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Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids Read chapter Read first chapter

2018 | OriginalPaper | Chapter

3. The Stokes Equation in the L p -Setting: Well-Posedness and Regularity Properties

Authors : Matthias Hieber, Jürgen Saal

Published in: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Publisher: Springer International Publishing

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Abstract

This article discusses the Stokes equation in various classes of domains \(\Omega \subset \mathbb{R}^{n}\) within the L p -setting for 1 ≤ p ≤ ∞ from the point of view of evolution equations. Classical as well as modern approaches to well-posedness results for strong solutions to the Stokes equation, to the Helmholtz decomposition, to the Stokes semigroup, and to mixed maximal L q − L p -regularity results for 1 ¡ p, q < ∞ are presented via the theory of \(\mathcal{R}\)-sectorial operators. Of concern are domains having compact or noncompact, smooth or nonsmooth boundaries, as well as various classes of boundary conditions including energy preserving boundary conditions. In addition, the endpoints of the L p -scale, i.e., p = 1 and p = ∞ are considered and recent well-posedness results for the case p = ∞ are described. Results on L p − L q -smoothing properties of the associated Stokes semigroups and on variants of the Stokes equation (e.g., nonconstant viscosity, Lorentz spaces, Stokes-Oseen system, flow past rotating obstacles, hydrostatic Stokes equation) complete this survey article.

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previous chapter Variational Modeling and Complex Fluids
next chapter Stokes Problems in Irregular Domains with Various Boundary Conditions
Literature
1.
2.
3.
K. Abe, Y. Giga, M. Hieber, Stokes resolvent estimates in spaces of bounded functions. Ann. Sci. Ec. Norm. Super. 48, 521–543 (2015)MathSciNetMATH
4.
5.
K. Abe, Y. Giga, K. Schade, T. Suzuki, On the Stokes resolvent estimate for cylindrical domains. J. Evol. Equ. (to appear)
6.
T. Abe, Y. Shibata, On a resolvent estimate of the Stokes equation on an infinite layer. J. Math. Soc. Jpn. 55(2), 469–497 (2003)MathSciNetMATHCrossRef
7.
T. Abe, Y. Shibata, On a resolvent estimate of the Stokes equation on an infinite layer II. J. Math. Fluid Mech. 5, 245–274 (2003)MathSciNetMATHCrossRef
8.
9.
H. Abels, Bounded imaginary powers and H ∞ -calculus for the Stokes operator in two-dimensional exterior domains. Math. Z. 251(3), 589–605 (2005)MathSciNetMATHCrossRef
10.
H. Abels, Reduced and generalized Stokes resolvent equations in asymptotically flat layers. I. Unique solvability. J. Math. Fluid Mech. 7(2), 201–222 (2005)MathSciNetMATHCrossRef
11.
H. Abels, Reduced and generalized Stokes resolvent equations in asymptotically flat layers. II. H ∞ -calculus. J. Math. Fluid Mech. 7(2), 223–260 (2005)MathSciNetMATHCrossRef
12.
H. Abels, On generalized solutions of two-phase flows for viscous incompressible fluids. Interfaces Free Bound. 9, 31—65 (2007)MathSciNetMATHCrossRef
13.
H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains. Discret. Contin. Dyn. Syst. Ser. S 3, 141–157 (2010)MathSciNetMATHCrossRef
14.
H. Abels, Y. Terasawa, On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. 344(2), 381–429 (2009)MathSciNetMATHCrossRef
15.
H. Abels, M. Wiegner, Resolvent estimates for the Stokes operator on an infinite layer. Differ. Integral Equ. 18(10), 1081–1110 (2005)MathSciNetMATH
16.
17.
H. Amann, Linear and Quasilinear Parabolic Problems, vol. I (Birkhäuser, Basel, 1995)MATHCrossRef
18.
19.
20.
H. Amann, Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part I. Necas Center for Mathematical Modeling. Lecture Notes, vol. 6 (Matfyzpress, Praha, 2009)
21.
W. Arendt, C. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96, 2nd edn. (Birkhäuser, Basel, 2011)MATHCrossRef
22.
H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations (Springer, Grundlehren, 2011)MATHCrossRef
23.
L. von Below, The Stokes and Navier-Stokes equations in layer domains with and without a free surface. PhD thesis, TU Darmstadt, Darmstadt, 2014
24.
25.
