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On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscous fluid

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References

  1. L. I. Sedov, Continuum Mechanics. Vol. 1 [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  2. L. G. Loitsyanskii, Mechanics of Liquids and Gases [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  3. Ya. I. Kanel', “On a model system of equations of one-dimensional gas motion,” Differentsial'nye Uravneniya, No. 4, 721–734 (1968).

    Google Scholar 

  4. N. Itaya, “On the temporally global problem of the generalized Burgers equation,” J. Math. Kyoto Univ.,14, No. 1, 129–177 (1974).

    Google Scholar 

  5. A. V. Kazhikhov, “On global solvability of one-dimensional boundary value problems for the equations of a viscous heat-conductive gas,” Dinamika Sploshn. Sredy (Novosibirsk),24, 45–61 (1976).

    Google Scholar 

  6. A. V. Kazhikhov and V. V. Shelukhin, “Global unique solvability (in time) of initial-boundary value problems for one-dimensional equations of a viscous gas,” Prikl. Mat. i Mekh.,41, No. 2, 282–291 (1977).

    Google Scholar 

  7. S. Ya. Belov, “Global solvability of the flowing problem for the Burgers equations of a compressible fluid,” in: Boundary Value Problems for the Equations of Hydrodynamics [in Russian], Inst. Gidrodinamiki (Novosibirsk), Novosibirsk, 1981, pp. 3–14. (Dinamika Sploshn. Sredy,50.)

    Google Scholar 

  8. V. V. Shelukhin, “Periodic flows of a viscous gas,” in: Dynamics of an Inhomogeneous Fluid [in Russian], Inst. Gidrodinamiki (Novosibirsk), Novosibirsk, 1979, pp. 80–102. (Dinamika Sploshn. Sredy,42.)

    Google Scholar 

  9. V. A. Vaigant, “Inhomogeneous boundary value problems for the equations of a viscous heat-conductive gas,” in: Mathematical Problems in Continuum Mechanics [in Russian], Inst. Gidrodinamiki (Novosibirsk), Novosibirsk, 1990, pp. 3–21. (Dinamika Sploshn. Sredy,97.)

    Google Scholar 

  10. V. A. Vaigant, “Stabilization of solutions to an inhomogeneous boundary value problem for the equations of a viscous heat-conductive gas,” in: Free Boundary Problems in Continuum Mechanics [in Russian], Inst. Gidrodinamiki (Novosibirsk), Novosibirsk, 1991, pp. 31–52. (Dinamika Sploshn. Sredy,101.)

    Google Scholar 

  11. V. A. Solonnikov, “On solvability of an initial-boundary value problem for the equations of motion of a viscous compressible fluid,” in: Studies on Linear Operators and Function Theory. Vol. 6 [in Russain], Nauka, Leningrad, 1976, pp. 128–142. (Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI),56.)

    Google Scholar 

  12. A. Tani, “On the first initial-boundary value problem of compressible viscous fluid motion,” Publ. Res. Inst. Math. Sci.,13, No. 1, 193–253 (1977).

    Google Scholar 

  13. A. Matsumura and T. Nishida, “The initial value problem for the equations of motions of viscous and heat-conductive gases,” J. Math. Kyoto Univ.,20, No. 1, 67–104 (1980).

    Google Scholar 

  14. M. Padula, “Existence of global solutions for two-dimensional viscous compressible flows,” J. Funct. Anal.,69, No. 1, 1–20 (1986).

    Google Scholar 

  15. P. L. Lions, “Compacité des solutions des equations de Navier-Stokes compressible isentropiques,” Comp. Res. Acad. Sci.,317, No. 2, 115–120 (1993).

    Google Scholar 

  16. A. V. Kazhikhov, “The equations of potential flows of compressible viscous fluid at low Reynolds numbers,” Acta. Math. Appl.,37, No. 1, 77–81 (1994).

    Google Scholar 

  17. V. A. Vaigant and A. V. Kazhikhov, “Global solutions to the equations of potential flows of a viscous compressible fluid at low Reynolds numbers,” Differentsial'nye Uravneniya,30, No. 6, 1010–1022 (1994).

