Tokyo Journal of Mathematics

Leray's Inequality for Fluid Flow in Symmetric Multi-connected Two-dimensional Domains

Reinhard FARWIG and Hiroko MORIMOTO

Full-text: Open access

Abstract

We consider the stationary Navier-Stokes equations with nonhomogeneous boundary condition in a domain with several boundary components. If the boundary value satisfies only the necessary flux condition (GOC), Leray's inequality does not holds true in general and we cannot prove the existence of a solution. But for a 2-D domain which is symmetric with respect to a line and where the data is also symmetric, C. Amick showed the existence of solutions by reduction to absurdity; later H. Fujita proved Leray-Fujita's inequality and hence the existence of symmetric solutions. In this paper we give a new short proof of Leray-Fujita's inequality and hence a proof of the existence of weak solutions.

Article information

Source
Tokyo J. Math., Volume 35, Number 1 (2012), 63-70.

Dates
First available in Project Euclid: 19 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1342701344

Digital Object Identifier
doi:10.3836/tjm/1342701344

Mathematical Reviews number (MathSciNet)
MR2977445

Zentralblatt MATH identifier
1256.35053

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 76D05: Navier-Stokes equations [See also 35Q30]

Citation

FARWIG, Reinhard; MORIMOTO, Hiroko. Leray's Inequality for Fluid Flow in Symmetric Multi-connected Two-dimensional Domains. Tokyo J. Math. 35 (2012), no. 1, 63--70. doi:10.3836/tjm/1342701344. https://projecteuclid.org/euclid.tjm/1342701344


Export citation

References

  • C. J. Amick, Existence of solutions to the nonhomogeneous steady Navier-Stokes equations, Indiana Univ. Math. J. 33 (1984), 817–830.
  • R. Farwig, H. Kozono, T. Yanagisawa, Leray's inequality in general multi-connected domains in ${\mathbb R}^n$, Math. Ann. (to appear).
  • H. Fujita, On the existence and regularity of the steady-state solutions of the Navier-Stokes equation, J. Fac. Sci., Univ.Tokyo, Sec. I, 9 (1961), 59–102.
  • H. Fujita, On stationary solutions to Navier-Stokes equations in symmetric plane domains under general out-flow condition, Proceedings of International Conference on Navier-Stokes Equations, Theory and Numerical Methods, June 1997, Varenna Italy, Pitman Research Notes in Mathematics 388, pp. 16–30.
  • H. Fujita and H. Morimoto, A remark on the existence of the Navier-Stokes flow with non-vanishing outflow condition, Gakuto International Series Mathematical Science and Applications, Vol. 10 (1997) Nonlinear Waves, pp. 53–61.
  • E. Hopf, Ein allgemeiner Endlichkeitssatz der Hydrodynamik, Math. Ann. 117 (1941), 764–775.
  • H. Kozono and T. Yanagisawa, Leray's problem on the stationary Navier-Stokes equations with inhomogeneous boundary data, Math. Z. 262 (2009), 27–39.
  • O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, 1969.
  • J. Leray, Etude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pure Appl. 12 (1933), 1–82.
  • H. Morimoto, A remark on the existence of 2-D steady Navier-Stokes flow in symmetric domain under general outflow condition, J. Math. Fluid Mech. 9 (2007), 411–418.
  • H. Morimoto and S. Ukai, Perturbation of the Navier-Stokes flow in an annular domain with the non-vanishing outflow condition, J. Math. Sci., Univ. Tokyo 3 (1996), 73–82.
  • E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.
  • A. Takeshita, A remark on Leray's inequality, Pacific J. Math., 157 (1993), 151–158.
  • P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge. Philos. Soc. 53(1957), 568-575.