M. E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR. 248, 1037–1040 (1979)MathSciNet
26.
M.E. Bogovskiĭ, Decomposition of \(L_{p}(\Omega, \mathbb{R}^{n})\) into the direct sum of subspaces of solenoidal and potential vector fields. Dokl. Akad. Nauk SSSR 286, 781–786 (1986) (Russian); English translation in Sov. Math. Dokl. 33, 161–165 (1986)
27.
M. Bolkart, The Stokes equation in spaces of bounded functions and spaces of bounded mean oscillations. PhD thesis, TU Darmstadt, Darmstadt, 2016
28.
M. Bolkart, Y. Giga, On L ∞ − BMO-estimates for derivatives of the Stokes semigroup. Math. Z. (to appear)
29.
M. Bolkart, Y. Giga, T. Miura, T. Suzuki, Y. Tsutsui, On analyticity of the L p -Stokes semigroup for some non-Helmholtz domains. Math. Nachrichten (to appear)
30.
M. Bolkart, Y. Giga, T. Suzuki, Analyticity of the Stokes semigroup in BMO-type spaces. J. Math. Soc. Jpn. (to appear)
31.
M. Bolkart, M. Hieber, Pointwise upper bounds for the solution of the Stokes equation on \(L_{\sigma }^{\infty }(\Omega )\) and applications. J. Funct. Anal. 268, 1678–1710 (2015)MathSciNetMATHCrossRef
32.
33.
34.
W. Borchers, H. Sohr, On the semigroup of the Stokes operator for exterior domains in L q -spaces. Math. Z. 196(3), 415–425 (1987)MathSciNetMATHCrossRef
35.
W. Borchers, W. Varnhorn, On the boundedness of Stokes semigroup in two-dimensional exterior domains. Math. Z. 213, 275–299 (1993)MathSciNetMATHCrossRef
36.
D. Bothe, M. Köhne, J. Prüss, On a class of energy preserving boundary conditions for incompressible Newtonian flows. SIAM J. Math. Anal. 45(6), 3768–3822 (2013)MathSciNetMATHCrossRef
37.
38.
D. Bothe, J. Prüss, On the interface formation model for dynamic triple lines. (2015). arXiv:1504.04758v1
39.
J. Bourgain, H. Brezis, On the equation div Y = f and application to control of phases. J. Am. Math. Soc. 16, 393–426 (2003)MathSciNetMATHCrossRef
40.
F. Boyer, P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Applied Mathematical Sciences (Springer, New York, 2013)MATHCrossRef
41.
C. Cao, E.S. Titi, Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. Ann. Math. 166, 245–267 (2007)MathSciNetMATHCrossRef
42.
L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Padova 31, 308–340 (1961)MathSciNetMATH
43.
J.-Y. Chemin, B. Desjardin, I. Gallagher, E. Grenier, Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations (Oxford University Press, Oxford, 2006)
44.
P. Constantin, C. Foias, Navier-Stokes Equations. Chicago Lectures in Mathematics (University of Chicago Press, Chicago 1988)
45.
M. Costabel, A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. Math. Z. 265, 297–320 (2010)MathSciNetMATHCrossRef
46.
W. Dan, T. Kobayashi, Y. Shibata, On the local energy decay approach to some fluid flow in an exterior domain. Lect. Notes in Num. Appl. Anal. 16, 1–51 (1998)MathSciNetMATH
47.
W. Dan, Y. Shibata, On the L q − L r estimates of the Stokes semigroup in a two dimensional exterior domain. J. Math. Soc. Jpn. 51, 181–207 (1999)MATHMathSciNetCrossRef
48.
W. Dan, Y. Shibata, Remark on the L q − L ∞ estimate of the Stokes semigroup in a 2-dimensional exterior domain. Pac. J. Math. 189, 223–239 (1999)MATHMathSciNetCrossRef
49.
R. Danchin, P. Mucha, Critical functional framework and maximal regularity in action on systems of incompressible flows. Mem. Soc. Math. Fr. 143 (2015)
50.
O. Darrigol, Between hydrodynamics and elasticity theory: the first five births of the Navier-Stokes equations. Arch. Hist. Exact Sci. 56, 95–150 (2002)MathSciNetMATHCrossRef
51.
M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I: Linearized equations. SIAM J. Math. Anal. 20(2), 74–97 (1989)MathSciNetMATH
52.
