Abstract.
We develop a theory for a general class of very weak solutions to stationary Stokes and Navier-Stokes equations in a bounded domain Ω with boundary ∂Ω of class C2,1, corresponding to boundary data in the distribution space W−1/q,q(∂Ω), 1<q<∞. These solutions exist and are unique (for small data, in the nonlinear case) in their class of existence, and satisfy a correponding estimate in terms of the data. Moreover, they become regular if the data are regular. To our knowledge, the only existence result for solutions attaining such boundary data is due to Giga, [16], Proposition 2.2, for the Stokes case. However, the methods and the approach used in the present paper are different than Giga’s and cover more general issues, including the nonlinear Navier-Stokes equations and the precise way in which the boundary data are attained by the solutions. We also introduce, in the last section, a further generalization of the solution class.
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Adams, R.A.: Sobolev Spaces. Academic Press, New York, 1975
Amann, H.: Linear and Quasilinear Parabolic Equations. Birkhäuser Verlag, Basel, 1995
Amann, H.: On the Strong Solvability of the Navier-Stokes Equations. J. Math. Fluid Mech. 2, 16–98 (2000)
Amann, H.: Nonhomogeneous Navier-Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. Kluwer Academic/Plenum Publishing, New York, 2002, pp. 1–26
Bogovskii, M.E.: Solution of the First Boundary Value Problem for the Equation of Continuity of an Incompressible Medium. Soviet Math Dokl. 20, 1094–1098 (1979)
Cannone, M.: Viscous flows in Besov spaces. Advances in mathematical fluid mechanics. (Paseky, 1999), Springer, Berlin, 2000, pp. 1–34
Fabes, E.B., Jones, B.F., Rivière, N.M.: The Initial Value Problem for the Navier-Stokes Equations with Data in Lp. Arch. Ration. Mech. Anal. 45, 222–240 (1972)
Foiaş, C.: Une Remarque sur l’Unicité des Solutions des Equations de Navier-Stokes en dimension n. Bull. Soc. Math. France, 89, 1–6 (1961)
Farwig, R., Sohr: Generalized Resolvent Estimates for the Stokes System in Bounded and Unbounded Domains. J. Math. Soc. Japan, 46, 607–643 (1994)
Fujiwara, D., Morimoto, H.: An L r -Theorem of the Helmholtz Decomposition of Vector Fields. J. Fac. Sci. Univ. Tokyo (1A), 24, 685–700 (1977)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Linearized Steady Problems. Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York, Revised Edition, 1998
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Nonlinear Steady Problems. Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New York, Revised Edition, 1998
Galdi, G.P., Simader, C.G., Sohr, H.:On the Stokes Problem in Lipschitz Domains. Ann. Mat. Pura Appl. 167, 147–163 (1994)
Galdi, G.P.: On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Ration. Mech. Anal. 148 (1), 53–88 (1999)
Galdi, G.P., Sohr, H.: A New Class of Very Weak Solutions for Unsteady Stokes and Navier-Stokes Problems. J. Math. Fluid Mech. To appear
Giga, Y.: Analyticity of the Semigroup Generated by the Stokes Operator in L r -Spaces. Math. Z. 178, 287–329 (1981)
Giga, Y.: Domains of Fractional Powers of the Stokes Operator in L r -spaces. Arch. Ration. Mech. Anal. 89, 251–265 (1985)
Giga, Y., Sohr, H.: On the Stokes Operator in Exterior Domains. J. Fac. Sci. Univ. Tokyo, Sec. IA, 36, 103–130 (1989)
Grubb, G.: Nonhomogeneous Dirichlet Navier-Stokes problems in low regularity L p Sobolev spaces. J. Math. Fluid Mech.3 (1), 57–81 (2001)
Kato, T.: Strong Lp-Solutions to the Navier-Stokes Equations in ℝm, with Applications to Weak Solutions. Math. Z. 187, 471–480 (1984)
Kozono, H., Yamazaki, M.: Local and Global Unique Solvability of the Navier-Stokes Exterior Problem with Cauchy Data in the Space Ln,∞. Houston J. Math, 21, 755–739 (1995)
Necas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques. Masson et Cie, 1967
Simader, C.G., Sohr, H.: The Helmholtz Decomposition in Lq and Related Topics. Mathematical Problems Related to the Navier-Stokes Equation, Galdi, G.P. (ed.), Advances in Mathematics for Applied Science, World Scientific, 11, 1–35 (1992)
Simader, C.G., Sohr, H.: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Mathematics Series, Longman Scientific and Technical, 360, 1997
Solonnikov, V.A.: Estimates for Solutions of Nonstationary Navier-Stokes Equations. J. Soviet Math. 8, 467–528 (1977)
Sohr, H.: The Navier-Stokes equations. An elementary functional analytic approach. Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2001
Temam, R.: Navier-Stokes Equations. North-Holland Pub. Co. Amsterdam-New York-Tokyo, 1977
Varnhorn, W.: The Stokes Equations. Mathematical Research, 76, Akademie Verlag, 1994
Yamazaki, M.: The Navier-Stokes equations in the weak-Ln space with time-dependent external force. Math. Ann. 317, 635–675 (2000)
von Wahl, W.: The Equations of Navier-Stokes and Abstract Parabolic Equations. Vieweg, Braunschweig, 1985
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Acknowledgement We would like to thank Professors Herbert Amann and Remigio Russo for helpful conversations on aspects of this research. In particular, Amann’s recent paper [4] for the nonstationary case was for us a motivation to study the steady case. We also would like to thank an anonymous referee whose comments helped the clarity of the formulation of the trace theorem. This paper was completed while G.P.Galdi was a Deutsche Forschungsgemeinschaft (DFG) Mercator Professor at the University of Paderborn in the period May-August 2003.
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Galdi, G., Simader, C. & Sohr, H. A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W−1/q,q. Math. Ann. 331, 41–74 (2005). https://doi.org/10.1007/s00208-004-0573-7
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DOI: https://doi.org/10.1007/s00208-004-0573-7