An existence theorem for compressible viscous fluids. (English) Zbl 0599.76081

Valli, Alberto

Ann. Mat. Pura Appl., IV. Ser. 130, 197-213 (1982).

Existence and uniqueness of compressible viscous fluid motions have been studied by several researchers, and no global result is known for the initial-boundary value problem in dimension greater than one. The author presents an existence theorem (local in time) for some initial-boundary value problems which are physically reasonable. The governing equations are written in general form, and the solution is found in Sobolev spaces of Hilbert type, using the method of successive approximation. The basic estimates are obtained by using some well-known theorems of J.-L. Lions and E. Magenes [Problèmes aux limites non homogènes et applications, Vol. 1, 2 (1968; Zbl 0165.108)] and it is shown that the result is strictly related to the general theory of parabolic equations.

MSC:

76N10	Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

Existence; uniqueness; compressible viscous fluid motions; initial- boundary value problem; existence theorem; Sobolev spaces of Hilbert type; method of successive approximation

Citations:

Zbl 0599.76082; Zbl 0165.108

References:

[1]	M.Böhm,Existence of solutions to equations describing the temperature-dependent motion of a non-homogeneous viscous flow, inStudies on Some Nonlinear Evolution Equations, Seminarbericht Nr. 17, 1979, Humboldt-Universität zu Berlin.
[2]	Bourguignon, J. P.; Brezis, H., Remarks on the Euler equation, J. Functional Analysis, 15, 341-363 (1974) · Zbl 0279.58005
[3]	Giusti, E., Equazioni ellittiche del secondo ordine (1978), Bologna: Pitagora Editrice, Bologna · Zbl 1308.35001
[4]	Graffi, D., Il teorema di unicità nella dinamica dei fluidi compressibili, J. Rat. Mech. Anal., 2, 99-106 (1953) · Zbl 0050.19604
[5]	Itaya, N., On the Cauchy problem for the system of fundamental equations describing the movement of compressible viscous fluids, Kodai Math. Sem. Rep., 23, 60-120 (1971) · Zbl 0219.76080
[6]	Itaya, N., On the initial value problem of the motion of compressible viscous fluid, especially on the problem of uniqueness, J. Math. Kyoto Univ., 16, 413-427 (1976) · Zbl 0335.35029
[7]	O. A.Ladyzenskaja - V. A.Solonnikov - N. N.Ural’ceva,Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, vol. 23, Providence, 1968 (translated from russian).
[8]	Lions, J. L.; Magenes, E., Problèmes aux limites non homogènes et applications, vol. 1 and 2 (1968), Paris: Dunod, Paris · Zbl 0165.10801
[9]	Matsumura, A.; Nishida, T., The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20, 67-104 (1980) · Zbl 0429.76040
[10]	A.Matsumura - T.Nishida,The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad., to appear. · Zbl 0447.76053
[11]	Morrey, Ch. B., Multiple Integrals in the Calculus of Variations (1966), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York
[12]	Nash, J., Le problème de Cauchy pour les équations différentielles d’un fluide général, Bull. Soc. Math. France, 90, 487-497 (1962) · Zbl 0113.19405
[13]	Nečas, J., Les méthodes directes en théorie des équations elliptiques (1967), Paris-Prague: Masson et Cie-Academia, Paris-Prague · Zbl 1225.35003
[14]	Serrin, J., On the Uniqueness of Compressible Fluid Motions, Arch. Rat. Mech. Anal., 3, 271-288 (1959) · Zbl 0089.19103
[15]	Serrin, J., Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik, Bd. VIII/1 (1959), Berlin-Göttingen-Heidelberg: Springer-Verlag, Berlin-Göttingen-Heidelberg
[16]	Solonnikov, V. A., Solvability of the initial-boundary value problem for the equations of motion of a viscous compressible fluid, J. Soviet Math., 14, 1120-1133 (1980) · Zbl 0451.35092
[17]	Tani, A., On the First Initial-Boundary Value Problem of Compressible Viscous Fluid Motion, Publ. RIMS, Kyoto Univ., 13, 193-253 (1977) · Zbl 0366.35070
[18]	Ton, B. A., On the initial boundary-value problem for viscous heat conducting compressible fluids, Kodai Math. J., 4, 97-128 (1981) · Zbl 0473.76058
[19]	Valli, A., Uniqueness theorems for compressible viscous fluids, especially when the Stokes relation holds, Boll. Un. Mat. It., Anal. Funz. Appl., 18-C, 317-325 (1981) · Zbl 0484.76075
[20]	Vol’Pert, A. I.; Hudjaev, S. I., On the Cauchy problem for composite systems of nonlinear differential equations, Math. USSR Sbornik, 16, 517-544 (1972) · Zbl 0251.35064