R. Denk, G. Dore, M. Hieber, J. Prüss, A. Venni, New thoughts on old results of R. T. Seeley. Math. Ann. 328(4), 545–583 (2004)MathSciNetMATHCrossRef
53.
R. Denk, M. Geissert, M. Hieber, J. Saal, O. Sawada, The spin-coating process: analysis of the free boundary value problem. Commun. Partial Differ. Equ. 36, 1145–1192 (2011)MathSciNetMATHCrossRef
54.
R. Denk, M. Hieber, J. Prüss, \(\mathcal{R}\)-Boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166, viii+114 (2003)
55.
R. Denk, M. Hieber, J. Prüss, Optimal L p − L q -estimates for parabolic boundary value problems with inhomogeneous data. Math. Z. 257, 193–224 (2007)MathSciNetMATHCrossRef
56.
57.
P. Deuring, L p -theory for the Stokes system in 3D domains with conical boundary points. Indiana Univ. Math. J. 47, 11–47 (1998)MathSciNetMATHCrossRef
58.
P. Deuring, The Stokes resolvent in 3D domains with conical boundary points: non-regularity in L p -spaces. Adv. Differ. Equ. 6, 175–226 (2001)MATHMathSciNet
59.
P. Deuring, W. Varnhorn, On Oseen resolvent estimates. Differ. Integral Equ. 23, 1139–1149 (2010)MathSciNetMATH
60.
P. Deuring, W.v. Wahl, Strong solutions to the Navier-Stokes system in Lipschitz bounded domains. Math. Nachr. 171, 111–148 (1995)MathSciNetMATHCrossRef
61.
62.
63.
S. Ervedoza, M. Hillairet, C. Lacave, Long-time behaviour for the two dimensional motion of a disk in a viscous fluid. Commun. Math. Phys. 329, 325–382 (2014)MATHCrossRef
64.
E. Fabes, O. Mendez, M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159(2), 323–368 (1998)MathSciNetMATHCrossRef
65.
R. Farwig, H. Kozono, H. Sohr, An L q -approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195, 21–53 (2005)MathSciNetMATHCrossRef
66.
67.
R. Farwig, J. Neustupa, On the spectrum of a Stokes type operator arising from flow around a rotating body. Manuscripta Math. 122, 419–437 (2007)MathSciNetMATHCrossRef
68.
R. Farwig, H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Jpn. 46, 607–643 (1994)MathSciNetMATHCrossRef
69.
R. Farwig, H. Sohr, Helmholtz decomposition and Stokes resolvent estimates for aperture domains in L q -spaces. Analysis 16, 1–26 (1996)MathSciNetMATHCrossRef
70.
M. Franzke, Strong solution of the Navier-Stokes equations in aperture domains. Ann. Univ. Ferrara Sez. VII (N.S.) 46, 161–173 (2000)
71.
A. Fröhlich, The Stokes operator in weighted L q -spaces I: weighted estimates for the resolvent problem in a half space. J. Math. Fluid Mech. 5, 166–199 (2003)MathSciNetMATHCrossRef
72.
A. Fröhlich, The Stokes operator in weighted L q -spaces II: weighted resolvent estimates and maximal L p -regularity. Math. Ann. 339, 287–316 (2007)MathSciNetMATHCrossRef
73.
D. Fujiwara, H. Morimoto, An L r -theorem for the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo 24, 685–700 (1977)MathSciNetMATH
74.
G.P. Galdi, On the energy equation and on the uniqueness for D-solutions to steady Navier-Stokes Equations in exterior domains, in Mathematical Problems Related to Navier-Stokes Equations, ed. by G.P. Galdi. Advances in Mathematics for Applied Sciences, vol. 11 (World Scientific, Singapore, 1992), pp. 36–80
75.
G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady State Problems, 2nd edn. (Springer, New York, 2011)
76.
G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II, Nonlinear Steady Problems (Springer, New York, 1994)
77.
G.P. Galdi, A. Silvestre, The steady motion of a Navier-Stokes liquid around a rigid body. Arch. Ration. Mech. Anal. 184, 371–400 (2007)MathSciNetMATHCrossRef
78.
G.P. Galdi, C. Simader, Existence, uniqueness and L q -estimates for the Stokes problem in an exterior domain. Arch. Ration. Mech. Anal. 112, 291–318 (1990)MATHMathSciNetCrossRef
79.
80.
J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics (North-Holland, Amsterdam, 1985)MATH
81.
