On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds

Mitrea, Marius; Monniaux, Sylvie

doi:10.1090/S0002-9947-08-04827-7

On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds
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by Marius Mitrea and Sylvie Monniaux PDF

Trans. Amer. Math. Soc. 361 (2009), 3125-3157 Request permission

Abstract:

We study the analyticity of the semigroup generated by the Stokes operator equipped with Neumann-type boundary conditions on $L^p$ spaces in Lipschitz domains. Our strategy is to regularize this operator by considering the Hodge Laplacian, which has the additional property that it commutes with the Leray projection.

References

Shmuel Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math. 15 (1962), 119–147. MR 147774, DOI 10.1002/cpa.3160150203
Pascal Auscher, Steve Hofmann, John L. Lewis, and Philippe Tchamitchian, Extrapolation of Carleson measures and the analyticity of Kato’s square-root operators, Acta Math. 187 (2001), no. 2, 161–190. MR 1879847, DOI 10.1007/BF02392615
T. Akiyama, H. Kasai, Y. Shibata and M. Tsutsumi, The $L^p$ approach to the Laplace system with some boundary conditions and applying to the parabolic system, unpublished lecture notes, 1999.
T. Akiyama, H. Kasai, Y. Shibata, and M. Tsutsumi, On a resolvent estimate of a system of Laplace operators with perfect wall condition, Funkcial. Ekvac. 47 (2004), no. 3, 361–394. MR 2126323, DOI 10.1619/fesi.47.361
Viorel Barbu, Teodor Havârneanu, Cătălin Popa, and S. S. Sritharan, Exact controllability for the magnetohydrodynamic equations, Comm. Pure Appl. Math. 56 (2003), no. 6, 732–783. MR 1959739, DOI 10.1002/cpa.10072
P. E. Conner, The Neumann’s problem for differential forms on Riemannian manifolds, Mem. Amer. Math. Soc. 20 (1956), 56. MR 78467
T. G. Cowling, Magnetohydrodynamics, Interscience Tracts on Physics and Astronomy, No. 4, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1957. MR 0098556
Robert Dautray and Jacques-Louis Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 7, INSTN: Collection Enseignement. [INSTN: Teaching Collection], Masson, Paris, 1988 (French). Evolution: Fourier, Laplace; Reprint of the 1985 edition. MR 1016604
Paul Deuring, $L^p$-theory for the Stokes system in 3D domains with conical boundary points, Indiana Univ. Math. J. 47 (1998), no. 1, 11–47. MR 1631608, DOI 10.1512/iumj.1998.47.1439
E. B. Fabes, J. E. Lewis, and N. M. Rivière, Boundary value problems for the Navier-Stokes equations, Amer. J. Math. 99 (1977), no. 3, 626–668. MR 460928, DOI 10.2307/2373933
Eugene Fabes, Osvaldo Mendez, and Marius Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal. 159 (1998), no. 2, 323–368. MR 1658089, DOI 10.1006/jfan.1998.3316
Yoshikazu Giga, Analyticity of the semigroup generated by the Stokes operator in $L_{r}$ spaces, Math. Z. 178 (1981), no. 3, 297–329. MR 635201, DOI 10.1007/BF01214869
David Jerison and Carlos E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), no. 1, 161–219. MR 1331981, DOI 10.1006/jfan.1995.1067
Alf Jonsson and Hans Wallin, Function spaces on subsets of $\textbf {R}^n$, Math. Rep. 2 (1984), no. 1, xiv+221. MR 820626
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media, Course of Theoretical Physics, Vol. 8, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960. Translated from the Russian by J. B. Sykes and J. S. Bell. MR 0121049
Tetsuro Miyakawa, The $L^{p}$ approach to the Navier-Stokes equations with the Neumann boundary condition, Hiroshima Math. J. 10 (1980), no. 3, 517–537. MR 594132
Dorina Mitrea, Marius Mitrea, and Michael Taylor, Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds, Mem. Amer. Math. Soc. 150 (2001), no. 713, x+120. MR 1809655, DOI 10.1090/memo/0713
Dorina Mitrea and Marius Mitrea, Finite energy solutions of Maxwell’s equations and constructive Hodge decompositions on nonsmooth Riemannian manifolds, J. Funct. Anal. 190 (2002), no. 2, 339–417. MR 1899489, DOI 10.1006/jfan.2001.3870
Marius Mitrea, Sharp Hodge decompositions, Maxwell’s equations, and vector Poisson problems on nonsmooth, three-dimensional Riemannian manifolds, Duke Math. J. 125 (2004), no. 3, 467–547. MR 2166752, DOI 10.1215/S0012-7094-04-12322-1
M. Mitrea and S. Monniaux, The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains, to appear in the Journal of Differential and Integral Equations, (2008).
Michael Taylor, Marius Mitrea, and András Vasy, Lipschitz domains, domains with corners, and the Hodge Laplacian, Comm. Partial Differential Equations 30 (2005), no. 10-12, 1445–1462. MR 2182300, DOI 10.1080/03605300500299547
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Michel Sermange and Roger Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 36 (1983), no. 5, 635–664. MR 716200, DOI 10.1002/cpa.3160360506
Michael E. Taylor, Partial differential equations, Texts in Applied Mathematics, vol. 23, Springer-Verlag, New York, 1996. Basic theory. MR 1395147, DOI 10.1007/978-1-4684-9320-7
Michael E. Taylor, Incompressible fluid flows on rough domains, Semigroups of operators: theory and applications (Newport Beach, CA, 1998) Progr. Nonlinear Differential Equations Appl., vol. 42, Birkhäuser, Basel, 2000, pp. 320–334. MR 1788895
N. Yamaguchi, $L^q$-$L^r$ estimates of solution to the parabolic Maxwell equations and their application to the magnetohydrodynamic equations, in Proceedings of the 28th Sapporo Symposium on Partial Differential Equations, 2003.