    Google Scholar 

  18. V. A. Vaigant, “On the question of global solvability of a boundary value problem for the Navier-Stokes equations of a viscous compressible barotropic fluid,” in: Current Questions of Contemporary Mathematics [in Russian], Novosibirsk. Univ., Novosibirsk, 1995, 1, pp. 43–51.

    Google Scholar 

  19. V. A. Vaigant, “An example of global nonexistence (in time) of solutions to the Navier-Stokes equations of a viscous compressible barotropic fluid,” in: Fluid Dynamics with Free Boundaries [in Russian], Inst. Gidrodinamiki (Novosibirsk), Novosibirsk, 1993, pp. 39–48. (Dinamika Sploshn. Sredy,107.)

    Google Scholar 

  20. E. Cagliardo, “Ulterori proprieta' di alcune classi di funzioni in piu' variabili,” Ricerche Mat.,8, 24–51 (1959).

    Google Scholar 

  21. V. P. Il'in, “Some inequalities in function spaces and their application to studying convergence of variational methods,” Trudy Mat. Inst. Akad. Nauk SSSR,53, 64–127 (1959).

    Google Scholar 

  22. K. K. Golovkin, “On embedding theorems,” Dokl. Akad. Nauk SSSR,134, 19–22 (1960).

    Google Scholar 

  23. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  24. V. I. Yudovich, “On some estimates connected with integral operators and solutions of elliptic equations,” Dokl. Akad. Nauk SSSR,138, No. 4, 805–808 (1961).

    Google Scholar 

  25. S. I. Pokhozhaev, “On S. L. Sobolev's embedding theorem in the casepl=n,” in: Dokl. Nauchno-Tekhn. Konf. MèI, 1965, pp. 158–170.

  26. G. Talenti, “Best constant in Sobolev inequality,” Ann. Mat. Pura Appl. (4),110, 353–372 (1976).

    Google Scholar 

  27. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  28. E. M. Stein, Singular Integrals and Differentiability Properties of Functions [Russian translation], Mir, Moscow (1973).

    Google Scholar 

  29. I. N. Vekua, Generalized Analytic Functions [in Russian], Fizmatgiz, Moscow (1959).

    Google Scholar 

  30. V. I. Yudovich, “A two-dimensional nonstationary problem of the flowing of an ideal incompressible fluid through a fixed domain,” Mat. Sb.,64, No. 4, 562–588 (1964).

    Google Scholar 

  31. V. A. Solonnikov, “On boundary value problems for linear parabolic systems of general differential equations,” Trudy Mat. Inst. Akad. Nauk SSSR,83, 3–162 (1965).

    Google Scholar 

  32. O. A. Ladyzhenskaya and V. A. Solonnikov, “On unique solvability of an initial-boundary value problem for viscous inhomogeneous fluids,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI),52, 52–109 (1975).

    Google Scholar 

  33. D. Hoff, Strong Convergence to Global Solutions for Compressible, Viscous, Multidimensional Flow, with Polytropic Equations of State and Discontinuous Initial Data [Preprint, No. 9411], Indiana University (1994).

  34. V. I. Yudovich, “Nonstationary flows of an ideal incompressible fluid,” Zh. Vychisl. Matematiki i Mat. Fiziki,13, No. 6, 1032–1036 (1963).

    Google Scholar 

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Authors and Affiliations

  1. Barnaul

    V. A. Vaigant & A. V. Kazhikhov

  2. Novosibirsk

    V. A. Vaigant & A. V. Kazhikhov

Authors
  1. V. A. Vaigant
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  2. A. V. Kazhikhov
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Additional information

The research was financially supported by the Center for Mathematical Research at Novosibirsk State University.

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 36, No. 6, pp. 1283–1316, November–December, 1995.

The authors express their gratitude to Prof. V. V. Shelukhin for useful discussions, and to Prof. A. Novotný and Prof. D. Hoff who kindly make their results available to the authors prior to publication.

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Vaigant, V.A., Kazhikhov, A.V. On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscous fluid. Sib Math J 36, 1108–1141 (1995). https://doi.org/10.1007/BF02106835

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