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

On the Cauchy Problem for the Zakharov System

We study Trudinger type inequalities in RN and their best exponents αN . We show for α ∈ (0, αN ), αN = Nω N−1 (ωN−1 is the surface area of the unit sphere in RN ), there exists a constant Cα > 0 such that ∫ RN ΦN α( |u(x)| ‖∇u‖LN (RN ) ) N N−1  dx ≤ Cα ‖u‖NLN (RN ) ‖∇u‖N LN (RN ) (∗) for all u ∈W 1,N (RN ) \ {0}. Here ΦN (ξ) is defined by ΦN (ξ) = exp(ξ) − N−2 ∑ j=0 1 j! ξ . It is also shown that (∗) with α ≥ αN is false, which is different from the usual Trudinger’s inequalities in bounded domains. 0. Introduction In this note, we study the limit case of Sobolev’s inequalities; suppose N ≥ 2 and let D ⊂ R be an open set. We denote by W 1,N 0 (D) the usual Sobolev space, that is, the completion of C∞ 0 (D) with the norm ‖u‖W 1,p 0 (D) = ‖∇u‖p+‖u‖p. Here ‖u‖p = (∫

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Trudinger type inequalities in ^{} and their best exponents

Two regularity criteria for the 3D MHD equations

Chapter 7 – On the Motion of a Rigid Body in a Viscous Liquid: A Mathematical Analysis with Applications

We show the critical Sobolev inequalities in the Besov spaces with the logarithmic form such as Brezis-Gallouet-Wainger and Beale-Kato-Majda. As an application of those inequalities, the regularity problem under the critical condition to the Navier-Stokes equations, the Euler equations in 
${mathbb R}^n$
 and the gradient flow to the harmonic map to the sphere are discussed. Namely the Serrin-Ohyama type regularity criteria are improved in the terms of the Besov spaces.

The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations

Large time behavior and Lp−Lq estimate of solutions of 2-dimensional nonlinear damped wave equations

Interaction Equations for Short and Long Dispersive Waves

We prove a local existence theorem for the Navier-Stokes equations with the initial data in B0∞,∞ containing functions which do not decay at infinity. Then we establish an extension criterion on our local solutions in terms of the vorticity in the homogeneous Besov space B·0∞,∞.

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Navier-stokes equations in the besov space near l∞ and bmo

We discuss the existence of the blow-up solution for some nonlinear parabolic system called attractive drift-diffusion equation in two space dimensions. We show that if the initial data satisfies a threshold condition, the corresponding solution blows up in a finite time. This is a system case for the blow-up result of the chemotactic equation proved by Nagai [28] and Nagai-Senba-Suzuki [30] and gravitational interaction of particles by Biler-Nadzieja [7], [8].

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Takayoshi Ogawa

Papers

The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations

Large time behavior and Lp−Lq estimate of solutions of 2-dimensional nonlinear damped wave equations

Interaction Equations for Short and Long Dispersive Waves

Navier-stokes equations in the besov space near l∞ and bmo

Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type