A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in \(W^{-1/q,q}\). (English) Zbl 1064.35133

The following boundary value problem to the stationary Stokes and Navier-Stokes equations are considered \[ \begin{aligned} &-\Delta u+\nabla p =\operatorname{div}F,\quad \operatorname{div}u=h\;\text{in }\Omega, \quad u| _{\partial\Omega}=g \tag{1}\\ &-\Delta u+u\cdot\nabla u+\nabla p =\operatorname{div} F,\quad \operatorname{div}u=h\;\text{in }\Omega, \quad u| _{\partial\Omega}=g \tag{2} \end{aligned} \] Here \(\Omega\subset \mathbb R^n\) is a bounded domain with boundary \(\partial\Omega\) of class \(C^{2,1}\).
The authors introduce a very weak solution to the problems (1), (2). A very weak solution \(u\) is a function from \(L^q(\Omega)\) with \(\operatorname{div}u\in L^q(\Omega)\), and \(u\) satisfies a certain \(L^q - L^{q'}\) duality which does not include derivatives of \(u\). It is proved that for any data \[ F\in L^r(\Omega),\quad h\in L^q(\Omega),\quad g\in W^{-1/q,q}(\partial\Omega),\quad 1<r\leq q<\infty,\quad \frac 13+\frac 1q\geq \frac 1r \] problem (1) has a unique very weak solution. If \(3\leq q<\infty\), \(r\leq\frac q2\) and suitable norms of the data are sufficiently small then problem (2) has a unique very weak solution. The way in which the solution \(u\) attains the boundary data \(g\) is analyzed in detail. It is proved that very weak solutions become regular if the data are regular.


35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Adams, R.A.: Sobolev Spaces. Academic Press, New York, 1975 · Zbl 0314.46030
[2] Amann, H.: Linear and Quasilinear Parabolic Equations. Birkhäuser Verlag, Basel, 1995 · Zbl 0819.35001
[3] Amann, H.: On the Strong Solvability of the Navier-Stokes Equations. J. Math. Fluid Mech. 2, 16-98 (2000) · Zbl 0989.35107
[4] Amann, H.: Nonhomogeneous Navier-Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. Kluwer Academic/Plenum Publishing, New York, 2002, pp. 1-26 · Zbl 1201.76038
[5] 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) · Zbl 0499.35022
[6] Cannone, M.: Viscous flows in Besov spaces. Advances in mathematical fluid mechanics. (Paseky, 1999), Springer, Berlin, 2000, pp. 1-34 · Zbl 0980.35125
[7] 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) · Zbl 0254.35097
[8] Foia?, C.: Une Remarque sur l?Unicité des Solutions des Equations de Navier-Stokes en dimension n. Bull. Soc. Math. France, 89, 1-6 (1961) · Zbl 0107.07602
[9] Farwig, R., Sohr: Generalized Resolvent Estimates for the Stokes System in Bounded and Unbounded Domains. J. Math. Soc. Japan, 46, 607-643 (1994) · Zbl 0819.35109
[10] Fujiwara, D., Morimoto, H.: An Lr-Theorem of the Helmholtz Decomposition of Vector Fields. J. Fac. Sci. Univ. Tokyo (1A), 24, 685-700 (1977) · Zbl 0386.35038
[11] 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
[12] 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 · Zbl 0911.35085
[13] Galdi, G.P., Simader, C.G., Sohr, H.:On the Stokes Problem in Lipschitz Domains. Ann. Mat. Pura Appl. 167, 147-163 (1994) · Zbl 0824.35098
[14] 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) · Zbl 0957.76012
[15] 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 · Zbl 1104.35032
[16] Giga, Y.: Analyticity of the Semigroup Generated by the Stokes Operator in Lr-Spaces. Math. Z. 178, 287-329 (1981) · Zbl 0461.47019
[17] Giga, Y.: Domains of Fractional Powers of the Stokes Operator in Lr-spaces. Arch. Ration. Mech. Anal. 89, 251-265 (1985) · Zbl 0584.76037
[18] Giga, Y., Sohr, H.: On the Stokes Operator in Exterior Domains. J. Fac. Sci. Univ. Tokyo, Sec. IA, 36, 103-130 (1989)
[19] Grubb, G.: Nonhomogeneous Dirichlet Navier-Stokes problems in low regularity Lp Sobolev spaces. J. Math. Fluid Mech.3 (1), 57-81 (2001) · Zbl 0992.35065
[20] Kato, T.: Strong Lp-Solutions to the Navier-Stokes Equations in ?m, with Applications to Weak Solutions. Math. Z. 187, 471-480 (1984) · Zbl 0545.35073
[21] 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) · Zbl 0848.35099
[22] Necas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques. Masson et Cie, 1967
[23] 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) · Zbl 0791.35096
[24] 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 · Zbl 0868.35001
[25] Solonnikov, V.A.: Estimates for Solutions of Nonstationary Navier-Stokes Equations. J. Soviet Math. 8, 467-528 (1977) · Zbl 0404.35081
[26] Sohr, H.: The Navier-Stokes equations. An elementary functional analytic approach. Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2001 · Zbl 0983.35004
[27] Temam, R.: Navier-Stokes Equations. North-Holland Pub. Co. Amsterdam-New York-Tokyo, 1977 · Zbl 0383.35057
[28] Varnhorn, W.: The Stokes Equations. Mathematical Research, 76, Akademie Verlag, 1994 · Zbl 0813.35085
[29] Yamazaki, M.: The Navier-Stokes equations in the weak-Ln space with time-dependent external force. Math. Ann. 317, 635-675 (2000) · Zbl 0965.35118
[30] von Wahl, W.: The Equations of Navier-Stokes and Abstract Parabolic Equations. Vieweg, Braunschweig, 1985 · Zbl 0575.35074
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