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.
References
K. Abe, Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions. Acta Math. 211, 1–46 (2013)
K. Abe, Y. Giga, The L ∞-Stokes semigroup in exterior domains. J. Evol. Equ. 14, 1–28 (2014)
K. Abe, Y. Giga, M. Hieber, Stokes resolvent estimates in spaces of bounded functions. Ann. Sci. Ec. Norm. Super. 48, 521–543 (2015)
K. Abe, Y. Giga, K. Schade, T. Suzuki, On the Stokes semigroup in some non-Helmholtz domains. Arch. Math. 104, 177–187 (2015)
K. Abe, Y. Giga, K. Schade, T. Suzuki, On the Stokes resolvent estimate for cylindrical domains. J. Evol. Equ. (to appear)
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)
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)
H. Abels, Boundedness of imaginary powers of the Stokes operator in an infinite layer. J. Evol. Equ. 2(4), 439–457 (2002)
H. Abels, Bounded imaginary powers and H ∞ -calculus for the Stokes operator in two-dimensional exterior domains. Math. Z. 251(3), 589–605 (2005)
H. Abels, Reduced and generalized Stokes resolvent equations in asymptotically flat layers. I. Unique solvability. J. Math. Fluid Mech. 7(2), 201–222 (2005)
H. Abels, Reduced and generalized Stokes resolvent equations in asymptotically flat layers. II. H ∞-calculus. J. Math. Fluid Mech. 7(2), 223–260 (2005)
H. Abels, On generalized solutions of two-phase flows for viscous incompressible fluids. Interfaces Free Bound. 9, 31—65 (2007)
H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains. Discret. Contin. Dyn. Syst. Ser. S 3, 141–157 (2010)
H. Abels, Y. Terasawa, On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. 344(2), 381–429 (2009)
H. Abels, M. Wiegner, Resolvent estimates for the Stokes operator on an infinite layer. Differ. Integral Equ. 18(10), 1081–1110 (2005)
H. Amann, Stability of the rest state of a viscous incompressible fluid. Arch. Ration. Mech. Anal. 126, 231–242 (1994)
H. Amann, Linear and Quasilinear Parabolic Problems, vol. I (Birkhäuser, Basel, 1995)
H. Amann, On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech. 2, 16–98 (2000)
H. Amann, Navier-Stokes equations with nonhomogeneous Dirichlet data. Nonlinear Math. Phys. 10, 1–11 (2003)
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)
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)
H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations (Springer, Grundlehren, 2011)
L. von Below, The Stokes and Navier-Stokes equations in layer domains with and without a free surface. PhD thesis, TU Darmstadt, Darmstadt, 2014
D.N. Bock, On the Navier-Stokes equations in noncylindrical domains. J. Differ. Equ. 25, 151–162 (1977)
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)
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)
M. Bolkart, The Stokes equation in spaces of bounded functions and spaces of bounded mean oscillations. PhD thesis, TU Darmstadt, Darmstadt, 2016
M. Bolkart, Y. Giga, On L ∞− BMO-estimates for derivatives of the Stokes semigroup. Math. Z. (to appear)
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)
M. Bolkart, Y. Giga, T. Suzuki, Analyticity of the Stokes semigroup in BMO-type spaces. J. Math. Soc. Jpn. (to appear)
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)
W. Borchers, T. Miyakawa, L 2 decay for the Navier-Stokes flow in halfspaces. Math. Ann. 282, 139–155 (1988)
W. Borchers, T. Miyakawa, Algebraic L 2-decay for Navier-Stokes flows in exterior domains. Acta Math. 165, 189–227 (1990)
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)
W. Borchers, W. Varnhorn, On the boundedness of Stokes semigroup in two-dimensional exterior domains. Math. Z. 213, 275–299 (1993)
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)
D. Bothe, J. Prüss, L p -theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39(2), 379–421 (2007)
D. Bothe, J. Prüss, On the interface formation model for dynamic triple lines. (2015). arXiv:1504.04758v1
J. Bourgain, H. Brezis, On the equation div Y = f and application to control of phases. J. Am. Math. Soc. 16, 393–426 (2003)
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)
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)
L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Padova 31, 308–340 (1961)
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)
P. Constantin, C. Foias, Navier-Stokes Equations. Chicago Lectures in Mathematics (University of Chicago Press, Chicago 1988)
M. Costabel, A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. Math. Z. 265, 297–320 (2010)
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)
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)
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)
R. Danchin, P. Mucha, Critical functional framework and maximal regularity in action on systems of incompressible flows. Mem. Soc. Math. Fr. 143 (2015)
O. Darrigol, Between hydrodynamics and elasticity theory: the first five births of the Navier-Stokes equations. Arch. Hist. Exact Sci. 56, 95–150 (2002)
