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Methods Of Modern Mathematical Physics

EDDINGTON'S “Internal Constitution of the Stars” was published in 1926 and gives what now ranks as a classical account of his own researches and of the general state of the theory at that time. Since then, a tremendous amount of work has appeared. Much of it has to do with the construction of stellar models with different equations of state applying in different zones. Other parts deal with the effects of varying chemical composition, with pulsation and tidal and rotational distortion of stars, and with the precise relations between the interior and the atmosphere of a star. The striking feature of all this work is that so much can be done without assuming any particular mechanism of stellar energy-generation. Only such very comprehensive assumptions are made about the distribution and behaviour of the energy sources that we may expect future knowledge of their mechanism to lead mainly to more detailed results within the framework of the existing general theory. An Introduction to the Study of Stellar Structure By S. Chandrasekhar. (Astrophysical Monographs sponsored by The Astrophysical Journal.) Pp. ix+509. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 50s. net.

An Introduction to the Study of Stellar Structure

Preface 1. Fourier series and Fourier transforms 2. Distributions 3. Elliptic equations (fundamental theory) 4. Initial value problems (Cauchy problems) 5. Evolution equations 6. Hyperbolic equations 7. Semi-linear hyperbolic equations 8. Green's functions and spectra Supplementary remarks Guide to the literature Bibliography Symbols Index.

The theory of partial differential equations

Hydrodynamic and hydromagnetic stability

The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously.

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Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations

We consider a free boundary problem for the equation of the one-dimensional isentropic motion with density-dependent viscosity μ =b
 ϱ
 β, 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 One-Dimensional Motion of Compressible Gas with Density-Dependent Viscosity.

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 finite 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.

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

This work consider a more realistic equation of states. We generalize the method of DiPerna and G.-Q. Chen et al to show the existence of weak solution of the relativistic Euler equation with initial data containing the vacuum state

https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/2001/0008/0001/MAA-2001-0008-0001-a007.pdf

On the Relativistic Euler Equation

https://ir.nctu.edu.tw:443/bitstream/11536/26661/1/000221622700001.pdf

On spherically symmetric solutions of the relativistic Euler equation

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

Tetu Makino

Papers

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

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

On the Relativistic Euler Equation

On spherically symmetric solutions of the relativistic Euler equation

On Slowly Rotating Axisymmetric Solutions of the Euler–Poisson Equations