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Keywords:
Stokes equation; very weak solution; strong solution; domain of half space type
Stokes equation; very weak solution; strong solution; domain of half space type
Summary:
We consider the theory of very weak solutions of the stationary Stokes system with nonhomogeneous boundary data and divergence in domains of half space type, such as $\mathbb R^n_+$, bent half spaces whose boundary can be written as the graph of a Lipschitz function, perturbed half spaces as local but possibly large perturbations of $\mathbb R^n_+$, and in aperture domains. The proofs are based on duality arguments and corresponding results for strong solutions in these domains, which have to be constructed in homogeneous Sobolev spaces. In addition to very weak solutions we also construct corresponding pressure functions in negative homogeneous Sobolev spaces.
References:
[1] Amann, H.: Nonhomogeneous Navier-Stokes equations with integrable low-regularity data. Nonlinear Problems in Mathematical Physics and Related Topics II. In Honour of Professor O. A. Ladyzhenskaya Int. Math. Ser. (N.Y.) 2 Kluwer Academic Publishers, New York (2002), 1-28 M. S. Birman et al. MR 1971987 | Zbl 1201.76038
[2] Amann, H.: On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech. 2 (2000), 16-98. DOI 10.1007/s000210050018 | MR 1755865 | Zbl 0989.35107
[3] Nezza, E. Di, Palatucci, G., Valdinoci, E.: Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521-573. DOI 10.1016/j.bulsci.2011.12.004 | MR 2944369 | Zbl 1252.46023
[4] Farwig, R.: Note on the flux condition and pressure drop in the resolvent problem of the Stokes system. Manuscr. Math. 89 (1996), 139-158. DOI 10.1007/BF02567510 | MR 1371993 | Zbl 0847.35102
[5] 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
[6] Farwig, R., Galdi, G. P., Sohr, H.: Very weak solutions and large uniqueness classes of stationary Navier-Stokes equations in bounded domains of {${\mathbb R}^2$}. J. Differ. Equations 227 (2006), 564-580. DOI 10.1016/j.jde.2005.10.009 | MR 2237679
[7] Farwig, R., Galdi, G. P., Sohr, H.: Very weak solutions of stationary and instationary Navier-Stokes equations with nonhomogeneous data. Nonlinear Elliptic and Parabolic Problems. A Special Tribute to the Work of Herbert Amann, Zürich, Switzerland, 2004 Progr. Nonlinear Differential Equations Appl. 64 Birkhäuser, Basel (2005), 113-136 M. Chipot et al. MR 2185213 | Zbl 1246.35148
[8] Farwig, R., Kozono, H., Sohr, H.: Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data. J. Math. Soc. Japan 59 (2007), 127-150. DOI 10.2969/jmsj/1180135504 | MR 2302666 | Zbl 1107.76022
[9] Farwig, R., Kozono, H., Sohr, H.: Very weak, weak and strong solutions to the instationary Navier-Stokes system. Topics on Partial Differential Equations Jindřich Nečas Cent. Math. Model. Lect. Notes 2 Matfyzpress, Praha (2007), 1-54 P. Kaplický et al. MR 2856664
[10] Farwig, R., Sohr, H.: Helmholtz decomposition and Stokes resolvent system for aperture domains in {$L^q$}-spaces. Analysis 16 (1996), 1-26. DOI 10.1524/anly.1996.16.1.1 | MR 1384351
[11] 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
[12] Fichera, G.: The trace operator. Sobolev and Ehrling lemmas. Linear Elliptic Differential Systems and Eigenvalue Problems Lecture Notes in Mathematics 8 Springer, Berlin (1965), 24-29.
[13] Focardi, M.: Aperiodic fractional obstacle problems. Adv. Math. 225 (2010), 3502-3544. DOI 10.1016/j.aim.2010.06.014 | MR 2729014 | Zbl 1213.49022
[14] Franzke, M.: Die Navier-Stokes-Gleichungen in "Offnungsgebieten. PhD thesis Shaker, Aachen German (2000).
[15] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I: Linearized Steady Problems. Springer Tracts in Natural Philosophy 38 Springer, New York (1994). MR 1284205 | Zbl 0949.35004
[16] Galdi, G. P., Simader, C. G., Sohr, H.: A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in {$W^{-1/q,q}$}. Math. Ann. 331 (2005), 41-74. DOI 10.1007/s00208-004-0573-7 | MR 2107439 | Zbl 1064.35133
[17] Kudrjavcev, L. D.: An imbedding theorem for a class of functions defined in the whole space or in the half-space. I. Transl., Ser. 2, Am. Math. Soc. 74 (1968), 199-225 translation from Mat. Sb., N. Ser. 69 (1966), 616-639 Russian. MR 0206704
[18] Kudrjavcev, L. D.: Imbedding theorems for classes of functions defined in the whole space or in the half-space. {II}. Transl., Ser. 2, Am. Math. Soc. 74 (1968), 227-260 translation from Mat. Sb., N. Ser. 70 3-35 (1966), Russian. MR 0206705
[19] Riechwald, P. F.: Interpolation of sum and intersection spaces of {$L^q$}-type and applications to the Stokes problem in general unbounded domains. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 58 (2012), 167-181. DOI 10.1007/s11565-011-0140-6 | MR 2915345 | Zbl 1307.46012
[20] Riechwald, P. F.: Very Weak Solutions to the Navier-Stokes Equations in General Unbounded Domains. PhD thesis TU Darmstadt, Darmstadt; Fachbereich Mathematik (Diss.), München (2011). Zbl 1252.35005
[21] Schumacher, K.: Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 54 (2008), 123-144. DOI 10.1007/s11565-008-0038-0 | MR 2403378 | Zbl 1179.35225
[22] Schumacher, K.: The Navier-Stokes Equations with Low-Regularity Data in Weighted Function Spaces. PhD thesis TU Darmstadt, Fachbereich Mathematik (Diss.), Darmstadt (2007). Zbl 1134.35088
[23] Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications 2 North-Holland Publishing, Amsterdam (1977). MR 0769654 | Zbl 0383.35057
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