M. Geissert, H. Heck, M. Hieber, L p -theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596, 45–62 (2006)MathSciNetMATH
82.
M. Geissert, H. Heck, M. Hieber, On the equation div u = f and Bogovskiĭ’s operator in Sobolev spaces of negative order, in Operator Theory: Advances and Applications, vol. 168 (Birkhaüser, Basel, 2006), pp. 113–121MATH
83.
M. Geissert, H. Heck, M. Hieber, O. Sawada, Weak Neumann implies Stokes. J. Reine Angew. Math. 669, 75–100 (2012)MathSciNetMATH
84.
M. Geissert, M. Hess, M. Hieber, C. Schwarz, K. Stavrakidis, Maximal L p − L q -estimates for the Stokes equation: a short proof of Solonnikov’s Theorem. J. Math. Fluid Mech. 12, 47–60 (2010)MathSciNetMATHCrossRef
85.
M. Geissert, M. Hieber, H. Nguyen, A general approach to time periodic incompressible fluid flow problems. Arch. Ration. Mech. Anal. 220, 1095–1118 (2016)MathSciNetMATHCrossRef
86.
87.
88.
M.-H. Giga, Y. Giga, J. Saal, Nonlinear Partial Differential Equations – Asymptotic Behavior of Solutions and Self-Similar Solutions (Birkhäuser, Basel, 2010)MATH
89.
90.
Y. Giga, The nonstationary Navier-Stokes system with some first order boundary condition. Proc. Jpn. Acad. Ser. A Math. Sci. 58(3), 101–104 (1982)MathSciNetMATHCrossRef
91.
92.
Y. Giga, Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Differ. Equ. 62, 186–212 (1986)MATHMathSciNetCrossRef
93.
Y. Giga, M. Gries, M. Hieber, A. Hussein, T. Kashiwabara, Bounded H ∞ -calculus for the hydrostatic Stokes operator on L p -spaces with applications. Submitted
94.
Y. Giga, K. Inui, A. Mahalov, S. Matsui, Uniform local solvability of the Navier-Stokes equations with the Coriolis force. Methods Appl. Anal. 12(4), 381–394 (2005)MathSciNetMATH
95.
Y. Giga, K. Inui, A. Mahalov, J. Saal, Uniform global solvability of the rotating Navier-Stokes equations for nondecaying data. Indiana Univ. Math. J. 57, 2775–2791 (2008)MathSciNetMATHCrossRef
96.
Y. Giga, S. Matsui, Y. Shimizu, On estimates in Hardy spaces for the Stokes flow in a half space. Math. Z. 231, 383–396 (1999)MathSciNetMATHCrossRef
97.
Y. Giga, H. Sohr, On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36(1), 103–130 (1989)MathSciNet
98.
Y. Giga, H. Sohr, Abstract L p -estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)MathSciNetMATHCrossRef
99.
P. Grisvard, Elliptic Problems on Nonsmooth Domains. Classics in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, 2011)
100.
G. Grubb, V.A. Solonnikov, A pseudo-differential treatment of general inhomogeneous initial-boundary value problems for the Navier-Stokes equations. Journées Équations aux dérivées partielles 1–8 (1988)
101.
G. Grubb, V.A. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods. Math. Scand. 69(2), 217–290 (1991)MathSciNetMATHCrossRef
102.
B. Guo, C. Schwab, Analytic regularity of Stokes flow on polygonal domains in countably weighted Sobolev spaces. J. Comput. Appl. Math. 190(1–2), 487–519 (2006)MathSciNetMATHCrossRef
103.
104.
R. Haller-Dintelmann, H. Heck, M. Hieber, L p − L q -estimates for parabolic non-divergence form operators with VMO-coefficients. J. Lond. Math. Soc. 74, 717–736 (2006)MATHCrossRef
105.
T. Hansel, A. Rhandi, The Oseen-Nasvier-Stokes flow in the exterior of a rotating body: the non-autonomous case. J. Reine Angew. Math. 694, 1–26 (2014)MathSciNetMATHCrossRef
106.
107.
J. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980)MathSciNetMATHCrossRef
108.
M. Hieber, A. Hussein, T. Kashiwabara, Global strong L p well-posedness of the 3D primitive equations with heat and salinity diffusion. J. Differ. Equ. 261, 6950–6981, (2016)MATHMathSciNetCrossRef
109.
