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

Abstract

Up io the present day many kinds of mathematical discussions on incompressible viscous fluid motion have fully developed (cf. [32,36] 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 s due to Eidel'man [9,18] [1, 10-12, 14, 15, 22, 29-31, 54].We estimate the Green matrix in § 4 and the solution in § 5. Making use of these results we give a proof of the existence in § 6 by the method of successive approximation and that of the uniqueness in § 7, therefore the proof of Theorem 2 is completed.Acknowledgement. Deep gratitude is due to Professor N. Itaya

On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities

“…In case that ρ 0 has a positive lower bound and u 0 has the additional integrability condition u 0 ∈ L 2 , Theorem 1.1 can be proved applying the method of successive approximations or a fixed point argument as in [1,13,18,24,29]. Our proof of the theorem is based on the method of successive approximations, whose general strategy may be described as follows.…”

“…Under the crucial assumption that the initial density ρ 0 is bounded below away from zero, the first existence results for the IBVP (1.1)-(1.5) were obtained by Nash [20], Itaya [13] and Tani [24]. They applied a fixed point argument or the method of successive approximations in Hölder spaces to prove the local (in time) existence of classical solutions even for more general heat-conducting fluid models.…”

Global Solutions to the Three-Dimensional Full Compressible Navier--Stokes Equations with Vacuum at Infinity in Some Classes of Large Data

“…In this case, there have been a lot of works so far on the well-posedness of solutions to the Cauchy problem and the initialboundary-value problem for (1.1). Refer, for instance, to these elegant works [17,18,19,25,26,28,29,32] for local and global existence of classical solutions from one dimension to high dimensions. In particular, Matsumura and Nishida in [28,29] showed that the global classical solution in three dimensions exists provided that the initial data is small in H 3 .…”

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