Abstract
The present paper is devoted to the problem of global existence of sufficiently regular solutions to two- and three-dimensional equations of a compressible non-Newtonian fluid. In the case of the potential stress tensor, we develop a technique for deriving energy identities that do not contain derivatives of density. On the basis of these identities, in the case of sufficiently rapidly increasing potentials, we obtain an extended system ofa priori estimates for the equations mentioned above. We also study the related problem of estimating solutions to the nonlinear elliptic system generated by the stress tensor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Serrin,Mathematical Principles of Classical Fluid Mechanics, Handbuch für Physik, Bd. VIII, Berlin (1959).
V. A. Vaigant and A. V. Kazhikhov, “On the existence of global solutions of two-dimensional Navier-Stokes, equations of compressible viscous fluid,”Sibirsk. Mat. Zh. [Siberian Math. J.],36, No. 6, 1283–1316 (1995).
P.-L. Lions,Mathematical Topics in Fluid Mechanics, Vol. 2,Compressible Models, Clarendon Press, Oxford (1998).
J. Malek, J. Necas, M. Rokyta, and M. Ruzicka,Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman and Hall, London, Weinheim et al. (1996).
A. E. Mamontov, “Existence of global solutions to multidimensional Bürgers equations of compressible viscous fluid,”Dokl. Ross. Akad. Nauk [Russian Acad. Sci. Dokl. Math.],361, No. 2, 161–163 (1998).
A. E. Mamontov, “Global solvability of multidimensional Navier-Stoker equations of compressible nonlinearly viscous fluid. I.,”Sibirsk. Mat. Zh. [Siberian Math. J.],40, No. 2, 408–420 (1999).
A. E. Mamontov, “Global solvability of multidimensional Navier-Stoker equations of compressible nonlinearly viscous fluid. II,”Sibirsk. Mat. Zh. [Siberian Math. J.],40, No. 3, 635–649 (1999).
P. Kaplický, J. Malek, and J. Stará, “C 1, α-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem,” in:Sonderforschungsbereich “Nichtlineare partielle Differentialgleichungen,” No. 529, Rheinische Friedrich-Wilhelms-Universität, Bonn (1997).
A. V. Kazhikhov and A. E. Mamontov, “On a class of convex functions and exact classes of well-posedness of the Cauchy problem for the transport equation in Orlicz spaces,”Sibirsk. Mat. Zh. [Siberian Math. J.],39, No. 4, 831–850 (1998).
J. Bergh and J. Löfstrom,Interpolation Spaces. Introduction, Springer, Heidelberg (1976).
L. Tartar, “Interpolation nonlineaire et regularité,”J. Funct. Anal., No. 9, 469–489 (1972).
Author information
Authors and Affiliations
Additional information
Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 360–376, September, 2000.
Rights and permissions
About this article
Cite this article
Mamontov, A.E. Global regularity estimates for multidimensional equations of compressible non-Newtonian fluids. Math Notes 68, 312–325 (2000). https://doi.org/10.1007/BF02674554
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02674554