M. Hieber, T. Kashiwabara, Strong well-posedness of the three dimensional primitive equations in the L p -setting. Arch. Ration. Mech. Anal. 221, 1077–1115 (2016)MathSciNetMATHCrossRef
110.
M. Hieber, P. Maremonti. Bounded analyticity of the Stokes semigroup on spaces of bounded functions, in Recent Developments of Mathematical Fluid Mechanics, ed. by H. Amann et al. (Birkhäuser, Basel, 2016), pp. 275–289MATHCrossRef
111.
M. Hieber, H. Nguyen, A. Seyfert, On periodic and almost periodic solutions to incompressible viscous fluid flow problems on the whole line, in Mathematics of Nonlinear Phenomena, ed. by Y. Maekawa (Springer, to appear)
112.
M. Hieber, J. Prüss, Dynamics of nematic liquid crystal flows I: general isotropic incompressible fluids. Math. Ann. (in press)
113.
M. Hieber, J. Prüss, Modeling and analysis of the Ericksen-Leslie equations for nematic liquid crystal flows. in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, ed. by Y. Giga, A. Novotny (Springer, in press)
114.
M. Hieber, H. Saito, Strong solutions for the two-phase free boundary problem for a class of non-Newtonoan fluids. J. Evol. Equ. (in press)
115.
M. Hieber, O. Sawada, The equation of Navier-Stokes on \(\mathbb{R}^{n}\) with linearly growing initial data. Arch. Ration. Mech. Anal. 175, 269–285 (2005)MathSciNetMATHCrossRef
116.
M. Hieber, Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework. Math. Z. 265, 481–491 (2010)MathSciNetMATHCrossRef
117.
T. Hishida, An existence theorem for the Navier-Stokes flow in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 150, 307–348 (1999)MathSciNetMATHCrossRef
118.
T. Hishida, The nonstationary Stokes and Navier-Stokes flows through an aperture, in Contributions to Current Challenges in Mathematical Fluid Mechanics, ed. by G.P. Galdi et al. Advances in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2004), pp. 79–123CrossRef
119.
T. Hishida, Y. Shibata, L p − L q -estimates of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 193, 339–421 (2009)MathSciNetMATHCrossRef
120.
A. Inoue, M. Wakimoto, On existence of solutions of the Navier-Stokes equation in a time dependent domain. J. Fac. Sci. Univ. Tokyo, Sect. I A 24, 303–319 (1977)
121.
T. Iwabuchi, R. Takada, Global well-posedness and ill-posedness for the Navier-Stokes equations with Coriolis force in function spaces of Besov type. J. Funct. Anal. 267, 1321–1337 (2014)MathSciNetMATHCrossRef
122.
H. Iwashita, L q − L r -estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problem in L q -spaces. Math. Ann. 285, 265–288 (1989)MathSciNetMATHCrossRef
123.
124.
T. Kato, Strong L p -solutions of the Navier-Stokes equations in \(\mathbb{R}^{m}\) with applications to weak solutions. Math. Z. 187, 471–480 (1984)MathSciNetMATHCrossRef
125.
126.
127.
Y. Koh, S. Lee, R. Takada, Dispersive estimates for the Navier-Stokes equations in the rotational framework. Adv. Differ. Equ. 19, 857–878 (2014)MathSciNetMATH
128.
M. Köhne, L p -theory for incompressible Newtonian flows. PhD thesis, TU Darmstadt, Darmstadtm, 2013
129.
V.A. Kondrat’ev, Boundary problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 16, 227–313 (1967)MathSciNetMATH
130.
H. Kozono, M. Yamazaki, Exterior problems for the stationary Navier-Stokes equations in the Lorentz space. Math. Ann. 310, 279–305 (1998)MathSciNetMATHCrossRef
131.
H. Kozono, Y. Mashiko, R. Takada, Existence of periodic solutions and their asymptotic stability to the Navier-Stokes equations with Coriolis force. J. Evol. Equ. 14, 565–601 (2014)MathSciNetMATHCrossRef
132.
133.
P.C. Kunstmann, L. Weis, Functional analytic methods for evolution equations, Maximal L p -regularity for parabolic equations, Fourier multiplier theorems, and H ∞ -functional calculus. Lecture Notes in Mathematics, vol 1855 (Springer, Berlin, 2004)
134.
P.C. Kunstmann, L. Weis, New criteria for the H ∞ -calculus and the Stokes operator on bounded Lipschitz domains. J. Evol. Equ. to appear.
