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Estimate for the second-order derivatives of solutions to boundary-value problems of the theory of non-Newtonian liquids

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Abstract

The boundary regularity of solutions to some boundary-value problems describing stationary flow of generalized Newtonian liquids is studied. The dissipative potential is of quadratic growth at infinity. We prove that the second-order derivatives of the solution are pth power summable functions, where p is greater than two. The partial regularity of the strain velocity tensor is established. In the two-dimensional case, the complete regularity of the strain velocity tensor is also proved. Bibliography: 14 titles.

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References

  1. A. Astarita and G. Marrucci,Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, London (1974).

    Google Scholar 

  2. M. Fuchs and G. Seregin, “Some remarks on non-Newtonian fluids including nonconvex perturbations of the Bingham and Powell-Eyring model for viscoplastic fluids,”Math. Models Methods Appl. Sci.,7, No. 3, 405–433 (1997).

    MathSciNet  Google Scholar 

  3. M. Fuchs and G. Seregin, “Regularity results for the quasi-static Bingham variational inequality in dimensions two and three,”Math. Z.,227, No. 3., 525–541 (1998).

    MathSciNet  Google Scholar 

  4. G. Seregin,Continuity for the Strain Velocity Tensor in Two-Dimensional Variational Problems from the Theory of the Bingham Fluid, Preprint No. 402, SFB256, Bonn (1995).

  5. T. N. Shilkin, “Regularity up to the boundary for solutions to boundary-value problems of the theory of generalized Newtonian liquids,”Probl. Mat. Anal.,16, 239–265 (1997).

    MATH  Google Scholar 

  6. J. Malek, J. Necas, and M. Ruzicka,On Weak Solutions to a Class of Non-Newtonian Incompressible Fluids in Bounded Three-Dimensional Domains. Case p≥2, Preprint No. 481, SFB256, Bonn (1996).

  7. P. P. Mosolov and V. P. Myasnikov, “Correctness of boundary-value problems in mechanics of continuous media”,Mat. Sb.,88, No. 2, 256–267 (1972).

    MathSciNet  Google Scholar 

  8. O. A. Ladyzhenskaya and N. N. Ural'tseva,Linear and Quasilinear Elliptic Equations [in Russian], Nauka, Moscow (1973). [English Translation of the 1st ed.:Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968)].

    Google Scholar 

  9. O. A. Ladyzhenskaya,The Mathematical Questions of Dynamics of Viscous Incompressible Flow [in Russian], Nauka, Moscow (1970). [English Translation of the 1st ed.:The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York (1969)].

    Google Scholar 

  10. F. W. Gehring, “TheL p integrability of the partial derivatives of a quasiconformal mapping,”Acta Math.,130, 265–277 (1973).

    MATH  MathSciNet  Google Scholar 

  11. M. Giaquinta,Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton (1983).

    Google Scholar 

  12. B. Bojarski and T. Iwaniec, “Analytical foundations of the theory of quasiconformal mappings in ℝn”,Ann. Acad. Sci. Fenn. Ser. A I Math.,8, 257–324 (1983).

    MathSciNet  Google Scholar 

  13. O. A. Ladyzhenskaya and V. A. Solonnikov, “Some problems of vector analysis, and general formulations of boundary-value problems for the Navier-Stokes equations,”Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova,59, 81–116 (1976).

    Google Scholar 

  14. V. A. Solonnikov and V. E. Shchadilov, “On a boundary-value problem for a stationary system of Navier-Stokes equations,”Proc. Steklov Inst. Math.,125, 186–199 (1973).

    Google Scholar 

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Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 222–246.

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Shilkin, T.N. Estimate for the second-order derivatives of solutions to boundary-value problems of the theory of non-Newtonian liquids. J Math Sci 98, 781–797 (2000). https://doi.org/10.1007/BF02355390

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