Previous |  Up |  Next

Article

Full entry | PDF   (0.2 MB)
Keywords:
Navier-Stokes equations; mild solutions; Stokes operator; extrapolation spaces; $H^\infty$-functional calculus; general unbounded domains; pressure term
Summary:
We consider the Navier-Stokes equations in unbounded domains $\Omega \subseteq \mathbb R ^n$ of uniform $C^{1,1}$-type. We construct mild solutions for initial values in certain extrapolation spaces associated to the Stokes operator on these domains. Here we rely on recent results due to Farwig, Kozono and Sohr, the fact that the Stokes operator has a bounded $H^\infty$-calculus on such domains, and use a general form of Kato's method. We also obtain information on the corresponding pressure term.
References:
[1] 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
[2] Cannone, M.: Ondelettes, paraproduits, et Navier-Stokes. Nouveaux Essais, Paris, Diderot (1995). MR 1688096 | Zbl 1049.35517
[3] Constantin, P., Foias, C.: Navier-Stokes equations. Chicago Lectures in Mathematics, University of Chicago Press (1988). MR 0972259 | Zbl 0687.35071
[4] Farwig, R., Kozono, H., Sohr, H.: An $L^q$-approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195 (2005), 21-53. DOI 10.1007/BF02588049 | MR 2233684
[5] Farwig, R., Kozono, H., Sohr, H.: On the Helmholtz decomposition in general unbounded domains. Arch. Math. 88 (2007), 239-248. DOI 10.1007/s00013-006-1910-8 | MR 2305602 | Zbl 1121.35097
[6] Farwig, R., Kozono, H., Sohr, H.: Maximal regularity of the Stokes operator in general unbounded domains of $\Bbb R^n$. H. Amann Functional analysis and evolution equations. The Günter Lumer volume. Basel: Birkhäuser 257-272 (2008). MR 2402733
[7] Giga, Y.: Domains of fractional powers of the Stokes operator in $L_r$ spaces. Arch. Ration. Mech. Anal. 89 (1985), 251-265. DOI 10.1007/BF00276874 | MR 0786549
[8] Grisvard, P.: Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics 24, Pitman (1985). MR 0775683 | Zbl 0695.35060
[9] Haak, B. H., Kunstmann, P. C.: Weighted admissibility and wellposedness of linear systems in Banach spaces. SIAM J. Control Optim. 45 (2007), 2094-2118. DOI 10.1137/060656139 | MR 2285716 | Zbl 1126.93021
[10] Haak, B. H., Kunstmann, P. C.: On Kato's method for Navier Stokes equations. J. Math. Fluid Mech 11 (2009), 492-535. DOI 10.1007/s00021-008-0270-5 | MR 2574794
[11] Kalton, N. J., Kunstmann, P. C., Weis, L.: Perturbation and interpolation theorems for the $H^\infty$-calculus with applications to differential operators. Math. Ann. 336 (2006), 747-801. DOI 10.1007/s00208-005-0742-3 | MR 2255174
[12] Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Commun. Partial Differ. Equations 19 (1994), 959-1014. DOI 10.1080/03605309408821042 | MR 1274547 | Zbl 0803.35068
[13] Kunstmann, P. C.: Maximal $L^p$-regularity for second order elliptic operators with uniformly continuous coefficients on domains, in Iannelli. Mimmo Evolution equations: applications to physics, industry, life sciences and economics, Basel, Birkhäuser., Prog. Nonlinear Differ. Equ. Appl. Vol. 55 293-305 (2003). MR 2013196
[14] Kunstmann, P. C.: $H^\infty$-calculus for the Stokes operator on unbounded domains. Arch. Math. 91 (2008), 178-186. DOI 10.1007/s00013-008-2621-0 | MR 2430801
[15] Kunstmann, P. C., Weis, L.: Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus. M. Iannelli, R. Nagel, S. Piazzera Functional Analytic Methods for Evolution Equations, Springer Lecture Notes Math. Vol. 1855 65-311 (2004). DOI 10.1007/978-3-540-44653-8_2 | MR 2108959
[16] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris, Dunod (1969). MR 0259693 | Zbl 0189.40603
[17] Meyer, Y.: Wavelets, paraproducts, and Navier-Stokes equations. R. Bott Current developments in mathematics, 1996. Proceedings of the joint seminar, Cambridge, MA, USA 1996. Cambridge, International Press 105-212 (1997). MR 1724946 | Zbl 0926.35115
[18] Prüss, J., Simonett, G.: Maximal regularity for evolution equations in weighted $L_p$-spaces. Arch. Math. 82 (2004), 415-431. DOI 10.1007/s00013-004-0585-2 | MR 2061448
[19] Sohr, H.: The Navier-Stokes equations. An elementary functional analytic approach, Basel, Birkhäuser (2001). MR 1928881 | Zbl 1007.35051
[20] Triebel, H.: Interpolation, Function Spaces, Differential Operators. North-Holland Mathematical Library. Vol. 18. Amsterdam-New York-Oxford (1978). MR 0503903

Partner of