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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