Geodesic
Navier-Stokes equations on unbounded domains with rough initial data
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 2, pp. 297-313 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 35K55, 35Q30, 76D05
Keywords: Navier-Stokes equations; mild solutions; Stokes operator; extrapolation spaces; $H^\infty $-functional calculus; general unbounded domains; pressure term 
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Kunstmann, Peer Christian. Navier-Stokes equations on unbounded domains with rough initial data. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 2, pp. 297-313. http://geodesic.mathdoc.fr/item/CMJ_2010_60_2_a0/

[1] Amann, H.: On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech. 2 (2000), 16-98. | DOI | MR | Zbl

[2] Cannone, M.: Ondelettes, paraproduits, et Navier-Stokes. Nouveaux Essais, Paris, Diderot (1995). | MR | Zbl

[3] Constantin, P., Foias, C.: Navier-Stokes equations. Chicago Lectures in Mathematics, University of Chicago Press (1988). | MR | Zbl

[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 | MR

[5] Farwig, R., Kozono, H., Sohr, H.: On the Helmholtz decomposition in general unbounded domains. Arch. Math. 88 (2007), 239-248. | DOI | MR | Zbl

[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

[7] Giga, Y.: Domains of fractional powers of the Stokes operator in $L_r$ spaces. Arch. Ration. Mech. Anal. 89 (1985), 251-265. | DOI | MR

[8] Grisvard, P.: Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics 24, Pitman (1985). | MR | Zbl

[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 | MR | Zbl

[10] Haak, B. H., Kunstmann, P. C.: On Kato's method for Navier Stokes equations. J. Math. Fluid Mech 11 (2009), 492-535. | DOI | MR

[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 | MR

[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 | MR | Zbl

[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

[14] Kunstmann, P. C.: $H^\infty$-calculus for the Stokes operator on unbounded domains. Arch. Math. 91 (2008), 178-186. | DOI | MR

[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 | MR

[16] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris, Dunod (1969). | MR | Zbl

[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 | Zbl

[18] Prüss, J., Simonett, G.: Maximal regularity for evolution equations in weighted $L_p$-spaces. Arch. Math. 82 (2004), 415-431. | DOI | MR

[19] Sohr, H.: The Navier-Stokes equations. An elementary functional analytic approach, Basel, Birkhäuser (2001). | MR | Zbl

[20] Triebel, H.: Interpolation, Function Spaces, Differential Operators. North-Holland Mathematical Library. Vol. 18. Amsterdam-New York-Oxford (1978). | MR