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)
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)
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)
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)
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)
W. Desch, M. Hieber, J. Prüss, L p-theory of the Stokes equation in a half space. J. Evol. Equ. 1, 115–142 (2001)
P. Deuring, L p-theory for the Stokes system in 3D domains with conical boundary points. Indiana Univ. Math. J. 47, 11–47 (1998)
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)
P. Deuring, W. Varnhorn, On Oseen resolvent estimates. Differ. Integral Equ. 23, 1139–1149 (2010)
P. Deuring, W.v. Wahl, Strong solutions to the Navier-Stokes system in Lipschitz bounded domains. Math. Nachr. 171, 111–148 (1995)
L. Dienig, M. Ruzicka, Strong solutions to generalized Newtonian fluids. J. Math. Fluid Mech. 7, 413–450 (2005)
G. Dore, A. Venni, On the closedness of the sum of two closed operators. Math. Z. 196, 189–201 (1987)
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)
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)
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)
R. Farwig, H. Kozono, H. Sohr, The Stokes operator in general unbounded domains. Hokkaido Math. J. 38, 111–136 (2009)
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)
R. Farwig, H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Jpn. 46, 607–643 (1994)
R. Farwig, H. Sohr, Helmholtz decomposition and Stokes resolvent estimates for aperture domains in L q-spaces. Analysis 16, 1–26 (1996)
M. Franzke, Strong solution of the Navier-Stokes equations in aperture domains. Ann. Univ. Ferrara Sez. VII (N.S.) 46, 161–173 (2000)
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)
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)
D. Fujiwara, H. Morimoto, An L r -theorem for the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo 24, 685–700 (1977)
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
G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady State Problems, 2nd edn. (Springer, New York, 2011)
G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II, Nonlinear Steady Problems (Springer, New York, 1994)
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)
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)
G.P. Galdi, C. Simader, H. Sohr, On the Stokes problem in Lipschitz domains. Ann. Mat. Pura Appl. CLXVII, 147–163 (1994)
J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics (North-Holland, Amsterdam, 1985)
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)
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. 113–121
M. Geissert, H. Heck, M. Hieber, O. Sawada, Weak Neumann implies Stokes. J. Reine Angew. Math. 669, 75–100 (2012)
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)
M. Geissert, M. Hieber, H. Nguyen, A general approach to time periodic incompressible fluid flow problems. Arch. Ration. Mech. Anal. 220, 1095–1118 (2016)
M. Geissert, P. Kunstmann. Weak Neumann implies H ∞ for Stokes. J. Math. Soc. Jpn. 67, 183–193 (2015)
J. Geng, Z. Shen, The Neumann problem and Helmholtz decompostion in convex domains. J. Funct. Anal. 259, 2147–2164 (2010)
M.-H. Giga, Y. Giga, J. Saal, Nonlinear Partial Differential Equations – Asymptotic Behavior of Solutions and Self-Similar Solutions (Birkhäuser, Basel, 2010)
Y. Giga, Analyticity of the semigroup generated by the Stokes operator on L r -spaces. Math. Z. 178, 297–329 (1981)
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)
Y. Giga, Domains of fractional powers of the Stokes operator in L r spaces. Arch. Ration. Mech. Anal. 89, 251–265 (1985)
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)
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
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)
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)
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)
Y. Giga, H. Sohr, On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36(1), 103–130 (1989)
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)
P. Grisvard, Elliptic Problems on Nonsmooth Domains. Classics in Applied Mathematics (Society for Industrial and Applied Mathematics, Philadelphia, 2011)
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)
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)
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)
R. Haller, H. Heck, M. Hieber, Muckenhoupt weights and maximal L p-regularity. Arch. Math. 81, 422–430 (2003)
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)
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)
J. Heywood, On uniqueness questions in the theory of viscous flow. Acta Math. 136, 61–102 (1976)
J. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980)
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)
