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Global regularity estimates for multidimensional equations of compressible non-newtonian fluids

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Il presente lavoro è dedicato al problema dell’esistenza globale di soluzioni sufficientemente regolari delle equazioni del moto di fluidi comprimibili non-Newtoniani in due e tre dimensioni. Nel caso di tensore degli sforzi potenziale, viene sviluppata una tecnica per la derivazione di identità dell’energia che non includono derivate della densità. Basandosi su tali identità, nel caso di potenziali che crescono abbastanza rapidamente viene ottenuto un sistema esteso di stime a priori per le equazioni del moto. Viene inoltre studiato il problema correlato delle stime di soluzioni di sistemi ellittici non lineari generati dal tensore degli sforzi.

Abstract

The present paper is devoted to the problem of global existence of sufficiently regular solutions to two- and three-dimensional equations of compressible nonnewtonian fluids. In the case of potential stress tensor, we develop the technique of derivation of energy identities that do not include the density derivatives. Basing on these identities, in the case of sufficiently fast-increasing potentials, we obtain an extended system of a priori estimates for the named equations. We also study the related problem of estimates for the solutions to non-linear elliptic system generated by stress tensor.

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References

  1. J. Serrin,Mathematical Principles of Classical Fluid Mechanics. Handbuch der Physik, Band VIII/I, Strömungsmechanik I, Berlin-Göttingen-Heidelberg, 1959.

    Google Scholar 

  2. A. V. Kazhikhov—V. A. Vaigant,On the Existence of Global Solutions to Two-Dimensional Navier-Stokes Equations of Compressible Viscous Fluid, Siberian Mathematical Journal,36, N. 6 (1995), pp. 1283–1316.

    MathSciNet  Google Scholar 

  3. P.-L. Lions,Mathematical Topics in Fluid Mechanics, Volume 2, Compressible models, Oxford, Clarendon Press, 1998.

    Google Scholar 

  4. J. Malek—J. Necas—M. Rokyta—M. Ruzicka,Weak and Measure-Valued Solutions to Evolutionary PDEs, London, Weinheim etc., Chapman & Hall, 1996.

    MATH  Google Scholar 

  5. A. E. Mamontov,Existence of Global Solutions to Multidimensional Burgers Equations of Compressible Viscous Fluid, Sbornik Mathematics,190, N. 8 (1999), pp. 1131–1150.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. E. Mamontov,Global Solvability of the Multidimensional Navier-Stokes Equations of a Compressible Fluid with Nonlinear Viscosity, I. Siberian Mathematical Journal, V.40, N. 2 (1999), pp. 351–362.

    MATH  MathSciNet  Google Scholar 

  7. A. E. Mamontov,Global Solvability of the Multidimensional Navier-Stokes Equations of a Compressible Fluid with Nonlinear Viscosity. II, Siberian Mathematical Journal,40, N. 3 (1999), pp. 541–555.

    Article  MathSciNet  Google Scholar 

  8. P. Kaplický—J. Malek—J. Stará,C 1, a-Solutions to a Class of Nonlinear Fluids in Two Dimensions—Stationary Dirichlet Problem. Sonderforschungsbereich 256 nichtlineare partielle differentialgleichungen, no. 529, Reinische Friedrich-Wilhelms-Universität, Bonn, 1997.

    Google Scholar 

  9. A. V. Kazhikhov—A. E. Mamontov,On a Certain Class of Convex Functions and the Exact Well-Posedness Classes of the Cauchy Problem for the Transport Equations in Orlicz Spaces, Siberian Mathematical Journal,39, N. 4 (1998), pp. 716–734.

    Article  MathSciNet  Google Scholar 

  10. J. Bergh—J. Löfström,Interpolation Spaces. The Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1978.

    Google Scholar 

  11. L. Tartar,Interpolation Non Lineaire et Regularité//J. of func. anal., N. 9 (1972), pp. 469–489.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. V. Kazhikhov—A. E. Mamontov,Transport Equations and Orlicz Spaces,in the book: «Hyperbolic Problems: Theory, Numerics, Applications, Seventh International Conference in Zürich, February 1998. Volume II» (International Series of Numerical Mathematics, Vol. 130), 1999, Birkhäuser, Basel-Boston-Berlin, pp. 535–544. Editors: Michael Fey and Rolf Jeltsch.

    Google Scholar 

  13. A. E. Mamontov,Global Regularity Estimates for Multidimensional Equations of Compressible Non-Newtonian Fluid, Matematicheskie Zametki, 2000. To appear.

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Author notes

  1. Alexander E. Mamontov

    Present address: Layrentyev Institute for Hydrodynamics, Lavrentyev Prospekt 15, 630090, Novosibirsk, Russia

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    1. Alexander E. Mamontov
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    This work is supported by Russian Foundation for Basic Research (Grant 99-01-10817), Cariplo Foundation and Volta Center of Landau Society.

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    Mamontov, A.E. Global regularity estimates for multidimensional equations of compressible non-newtonian fluids. Ann. Univ. Ferrara 46, 139–160 (2000). https://doi.org/10.1007/BF02837294

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