Global weak solvability of a regularized system of the Navier-Stokes equations for compressible fluid. (English) Zbl 0668.76074
The paper contains the proof of global existence of weak solutions to the mixed initial-boundary value problem for a certain modification of a system of equations of motion of viscous compressible fluid. The modification is based on an application of an operator of regularization to some terms appearing in the system of equations and it does not contradict the laws of fluid mechanics. It is assumed that pressure is a known function of density. The method of discretization in time is used and finally, a so called energy inequality is derived. The inequality is independent on the regularization used.
MSC:
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
| 35Q30 | Navier-Stokes equations |
Keywords:
Navier-Stoke equations; method of a discretization in time; global existence of weak solutions; mixed initial-boundary value problem; viscous compressible fluidReferences:
| [1] | O. A. Ladyzhenskaya N. N. Uralceva: Linear and Quasilinear Equations of the Elliptic Type. Nauka, Moscow, 1973 |
| [2] | L. G. Loicianskij: Mechanics of Liquids and Gases. Nauka, Moscow, 1973 |
| [3] | A. Matsumura T. Nishida: The Initical Value Problem for the Equations of Motion of Viscous and Heat-Conductive Gases. J. Math. Kyoto Univ. 20 (1980), 67-104, · Zbl 0429.76040 |
| [4] | J. Neustupa: A Note to the Global Weak Solvability of the Navier-Stokes Equations for Compressible Fluid. to appear prob. in Apl. mat. · Zbl 0668.76074 |
| [5] | R. Rautmann: The Uniqueness and Regularity of the Solutions of Navier-Stokes Problems. Functional Theoretic Methods for Partial Differential Equations, Proc. conf. Darmstadt 1976, Lecture Notes in Mathematics, Vol. 561, Berlin-Heidelberg-New York, Springer-Verlag, 1976, 378-393. · Zbl 0383.35059 |
| [6] | V. A. Sollonikov: The Solvability of an Initial-Boundary Value Problem for the Equations of Motion of Viscous Compressible Fluid. J. Soviet Math. 14 (1980), 1120-1133 (previously in Zap. Nauchn. Sem. LOMI 56 (1976), 128-142 (Russian)). · Zbl 0451.35092 · doi:10.1007/BF01562053 |
| [7] | R. Temam: Navier-Stokes Equations. North-Holland Publishing Company, Amsterdam- New York-Oxford, 1977. · Zbl 0383.35057 |
| [8] | A. Valli: An Existence Theorem for Compressible Viscous Fluids. Ann. Mat. Pura Appl. 130 (1982), 197-213. · Zbl 0599.76082 · doi:10.1007/BF01760990 |
| [9] | A. Valli: Periodic and Stationary Solutions for Compressible Navier-Stokes Equations via a Stability Method. Ann. Scuola Norm. Sup. Pisa, (IV) 10 (1983), 607-647. · Zbl 0542.35062 |
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