundary touching the vacuum. We solve this problem by the Nash-Moser theorem.
We construct spherically symmetric solutions to the Einstein-Euler equations, which give models o... more 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.
Japan Journal of Industrial and Applied Mathematics, Feb 1, 1994
We study the spherically symmetric motion of an ideal gas surrounding a solid ball. This is gover... more We study the spherically symmetric motion of an ideal gas surrounding a solid ball. This is governed by the compressible Euler equation of isentropic gas dynamics. The associated initial boundary value problem is solved by using the compensated compactness method for initial data containing the vacuum. The constructed weak solutions are temporally local but the class of initial data includes the stationaxy solutions.
In meteorology the analysis of motions of the atmosphere on the Earth has been done using various... more 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 coordinate system which rotates with constant angular velocity. Mathematically rigorous inquiry is tried. Although problems remain to be open, some fundamental results are exhibited.
We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball u... more We consider spherically symmetric motions of inviscid compressible gas surrounding a solid ball under the gravity of the core. Equilibria touch the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this timeperiodic solution plus the equilibrium is the first approximation. But this leads us to difficulty caused by singularities at the free boundary touching the vacuum. We solve this problem by the Nash-Moser theorem.
Abstract The equation of the hydrodynamical evolution of an adiabatic gaseous star, which consist... more Abstract The equation of the hydrodynamical evolution of an adiabatic gaseous star, which consists of the compressible Euler equation coupled with Poisson's equation, is waiting for mathematical treatises. Although many interesting studies from the numerical point of view supported by the development of big computers can be found among astrophysicists, rigorous mathematics have not caught up with them. The author has heard nothing about even the proof of the existence of solutions for the associated Cauchy problem. Now this article is devoted to establishing the existence of local solutions under a suitable condition on the initial data. The discussion is along a standard line except for the crucial difficulty of integrating the Euler equation for solutions with compact support, which does not appear in the hydrodynamics on the Earth. However the result is too restrictive so that the author hopes that he has presented a tentative treatise initiating the further development of mathematical theory on this interesting equation.
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