135.
O. Ladyzhenskaja, The Mathematical Theory of Viscous Incompressible Flow (Gordon and Beach, New York, 1969)
136.
P.G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem (Chapman and Hill, Boca Raton, 2002)MATHCrossRef
137.
P.G. Lemarié-Rieusset, The Navier-Stokes Equations in the 21th Century (Chapman and Hill, Boca Raton, 2016)MATH
138.
P.-L. Lions, Mathematical Topics in Fluid Mechanics I: Incompressible Models (Oxford University Press, Oxford, 1996)MATH
139.
140.
Y. Maekawa, H. Miura, Remark on the Helmholtz decomposition in domains with noncompact boundaries. Math. Ann. 359, 1077–1095 (2014)MathSciNetMATHCrossRef
141.
Y. Maekawa, H. Miura, On isomorphism for the space of solenoidal vector fields and its application to the Stokes problem. Hokkaido University Preprint Series, no. 1076, 2015
142.
A. Mahalov, B. Nicolaenko, Global solvability of the three-dimensional Navier-Stokes equations with uniformly large initial vorticity. Russ. Math. Surv. 58(2), 287–318 (2003)CrossRef
143.
S. Maier, J. Saal, Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discret. Contin. Dyn. Syst. Ser. S 7(5), 1045–1063 (2014)MathSciNetMATHCrossRef
144.
P. Maremonti. A remark on the Stokes problem with initial data in L1. J. Math. Fluid Mech. 13, 469–480 (2012)MATHCrossRef
145.
P. Maremonti, On the Stokes problem in exterior domains: the maximum modulus theorem. Discret. Contin. Dyn. Syst. Ser. A 34, 2135–2171 (2014)MathSciNetMATHCrossRef
146.
P. Maremonti, V. Solonnikov, On nonstationary Stokes problem in exterior domains. Ann. Sc. Norm. Super. Pisa 24, 395–449 (1997)MathSciNetMATH
147.
V. Maslennikova, M. Bogovskii, Elliptic boundary values in unbounded domains with noncompact and nonsmooth boundaries. Rend. Sem. Mat. Fis. Milano 56, 125–138 (1986)MathSciNetMATHCrossRef
148.
K. Masuda, On the generation of analytic semigroups of higher-order elliptic operators in spaces of continuous functions (In Japanese), in Proceeding of the Katata Symposium on PDE, 1972, pp. 144–149
149.
150.
V. Maz’ya, J. Rossmann, Elliptic Equations in Polyhedral Domains (American Mathematical Society, Providence, 2010)MATHCrossRef
151.
D. Mitrea, M. Mitrea, S. Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition on nonsmooth domains. Commun. Pure Appl. Anal. 7, 1295–1333 (2008)MathSciNetMATHCrossRef
152.
M. Mitrea, S. Monniaux, On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds. Trans. Am. Math. Soc. 361(6), 3125–3157 (2009)MathSciNetMATHCrossRef
153.
D. Mitrea, M. Mitrea, M. Taylor, Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds. Mem. Amer. Math. Soc. 150(713), x+120 (2001)
154.
T. Miyakawa, The L p approach to the Navier-Stokes equations with the Neumann boundary condition. Hiroshima Math. J. 10(3), 517–537 (1980)MathSciNetMATH
155.
T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain. Hiroshima Math. J. 12, 115–140 (1982)MathSciNetMATH
156.
T. Miyakawa, The Helmholtz decompostion of vector fields in some unbounded domains. Math. J. Toyama Univ. 17, 115–149 (1994)MathSciNetMATH
157.
S. Monniaux, J. Saal, Analyticity of the Hodge-Stokes semigroup with partial slip type boundary conditions in Lipschitz domains. In preparation
158.
S. A. Nazarov, K. Pileckas, On the solvability of the Stokes and Navier-Stokes problems in the domains that are layer-like at infinity. J. Math. Fluid Mech. 1(1), 78–116 (1999)MathSciNetMATHCrossRef
159.
S. A. Nazarov, K. Pileckas, On the Fredholm property of the Stokes operator in a layer-like domain. Z. Anal. Anwendungen 20(1), 155–182 (2001)MathSciNetMATHCrossRef
160.
161.
K. Pileckas, On the nonstationary linearized Navier-Stokes problem in domains with cylindrical outlets to infinity. Math. Ann. 332, 395–419 (2005)MathSciNetMATHCrossRef
162.
