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Keywords:
stationary Stokes and Navier-Stokes system; very weak solutions; existence and uniqueness in higher dimensions; regularity classes in higher dimensions
Summary:
For a bounded domain $\Omega \subset \Bbb R ^n$, $n\geq 3,$ we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $-\Delta u + u \cdot \nabla u + \nabla p=f$, $\div u = k$, $u_{|_{\partial \Omega }}=g$ with $u \in L^q$, $q \geq n$, and very general data classes for $f$, $k$, $g$ such that $u$ may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse & Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669--717, where the existence of a weak solution which is locally regular is proved.
References:
[1] Adams, R. A.: Sobolev Spaces. Academic Press, New York (1975). MR 0450957 | Zbl 0314.46030
[2] Amann, H.: Nonhomogeneous Navier-Stokes equations with integrable low-regularity data. Int. Math. Ser., Kluwer Academic/Plenum Publishing, New York (2002), 1-28. DOI 10.1007/978-1-4615-0701-7_1 | MR 1971987
[3] Amann, H.: Navier-Stokes equations with nonhomogeneous Dirichlet data. J. Nonlinear Math. Physics 10 (2003), 1-11. DOI 10.2991/jnmp.2003.10.s1.1 | MR 2063541
[4] Borchers, W., Miyakawa, T.: Algebraic $L^2$ decay for Navier-Stokes flows in exterior domains. Hiroshima Math. J. 21 (1991), 621-640. MR 1148998
[5] Bogovskij, M. E.: Solution of the first boundary value problem for the equation of continuity of an incompressible medium. Soviet Math. Dokl. 20 (1979), 1094-1098. Zbl 0499.35022
[6] Cannone, M.: Viscous flows in Besov spaces. Advances in Math. Fluid Mech., Springer, Berlin (2000), 1-34. MR 1863208 | 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 $L^p$. Arch. Rational Mech. Anal. 45 (1972), 222-240. DOI 10.1007/BF00281533 | MR 0316915
[8] Farwig, R., Sohr, H.: Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Japan 46 (1994), 607-643. DOI 10.2969/jmsj/04640607 | MR 1291109 | Zbl 0819.35109
[9] Farwig, R., Galdi, G. P., Sohr, H.: A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data. J. Math. Fluid Mech. 8 (2006), 423-444. DOI 10.1007/s00021-005-0182-6 | MR 2258419 | Zbl 1104.35032
[10] Frehse, J., Růžička, M.: Weighted estimates for the stationary Navier-Stokes equations. Acta Appl. Math. 37 53-66 (1994). DOI 10.1007/BF00995129 | MR 1308745
[11] Frehse, J., Růžička, M.: Regularity for the stationary Navier-Stokes equations in bounded domains. Arch. Rational Mech. Anal. 128 361-380 (1994). DOI 10.1007/BF00387714 | MR 1308859
[12] Frehse, J., Růžička, M.: On the regularity of the stationary Navier-Stokes equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (IV) 21 63-95 (1994). MR 1276763
[13] Frehse, J., Růžička, M.: Existence of regular solutions to the stationary Navier-Stokes equations. Math. Ann. 302 669-717 (1995). DOI 10.1007/BF01444513 | MR 1343646
[14] Frehse, J., Růžička, M.: Existence of regular solutions to the steady Navier-Stokes equations in bounded six-dimensional domains. Ann. Sc. Norm. Super. Pisa Cl. Sci. (IV) 23 701-719 (1996). MR 1469571
[15] Frehse, J., Růžička, M.: Regularity for steady solutions of the Navier-Stokes equations J. G. Heywood, et al. (eds.), Theory of the Navier-Stokes equations. Proc. 3rd Intern. Conf. Navier-Stokes Equations: theory and numerical methods. World Scientific Ser. Adv. Math. Appl. Sci., Singapore 47 159-178 (1998). DOI 10.1142/9789812816740_0013 | MR 1643033
[16] Frehse, J., Růžička, M.: A new regularity criterion for steady Navier-Stokes equations. Differential Integral Equations 11 (1998), 361-368. MR 1741851
[17] Fujiwara, D., Morimoto, H.: An $L_r$-theory of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo (1A) 24 (1977), 685-700. MR 0492980
[18] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations; Linearized Steady Problems. Springer Tracts in Natural Philosophy, Vol. 38, Springer-Verlag, New York (1998). MR 2808162
[19] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations; Nonlinear Steady Problems. Springer Tracts in Natural Philosophy, Vol. 39, New York (1998). MR 2808162
[20] Galdi, G. P., Simader, C. G., Sohr, H.: A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}(\partial \Omega)$. Math. Ann. 331 (2005), 41-74. DOI 10.1007/s00208-004-0573-7 | MR 2107439 | Zbl 1064.35133
[21] Gerhardt, C.: Stationary solutions of the Navier-Stokes equations in dimension four. Math. Z. 165 (1979), 193-197. DOI 10.1007/BF01182469 | MR 0520820
[22] Giga, Y.: Analyticity of the semigroup generated by the Stokes operator in $L_r$-spaces. Math. Z. 178 (1981), 287-329. DOI 10.1007/BF01214869 | MR 0635201 | Zbl 0473.35064
[23] Giga, Y.: Domains of fractional powers of the Stokes operator in $L_r$-spaces. Arch. Rational Mech. Anal. 89 (1985), 251-265. DOI 10.1007/BF00276874 | MR 0786549
[24] Giga, Y., Sohr, H.: On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo, Sec. IA 36 (1989), 103-130. MR 0991022
[25] Giga, Y., Sohr, H.: Abstract $L^q$-estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102 (1991), 72-94. DOI 10.1016/0022-1236(91)90136-S | MR 1138838 | Zbl 0739.35067
[26] Kato, T.: Strong $L^p$-solutions to the Navier-Stokes equations in $\Bbb R^m$ with applications to weak solutions. Math. Z. 187 (1984), 471-480. DOI 10.1007/BF01174182 | MR 0760047
[27] Kozono, H., Yamazaki, M.: Local and global solvability of the Navier-Stokes exterior problem with Cauchy data in the space $L^{n,\infty}$. Houston J. Math. 21 (1995), 755-799. MR 1368344
[28] Nečas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques. Academia, Prague (1967). MR 0227584
[29] Simader, C. G., Sohr, H.: A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$-spaces for bounded and exterior domains. Adv. Math. Appl. Sci., World Scientific 11 (1992), 1-35. DOI 10.1142/9789814503594_0001 | MR 1190728
[30] Solonnikov, V. A.: Estimates for solutions of nonstationary Navier-Stokes equations. J. Soviet Math. 8 (1977), 467-528. DOI 10.1007/BF01084616 | Zbl 0404.35081
[31] Sohr, H.: The Navier-Stokes equations. An elementary functional analytic approach. Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel (2001). MR 1928881 | Zbl 0983.35004
[32] Temam, R.: Navier-Stokes Equations. Theory and numerical analysis. North-Holland, Amsterdam, New York, Tokyo (1984). MR 0769654 | Zbl 0568.35002
[33] Triebel, H.: Interpolation Theory, Function Spaces. Differential Operators. North-Holland, Amsterdam (1978). MR 0503903 | Zbl 0387.46033
[34] Wahl, W. von: Regularity of weak solutions of the Navier-Stokes equations. Proc. Symp. Pure Math. 45 (1986), 497-503. MR 0843635
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