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Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 899–927 | Cite as

Global Existence of Solutions to 2-D Navier–Stokes Flow with Non-decaying Initial Data in Exterior Domains

  • Paolo Maremonti
  • Senjo Shimizu
Article

Abstract

We study the two dimensional Navier–Stokes initial boundary value problem in exterior domains assuming that the initial data \(u_0\) belongs \(L^\infty \). The global (in time) unique existence of this problem is furnished. The behavior of \(||u(t)||_\infty \) for large t is of double exponential kind.

Keywords

Non-decaying initial velocities 2-D Navier–Stokes flows exterior domains global existence 

Mathematics Subject Classification

35Q30 76D05 35K55 

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References

  1. 1.
    Abe K.: Exterior Navier–Stokes flows for bounded data. Math. Nachr. 290(7), 972–985 (2017)Google Scholar
  2. 2.
    Abe, K.: Global well-psedeness of the two-dimensional exterior Navier–Stokes equations for non-decaying data. arXiv:1608.06424
  3. 3.
    Bogovskiǐ, M.E.: Solution of the first boundary value problem for the equation of continuity of an incompressible medium. Dokl. Acad. Nauk. SSSR 248, 1037–1049 (1979) (English transl., Soviet Math. Dokl., 20, 1094–1098 (1979))Google Scholar
  4. 4.
    Bogovskiǐ, M.E.: Solution of some vector analysis problems connected with operators div and grad. In: Sobolev, S.L. (eds.) Trudy Seminar, vol. 80. Akademia Nauk SSSR. Sibirskoe Otdelnie Matematik, Nowosibirsk in Russian, pp. 5–40 (1980)Google Scholar
  5. 5.
    Cattabriga, L.: Su un problema al contorno relativo al sistema di equazioni di Stokes, (Italian). Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Dan, W., Shibata, Y.: Remark on the \(L_q-L_\infty \) estimate of the Stokes semigroup in a 2-dimensional exterior domain. Pac. J. Math. 189(n02), 223–239 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Crispo, F., Maremonti, P.: An interpolation inequality in exterior domains. Rend. Sem. Mat. Univ. Padova 112, 11–39 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Gallay, T.: Infinite energy solutions of the two-dimensional Navier–Stokes equations. arXiv:1411.5156
  9. 9.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-state Problems, 2nd edn. Springer Monographs in Mathematics. Springer, New-York (2011)zbMATHGoogle Scholar
  10. 10.
    Galdi, G.P., Maremonti, P., Zhou, Y.: On the Navier-Stokes problem in exterior domains with non decaying initial data. J. Math. Fluid Mech. 14, 633–652 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Giga, Y., Inui, K., Matsui, S.: On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data. Advances in Fluid Dynamics, Quad. Mathematics, vol. 4, pp. 27–68. Seconda University Napoli, Caserta, Department of Mathematics (1999)Google Scholar
  12. 12.
    Giga, Y., Matsui, S., Sawada, O.: Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity. J. Math. Fluid Mech. 3, 302–315 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Giga, Y., Sohr, H.: Abstract \(L^p\) estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102(1), 72–94 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Heywood, J.G.: The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29(5), 639–681 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow, Mathematics and Its Applications, vol. 2. Golden and Breach Science Publishers, New York (1963)Google Scholar
  16. 16.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations (Transl. from the Russian by Scripta Technica, Inc. Trans ed.: L. Ehrenpreis). Academic Press, New York (1968)Google Scholar
  17. 17.
    Maremonti, P.: On the Stokes equations: the maximum modulus theorem. Math. Models Methods Appl. Sci. 10(7), 1047–1072 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Maremonti, P.: Stokes and Navier-Stokes problems in the half-space: existence and uniqueness of solutions non converging to a limit at infinity. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 362 (2008) (Kraevye Zacachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 39, 176–240, 366; reprinted in J. Math. Sci. (N.Y.) 159 (2009), no. 4, 486–523)Google Scholar
  19. 19.
    Maremonti, P.: On the uniqueness of bounded very weak solutions to the Navier-Stokes Cauchy problem. Appl. Anal. 90(1), 125–139 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Maremonti, P., Russo, R.: On the maximum modulus theorem for the Stokes system. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21(4), 629–643 (1994)Google Scholar
  21. 21.
    Maremonti, P., Solonnikov, V.A.: An estimate for the solutions of a Stokes system in exterior domains, (Russian). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 180 (1990) (Voprosy Kvant. Teor. Polya i Statist. Fiz. 9, 105–120, 181; transl. J. Math. Sci. 68(2), 229–239 (1994))Google Scholar
  22. 22.
    Maremonti, P., Solonnikov, V.A.: On nonstationary Stokes problem in exterior domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24(3), 395–449 (1997)Google Scholar
  23. 23.
    Sawada, O., Taniuchi, Y.: A remark on \(L^\infty \) solutions to 2-D Navier-Stokes equations. J. Math. Fluid Mech. 3, 533–542 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Solonnikov V.A.: Estimates of the solutions of the nonstationary Navier–Stokes system. (Russian) Boundary value problems of mathematical physics and related questions in the theory of functions. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38, 153–231 (1973) ((e.t.) J. Soviet Math., 8 (1977), 467–529)Google Scholar
  25. 25.
    Zelik, S.: Infinite energy solutions for damped Navier-Stokes equations in \(\mathbb{R}^2\). J. Math. Fluid Mech. 15, 717–745 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità degli Studi della CampaniaCasertaItaly
  2. 2.Graduate School of Human and Environmental StudiesKyoto UniversitySakyo-kuJapan

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