K. Pileckas, Solvability in weighted spaces of a three-dimensional Navier-Stokes problem in domains with cylindrical outlets to infinity. Topol. Methods Nonlinear Anal. 29, 333–360 (2007)MathSciNetMATH
163.
K. Pileckas, Global solvability in W22,1-weighted spaces of a two-dimensional Navier-Stokes problem in domains with strip-like outlets to infinity. J. Math. Fluid Mech. 10, 272–309 (2008)MathSciNetMATHCrossRef
164.
J. Prüss, G. Simonett, On the two-phase Navier-Stokes equations with surface tension. Interfaces Free Bound. 10, 311–345 (2010)MathSciNetMATHCrossRef
165.
J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics (Birkhäuser, Basel, 2016)MATHCrossRef
166.
J. Saal, Robin boundary conditions and bounded H ∞ -calculus for the stokes operator. PhD Thesis, TU Darmstadt, Logos Verlag, Berlin, 2003
167.
J. Saal, Stokes and Navier-Stokes equations with Robin boundary conditions in a half-space. J. Math. Fluid Mech. 8, 211–241 (2006)MathSciNetMATHCrossRef
168.
J. Saal, Maximal regularity for the Stokes equations in non-cylindrical space-time domains. J. Math. Soc. Jpn. 58(3), 617–641 (2006)MATHMathSciNetCrossRef
169.
J. Saal, The Stokes operator with Robin boundary conditions in solenoidal subspaces of L1(R+ n ) and L ∞ (R+ n ). Commun. Partial Differ. Equ. 32(3), 343–373 (2007)MATHMathSciNetCrossRef
170.
J.O. Sather, The initial-boundary value problem for the Navier-Stokes equation in regions with moving boundaries. PhD thesis, University of Minnesota, 1963
171.
Z. Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains. Proc. Am. Math. Soc. 123(3), 801–811 (1995)MathSciNetMATHCrossRef
172.
Z. Shen, Resolvent estimates in L p for the Stokes operator in Lipschitz domains. Arch. Ration. Mech. Anal. 205(2), 395–424 (2012)MathSciNetMATHCrossRef
173.
Y. Shibata, On th \(\mathcal{R}\)-boundedness of solution operators for the Stokes equations with free boundary conditions. Differ. Integral Equ. 27, 313–368 (2014)
174.
Y. Shibata, R. Shimada, On a generalized resolvent estimate for the Stokes system with Robin boundary conditions. J. Math. Soc. Jpn. 59(2), 469–519 (2007)MathSciNetMATHCrossRef
175.
Y. Shibata, S. Shimizu, A decay property of the Fourier transform and its application to the Stokes problem. J. Math. Fluid Mech. 3, 213–230 (2001)MathSciNetMATHCrossRef
176.
Y. Shibata, S. Shimizu, Decay properties of the Stokes semigroup in exterior domains with Neumann boundary conditions. J. Math. Soc. Jpn. 59, 1–34 (2007)MathSciNetMATHCrossRef
177.
Y. Shibata, S. Shimizu, L p -L q maximal regularity of the Neumann problem for the Stokes equations in a bounded domain. Adv. Stud. Pure Math. 47, 349–362 (2007)MATHMathSciNet
178.
Y. Shibata, S. Shimizu, Maximal L p − L q -regularity for the two-phase Stokes equations; model problems. J. Differ. Equ. 251, 373–419 (2011)MATHMathSciNetCrossRef
179.
Y.D. Shikhmurzaev, Capillary Flows and Forming Interfaces (Chapmann and Hall/CRC, Boca Raton, 2006)MATH
180.
R. Shimada, On the L p -L q maximal regularity for Stokes equations with Robin boundary conditions in a bounded domain. Math. Methods Appl. Sci. 30, 257–289 (2007)MathSciNetMATHCrossRef
181.
C. Simader, H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in L q -spaces for bounded and exterior domains, in Mathematical Problems Relating to the Navier-Stokes Equations. Advances in Mathematics for Applied Sciences, vol. 11 (World Scientific, Singapore 1992), pp. 1–35
182.
C. Simader, H. Sohr, The Dirichlet problem for the Laplacian in Bounded and Unbounded Domains. Pitman Research Notes in Mathematics Series, vol. 360 (Longman, Harlow, 1996)
183.
