Abstract
A model of the one-dimensional motion of a viscous compressible fluid is considered. A class of nonlinear stressed-state equations for which the initial boundary-value problem has global (in time) solutions in the class of generalized solutions satisfying the energy identity is described. In particular, media exhibiting viscous properties only for large strain rates are studied.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Truesdell,A First Course in Rational Continuum Mechanics, Vol. 1, General Concepts (2nd edition), Academic Press, San Diego-London (1991).
S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov,Boundary-Value Problems in the Mechanics of Inhomogeneous Fluids [in Russian], Nauka, Novosibirsk (1983).
V. A. Vaigant, “Inhomogeneous problems for the equations of a viscous heat-conducting gas,” in:Mathematical Problems of Continuum Mechanics. The Dynamics of Continuum Media [in Russian], No. 97, Institute of Hydrodynamics of the Siberian Branch of the Academy of Sciences of the USSR, Novosibirsk (1990), pp. 3–21.
A. A. Amosov and A. A. Zlotnik, “Global solvability of a class of quasilinear systems of equations of composite type with nonsmooth data,”Differentsial’nye Uravneniya [Differential Equations],30 (4), 596–609 (1994).
V. A. Vaigant and A. V. Kazhikhov, “On the existence of global solutions to the two-dimensional Navier-Stokes equations of a compressible viscous fluid,”Sibirsk. Mat. Zh. [Siberian Math. J.],36 (6), 1283–1316 (1995).
J. Málek, J. Necas, M. Rokyta, and M. Růžička, “Weak and measure-valued solutions to evolutionary PDEs,” in:Applied Mathematics and Mathematical Computation, Vol. 13, Chapman & Hall, London (1996).
A. E. Mamontov,Orlicz Spaces in the Existence Problem of Global Solutions to Viscous Compressible Nonlinear Fluid Equations, Preprint No. 242 2-96, Institute of Hydrodynamics of the Siberian Branch of the Academy of Sciences of the USSR, Novosibirsk (1996).
W. R. Showalter,Mechanics of non-Newtonian Fluids, Pergamon Press, Oxford (1978).
V. V. Shelukhin, “A shear flow problem for the compressible Navier-Stokes equations,”Internat. J. Non-Linear Mech.,33 (2), 247–257 (1998).
J. Simon, “Compact sets in the spaceL p (0,T; B),”Ann. Matematica Pura ed Applicata (IV),VCXLVI, 65–69 (1987).
L. Tartar, “Compensated compactness and application to P.D.E.,” in:Nonlinear Analysand Mechanics, 4 (Heriot-Watt Symposium) (R. J. Knops, editor), Vol. 39, Pitman Reseach Notes in Math. Series (1979), pp. 136–212.
Author information
Authors and Affiliations
Additional information
Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 13–23, July, 2000.
Rights and permissions
About this article
Cite this article
Basov, I.V. On the solvability of equations of a nonlinear viscous-ideal fluid. Math Notes 68, 12–21 (2000). https://doi.org/10.1007/BF02674641
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02674641