On the First Initial-Boundary Value Problem of Compressible Viscous Fluid Motion

Up io the present day many kinds of mathematical discussions on incompressible viscous fluid motion have fully developed (cf. [32, 36]). As for compressible viscous one, however, there have been only a few works on it. In 1959 Serrin [50] proved the uniqueness theorem in a bounded domain, making use of the classical energy method. In 1962 Nash [44] tried to show the existence theorem in R, but it seems to the author that he has failed. Independently of them Itaya succeeded to prove the existence and the uniqueness theorems on the Cauchy problem for it in [24-28], using Tikhonov's fixed point theorem. Now in the present paper, we shall show that the first initial-boundary value problem for it can uniquely be solved under suitable assumptions for the initial-boundary data and for the boundary of the domain, from the classical point of view. In § 1 an exact statement and the main theorem (Theorem 1) will be found. In §2 we perform the characteristic transformation and mention the theorem of the transformed problem (Theorem 2). Firstly we prove Theorem 2 and then show that Theorem 2 implies Theorem 1 in the last section § 8. In §§ 3-5 linear equations connected with the transformed equations are treated. In more detail, in § 3. 1 we briefly state some basic results for a fundamental solution in the whole space R due to Eidel'man [9, 18] and Pogorzelski [46-48] (cf. [25]). In §3.2 we check the basic condition of uniform solvability due to Solomjak [52, 54], which is essential for the study of the boundary value problem in applied mathematics, corresponding to the Lopatinsky condition for the