S.L. Sobolev, Applications of Functional Analysis to Mathematical Physics. Translations of mathematical monographs, vol. 7 (American Mathematical Society, Providence, 1963)
184.
P.E. Sobolevskii, Study of the Navier-Stokes equations by the methods of the theory of parabolic equations in Banach spaces. Sov. Math. Dokl. 5, 720–723 (1964)
185.
H. Sohr, Navier-Stokes Equations: An Elementary Functional Analytic Approach (Birkhäuser, Basel/Boston, 2001)MATHCrossRef
186.
V.A. Solonnikov, Estimates of the solutions of a nonstationary linearized system of Navier-Stokes equations. Am. Math. Soc. Trans. II, 1–116 (1968)
187.
V.A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations. J. Sov. Math. 8, 467–529 (1977)MATHCrossRef
188.
V.A. Solonnikov, On the solvability of boundary and initial boundary value problems for the Navier-Stokes system in domains with noncompact boundaries. Pac. J. Math. 93, 213–317 (1981)MathSciNetMATHCrossRef
189.
V.A. Solonnikov, Solvability of a problem of evolution of an isolated amount of a viscous incompressible capillary fluid. Zap. Nauchn. Sem. LOMI 140, 179–186 (1984)MathSciNetMATH
190.
V.A. Solonnikov, Solvability of a problem of the evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval. Algebra i Analiz, 3, 222–257 (1991); English transl. in St. Petersburg Math. J. 3, 189–220 (1992)
191.
V.A. Solonnikov, On some free boundary problems for the Navier-Stokes equations with moving contact points and lines. Math. Ann. 302, 743–772 (1995)MathSciNetMATHCrossRef
192.
V.A. Solonnikov, Schauder estimates for the evolutionary Stokes problem. Ann. Univ. Ferrara. 53, 137–172 (1996)
193.
V.A. Solonnikov, L p -estimates for solutions to the initial-boundary value problem for the generalized Stokes system in a bounded domain. J. Math. Sci. 105, 2448–2484 (2001)MathSciNetCrossRef
194.
V.A. Solonnikov, On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity. J. Math. Sci. 114, 1726–1740 (2003)MathSciNetMATHCrossRef
195.
V.A. Solonnikov, Estimates of the solution of model evolution generalized Stokes problem in weighted Hölder spaces. Zap. Nauchn. Semin. POMI 336, 211–238 (2006)MATHMathSciNet
196.
E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals (Princeton University Press, Princeton, 1993)MATH
197.
H. B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Am. Math. Soc. 259(1), 299–310 (1980)MathSciNetMATHCrossRef
198.
M.E. Taylor, Incompressible fluid flows on rough domains, in Semigroups of Operators: Theory and Applications, ed. by A. V. Balakrishnan. Progress in Nonlinear Differential Equations and Their Applications, vol. 42 (Birkhäuser, Basel/Boston, 2000), pp. 320–334CrossRef
199.
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis (North-Holland, Amsterdam, 1977)MATH
200.
P. Tolksdorf, The L p -theory of the Navier-Stokes equations on Lipschitz domains. PhD Thesis, TU Darmstadt, Darmstadt, 2016
201.
H. Triebel, Theory of Function Spaces. (Reprint of 1983 edition) (Springer, Basel, 2010)
202.
203.
W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations. Aspects of Mathematics (Vieweg, Braunschweig, 1985)CrossRef
204.
205.
M. Wilke, Rayleigh-Taylor instability for the two-phase Navier-Stokes equations with surface tension in cylindrical domains. Habilitationschrift, University of Halle, Halle, 2013
206.
Y. Yamada, Periodic solutions of certain nonlinear parabolic differential equations in domains with periodically moving boundaries. Nagoya Math. J. 70, 111–123 (1978)MathSciNetMATHCrossRef
207.
M. Yamazaki, The Navier-Stokes equations in the weak L n -spaces with time dependent external force. Math. Ann. 317, 635–675 (2000)MathSciNetMATHCrossRef
208.
Y. Zheng, A. Tice, Local well-posedness of the contact line problem in 2-D Stokes flow. arXiv:1609.07085v1
Metadata
Title
The Stokes Equation in the L p -Setting: Well-Posedness and Regularity Properties
Authors
Matthias Hieber
Jürgen Saal
Copyright Year
2018
Book
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Print ISBN: 978-3-319-13343-0

Electronic ISBN: 978-3-319-13344-7

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-13344-7_3

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