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Example of zero viscosity limit for two dimensional nonstationary Navier-Stokes flows with boundary

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Abstract

We construct a Navier-Stokes flow in the unit disk, whose initial data have radially symmetric vorticity. Our goal is to show that this flow is convergent to some Euler flow as the viscosity tends to zero inL 2 norm. For the purpose we give necessary and sufficient conditions for this convergence inC([0,T];L 2 (μ)), where μ is a two dimensional bounded domain with smooth boundary.

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References

  1. K. Asano, Zero-viscosity limit of the incompressible Navier-Stokes equation 1. Preprint.

  2. K. Asano, Zero-viscosity limit of the incompressible Navier-Stokes equation 2. Preprint.

  3. P. C. Fife, Considerations regarding the mathematical basis for Prandtl’s boundary layer theory. Arch. Rational Mech. Anal.,28 (1968), 184–216.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Friedman, Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, 1964.

    MATH  Google Scholar 

  5. K.K. Golovkin, Vanishing viscosity in Cauchy’s problem for hydrodynamics equations. Proc. Steklov Inst. Math.,92 (1966), 33–53.

    Google Scholar 

  6. T. Kato, On classical solution for the two-dimensional non-stationary Euler equation. Arch. Rational Mech. Anal.,25 (1967), 188–200.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Kato, Non-stationary flows of viscous and ideal fluids inR 3. J. Funct. Anal.,9 (1972), 296–305.

    Article  MATH  Google Scholar 

  8. T. Kato, Quasi-linear equations of evolution with applications to partial differential equations. Lecture Notes in Math. vol. 448, Springer, 1975, 25–70.

  9. T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. Seminar on Nonlinear Partial Differential Equation (ed. S.S. Chern), Springer, 1982, 85–98.

  10. T. Kato and H. Fujita, On the nonstationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova,32 (1962), 243–260.

    MATH  MathSciNet  Google Scholar 

  11. O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., 1968.

  12. A. Majida, Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math.,39 (1986), 187–220.

    Article  MathSciNet  Google Scholar 

  13. S. Matsui and T. Shirota, On Prandtl boundary layer problem. Recent Topics in Nonlinear PDEII,. Kinokuniya/North-Holland Tokyo/Amsterdam, 1985, 81–105.

    Google Scholar 

  14. F.J. McGrath, Non-stationary plane flow of viscous and ideal fluids. Arch. Rational Mech. Anal.,27 (1968), 329–348.

    Article  MathSciNet  Google Scholar 

  15. O.A. Oleinik and S.N. Kruzhkov, Quasi-linear second order parabolic equations with many independent variables. Russian Math. Surveys,16 (1961), 106–146.

    Article  Google Scholar 

  16. J. Serrin, On the mathematical basis for Prandtl’s boundary layer theory: an example. Arch. Rational Mech. Anal.,28 (1968), 217–225.

    MATH  MathSciNet  Google Scholar 

  17. H. Swann, The convergence with vanishing viscosity of non-stationary Navier-Stokes flow to ideal flow inR 3. Trans. Amer. Math. Soc.,157 (1971), 373–397.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Hokhaido Information University, 59-2 Nishi-Nopporo, 069, Ebetsu, Japan

    Shin’ya Matsui

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  1. Shin’ya Matsui
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Matsui, S. Example of zero viscosity limit for two dimensional nonstationary Navier-Stokes flows with boundary. Japan J. Indust. Appl. Math. 11, 155–170 (1994). https://doi.org/10.1007/BF03167219

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  • DOI: https://doi.org/10.1007/BF03167219

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