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)
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. 275–289
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)
M. Hieber, J. Prüss, Dynamics of nematic liquid crystal flows I: general isotropic incompressible fluids. Math. Ann. (in press)
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)
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)
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)
M. Hieber, Y. Shibata, The Fujita-Kato approach to the Navier-Stokes equations in the rotational framework. Math. Z. 265, 481–491 (2010)
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)
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. 79–123
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)
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)
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)
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)
D. Jerison, C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)
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)
R.B. Kellogg, J.E. Osborn, A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal. 21, 397–431 (1976)
T. Kobayashi, Y. Shibata, On the Oseen equation in the three dimensional exterior domains. Math. Ann. 310, 1–45 (1998)
Y. Koh, S. Lee, R. Takada, Dispersive estimates for the Navier-Stokes equations in the rotational framework. Adv. Differ. Equ. 19, 857–878 (2014)
M. Köhne, L p-theory for incompressible Newtonian flows. PhD thesis, TU Darmstadt, Darmstadtm, 2013
V.A. Kondrat’ev, Boundary problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 16, 227–313 (1967)
H. Kozono, M. Yamazaki, Exterior problems for the stationary Navier-Stokes equations in the Lorentz space. Math. Ann. 310, 279–305 (1998)
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)
T. Kubo, The Stokes and Navier-Stokes equations in an aperture domain. J. Math. Soc. Jpn. 59, 837–859 (2007)
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)
P.C. Kunstmann, L. Weis, New criteria for the H ∞-calculus and the Stokes operator on bounded Lipschitz domains. J. Evol. Equ. to appear.
O. Ladyzhenskaja, The Mathematical Theory of Viscous Incompressible Flow (Gordon and Beach, New York, 1969)
P.G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem (Chapman and Hill, Boca Raton, 2002)
P.G. Lemarié-Rieusset, The Navier-Stokes Equations in the 21th Century (Chapman and Hill, Boca Raton, 2016)
P.-L. Lions, Mathematical Topics in Fluid Mechanics I: Incompressible Models (Oxford University Press, Oxford, 1996)
M. McCracken, The resolvent problem for the Stokes equation on halfspaces in L p . SIAM J. Math. Anal. 12, 201–228 (1981)
Y. Maekawa, H. Miura, Remark on the Helmholtz decomposition in domains with noncompact boundaries. Math. Ann. 359, 1077–1095 (2014)
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
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)
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)
P. Maremonti. A remark on the Stokes problem with initial data in L 1. J. Math. Fluid Mech. 13, 469–480 (2012)
P. Maremonti, On the Stokes problem in exterior domains: the maximum modulus theorem. Discret. Contin. Dyn. Syst. Ser. A 34, 2135–2171 (2014)
P. Maremonti, V. Solonnikov, On nonstationary Stokes problem in exterior domains. Ann. Sc. Norm. Super. Pisa 24, 395–449 (1997)
V. Maslennikova, M. Bogovskii, Elliptic boundary values in unbounded domains with noncompact and nonsmooth boundaries. Rend. Sem. Mat. Fis. Milano 56, 125–138 (1986)
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
K. Masuda, Weak solutions of Navier-Stokes equations. Tohuko Math. J. 36, 623–646 (1984)
V. Maz’ya, J. Rossmann, Elliptic Equations in Polyhedral Domains (American Mathematical Society, Providence, 2010)
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)
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)
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)
T. Miyakawa, The L p approach to the Navier-Stokes equations with the Neumann boundary condition. Hiroshima Math. J. 10(3), 517–537 (1980)
T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain. Hiroshima Math. J. 12, 115–140 (1982)
T. Miyakawa, The Helmholtz decompostion of vector fields in some unbounded domains. Math. J. Toyama Univ. 17, 115–149 (1994)
S. Monniaux, J. Saal, Analyticity of the Hodge-Stokes semigroup with partial slip type boundary conditions in Lipschitz domains. In preparation
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)
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)
A. Noll, J. Saal, H ∞-calculus for the Stokes operator on L q -spaces. Math. Z. 244, 651–688 (2003)
K. Pileckas, On the nonstationary linearized Navier-Stokes problem in domains with cylindrical outlets to infinity. Math. Ann. 332, 395–419 (2005)
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)
K. Pileckas, Global solvability in W 2 2, 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)
