Abstract
Consider the stationary Navier–Stokes equations in a bounded domain whose boundary consists of multi-connected components. We investigate the solvability under the general flux condition which implies that the total sum of the flux of the given data on each component of the boundary is equal to zero. Based on our Helmholtz–Weyl decomposition, we prove existence of solutions if the harmonic part of the solenoidal extension of the given boundary data is sufficiently small in L 3 compared with the viscosity constant.
Similar content being viewed by others
References
Amick C.J.: Existence of solutions to the nonhomogeneous steady Navier–Stokes equations. Indiana Univ. Math. J. 33, 817–830 (1984)
Bogovkii M.E.: Solution of the first boundary value problem for the equation of continuity of an incompressible medium. Sov. Math. Dokl. 20, 1094–1098 (1979)
Borchers W., Pileckas K.: Note on the flux problem for stationary incompressible Navier–Stokes equations in domains with a multiply connected boundary. Acta Appl. Math. 37, 21–30 (1994)
Borchers W., Sohr H.: On the equations rot v = g and div u = f with zero boundary conditions. Hokkaido Math. J. 19, 67–87 (1990)
Fujita H.: On the existence and regularity of the steady-state solutions of the Navier–Stokes equations. J. Fac. Sci. Univ. Tokyo Sec. I. A. 9, 59–102 (1960)
Fujita, H.: On stationary solutions to Navier–Stokes equation in symmetric plane domains under general outflow condition. In: Navier–Stokes equations: theory and numerical methods, Varenna 1997 Pitmann Res. Notes Math. Ser. vol. 388. Longman-Harlow, pp. 16–30 (1998)
Fujita, H., Morimoto, H.: A remark on the existence of the Navier–Stokes flow with non-vanishing outflow condition. In: Agemi, et al. (eds.) Proceeding of the 4th MSJ-IRI on Nonlinear Waves, pp. 53–61 (1997)
Fujita H., Morimoto H., Okamoto H.: Stability analysis of Navier–Stokes flows in annuli. Math. Methods Appl. Sci. 20, 959–978 (1997)
Galdi G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations vol. II. Springer, New York (1994)
Kozono, H., Yanagisawa, T.: L r-variational inequality for vector fields and the Helmholtz–Weyl decomposition in bounded domains (submitted)
Kozono, H., Yanagisawa, T.: Nonhomogeneous boundary value problems for stationary Navier–Stokes equations in a bounded domain with a multiply connected boundary (preprint)
Ladyzehnskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, London (1969)
Leray J.: Etude de diverses équations intégrals non linés et de quelques probléms que pose lH́ydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)
Morimoto, H.: General outflow condition for Navier–Stokes flow. In: Kozono, H., Shibata, Y. (eds.) Recent topics in mathematical theory of viscous incompressible fluids, Tsukuba 1996. Lec. Notes Num. Appl., vol. 16, pp. 209–224. Kinokuniya, Tokyo (1998)
Morimoto H., Ukai S.: Perturbation of the Navier–Stokes flow in an annular domain with the non-vanishing outflow condition. J. Math. Sci. Univ. Tokyo 3, 73–82 (1996)
Takeshita A.: A remark on Leray’s inequality. Pacific J. Math. 157, 151–158 (1993)
Temam R.: Navier–Stokes equations. Theory and Numerical Analysis. North-Holland Pub. Co., Amsterdam (1979)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Giovanni P. Galdi on the occasion of his 60th birthday.
Rights and permissions
About this article
Cite this article
Kozono, H., Yanagisawa, T. Leray’s problem on the stationary Navier–Stokes equations with inhomogeneous boundary data. Math. Z. 262, 27–39 (2009). https://doi.org/10.1007/s00209-008-0361-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-008-0361-2