J. Prüss, G. Simonett, On the two-phase Navier-Stokes equations with surface tension. Interfaces Free Bound. 10, 311–345 (2010)
J. Prüss, G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations. Monographs in Mathematics (Birkhäuser, Basel, 2016)
J. Saal, Robin boundary conditions and bounded H ∞-calculus for the stokes operator. PhD Thesis, TU Darmstadt, Logos Verlag, Berlin, 2003
J. Saal, Stokes and Navier-Stokes equations with Robin boundary conditions in a half-space. J. Math. Fluid Mech. 8, 211–241 (2006)
J. Saal, Maximal regularity for the Stokes equations in non-cylindrical space-time domains. J. Math. Soc. Jpn. 58(3), 617–641 (2006)
J. Saal, The Stokes operator with Robin boundary conditions in solenoidal subspaces of L 1(R + n) and L ∞(R + n). Commun. Partial Differ. Equ. 32(3), 343–373 (2007)
J.O. Sather, The initial-boundary value problem for the Navier-Stokes equation in regions with moving boundaries. PhD thesis, University of Minnesota, 1963
Z. Shen, A note on the Dirichlet problem for the Stokes system in Lipschitz domains. Proc. Am. Math. Soc. 123(3), 801–811 (1995)
Z. Shen, Resolvent estimates in L p for the Stokes operator in Lipschitz domains. Arch. Ration. Mech. Anal. 205(2), 395–424 (2012)
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)
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)
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)
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)
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)
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)
Y.D. Shikhmurzaev, Capillary Flows and Forming Interfaces (Chapmann and Hall/CRC, Boca Raton, 2006)
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)
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
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)
S.L. Sobolev, Applications of Functional Analysis to Mathematical Physics. Translations of mathematical monographs, vol. 7 (American Mathematical Society, Providence, 1963)
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)
H. Sohr, Navier-Stokes Equations: An Elementary Functional Analytic Approach (Birkhäuser, Basel/Boston, 2001)
V.A. Solonnikov, Estimates of the solutions of a nonstationary linearized system of Navier-Stokes equations. Am. Math. Soc. Trans. II, 1–116 (1968)
V.A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations. J. Sov. Math. 8, 467–529 (1977)
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)
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)
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)
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)
V.A. Solonnikov, Schauder estimates for the evolutionary Stokes problem. Ann. Univ. Ferrara. 53, 137–172 (1996)
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)
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)
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)
E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals (Princeton University Press, Princeton, 1993)
H. B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Am. Math. Soc. 259(1), 299–310 (1980)
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. 320–334
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis (North-Holland, Amsterdam, 1977)
P. Tolksdorf, The L p-theory of the Navier-Stokes equations on Lipschitz domains. PhD Thesis, TU Darmstadt, Darmstadt, 2016
H. Triebel, Theory of Function Spaces. (Reprint of 1983 edition) (Springer, Basel, 2010)
S. Ukai, A solution formula for the Stokes equation in \(\mathbb{R}_{+}^{n}\). Commun. Pure Appl. Math. 40(5), 611–621 (1987)
W. von Wahl, The Equations of Navier-Stokes and Abstract Parabolic Equations. Aspects of Mathematics (Vieweg, Braunschweig, 1985)
L. Weis, Operator-valued Fourier multiplier theorems and maximal L p-regularity. Math. Ann. 319, 735–758 (2001)
M. Wilke, Rayleigh-Taylor instability for the two-phase Navier-Stokes equations with surface tension in cylindrical domains. Habilitationschrift, University of Halle, Halle, 2013
Y. Yamada, Periodic solutions of certain nonlinear parabolic differential equations in domains with periodically moving boundaries. Nagoya Math. J. 70, 111–123 (1978)
M. Yamazaki, The Navier-Stokes equations in the weak L n-spaces with time dependent external force. Math. Ann. 317, 635–675 (2000)
Y. Zheng, A. Tice, Local well-posedness of the contact line problem in 2-D Stokes flow. arXiv:1609.07085v1
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this entry
Cite this entry
Hieber, M., Saal, J. (2016). The Stokes Equation in the L p-setting: Well Posedness and Regularity Properties. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_3-1
Download citation
DOI: https://doi.org/10.1007/978-3-319-10151-4_3-1
Received:
Accepted:
Published:
Publisher Name: Springer, Cham
Online ISBN: 978-3-319-10151-4
eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering