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

2018 | OriginalPaper | Chapter

8. Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains

Authors : Reinhard Farwig, Hideo Kozono, Hermann Sohr

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

Publisher: Springer International Publishing

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Abstract

To solve the (Navier-)Stokes equations in general smooth domains \(\Omega \subset \mathbb{R}^{n}\), the spaces \(\tilde{L}^{q}(\Omega )\) defined as L q ∩ L2 when 2 ≤ q < ∞ and L q + L2 when 1 < q < 2 have shown to be a successful strategy. First, the main properties of the spaces \(\tilde{L}^{q}(\Omega )\) and related concepts for solenoidal subspaces, Sobolev spaces, Bochner spaces, and the corresponding Helmholtz projection and Stokes operator will be discussed. Then these concepts are used to construct and analyze very weak, weak, mild, and strong solutions to the instationary (Navier-)Stokes equations in general domains. In particular, the strategy allows to find weak solutions of the (Navier-)Stokes system satisfying the localized energy inequality and the strong energy inequality which are important in the context of Leray structure theorem and partial regularity results.

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previous chapter Steady-State Navier-Stokes Flow Around a Moving Body
next chapter Self-Similar Solutions to the Nonstationary Navier-Stokes Equations
Literature
1.
H. Abels, Bounded Imaginary Powers and H ∞ -Calculus of the Stokes Operator in Unbounded Domains. Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 64 (Birkhäuser, Basel, 2005), pp. 1–15
2.
H. Abels, Y. Terasawa, On stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann. 344, 381–429 (2009)MathSciNetCrossRef
3.
H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I: Abstract Linear Theory (Birkhäuser, Basel, 1995)CrossRef
4.
H. Amann, Nonhomogeneous Navier-Stokes equations with integrable low-regularity data, in Nonlinear Problems in Mathematical Physics and Related Problems II, ed. by M.Sh. Birman, S. Hildebrandt, V.A. Solonnikov, N.N. Uraltseva. International Mathematical Series, vol. 2 (Kluwer Academic/Plenum Publication, New York, 2002), pp. 1–26
5.
H. Amann, Navier-Stokes equations with nonhomogeneous Dirichlet data. J. Nonlinear Math. Phys. 10 (Suppl. 1), 1–11 (2003)MathSciNetCrossRef
6.
7.
Ch. Amrouche, V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czech. Math. J. 44, 109–140 (1994)MathSciNetMATH
8.
9.
M.E. Bogovskiĭ, Decomposition of \(L_{p}(\Omega, \mathbb{R}^{n})\) into the direct sum of subspaces of solenoidal and potential vector fields. Sov. Math. Dokl. 33, 161–165 (1986)
10.
M. Bolkart, Y. Giga, On L ∞ − BMO estimates for derivatives of the Stokes semigroup. Math. Z. 284, 1163–1183 (2016)MathSciNetCrossRef
11.
M. Bolkart, Y. Giga, T.-H. Miura, T. Suzuzki, Y. Tsutsui, On analyticity of the L p -stokes semigroup for some non-Helmholtz domains. Hokkaido Univ. Preprint Ser. Math. 1082 (2015)
12.
M. Bolkart, Y. Giga, T. Suzuzki, Analyticity of the stokes semigroup in BMO type spaces. Hokkaido Univ. Preprint Ser. Math. 1078 (2015)
13.
W. Borchers, W. Varnhorn, On the boundedness of the stokes semigroup in two-dimensional exterior domains. Math. Z. 213, 275–299 (1993)MathSciNetCrossRef
14.
L. Caffarelli, R. Kohn, R. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)MathSciNetCrossRef
15.
P. Cannarsa, V. Vespri, On maximal L p regularity for the abstract Cauchy problem. Boll. Un. Mat. Ital. 6, 165–175 (1986)MATH
16.
R. Farwig, Local regularity results for the instationary Navier-Stokes equations based on Besov space type criteria, in Recent Developments of Mathematical Fluid Mechanics, ed. by H. Amann et al. Advances in Mathematical Fluid Mechanics (Springer, Basel, 2016), pp. 183–214
17.
R. Farwig, G.P. Galdi, H. Sohr, A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data. J. Math. Fluid Mech. 8, 423–444 (2006)MathSciNetCrossRef
18.
R. Farwig, G.P. Galdi, H. Sohr, Very weak solutions of stationary and instationary Navier-Stokes equations with nonhomogeneous data, in Nonlinear Elliptic and Parabolic Problems, ed. by M. Chipot, J. Escher (Birkhäuser, Basel/Boston/Berlin, 2005), pp. 113–136CrossRef
19.
R. Farwig, C. Komo, Regularity of weak solutions to the Navier-Stokes equations in exterior domains. Nonlinear Differ. Equ. Appl. 17, 303–321 (2010)MathSciNetCrossRef
20.
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)MathSciNetCrossRef
21.
R. Farwig, H. Kozono, H. Sohr, On the Helmholtz decomposition in general unbounded domains. Arch. Math. 88, 239–248 (2007)MathSciNetCrossRef
22.
R.R. Farwig, H. Kozono, H. Sohr, Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data. J. Math. Soc. Jpn. 59, 127–150 (2007)MathSciNetCrossRef
23.
R.R. Farwig, H. Kozono, H. Sohr, Local in time regularity properties of the Navier-Stokes equations. Indiana Univ. Math. J. 56, 2111–2131 (2007)MathSciNetCrossRef
24.
R. Farwig, H. Kozono, H. Sohr, Energy-based regularity criteria for the Navier-Stokes equations. J. Math. Fluid Mech. 11, 1–14 (2008)MathSciNetMATH
25.
R. Farwig, H. Kozono, H. Sohr, On the Stokes operator in general unbounded domains. Hokkaido Math. J. 38, 111–136 (2009)MathSciNetCrossRef
26.
R. Farwig, P.F. Riechwald, Very weak solutions to the Navier-Stokes system in general unbounded domains. J. Evol. Equ. 15, 253–279 (2015)MathSciNetCrossRef
27.
R. Farwig, P.F. Riechwald, Very weak solutions and the Fujita-Kato approach to the Navier-Stokes system in general unbounded domains. NoDEA Nonlinear Differ. Equ. Appl. 22, 1143–1165 (2015)MathSciNetCrossRef
28.
R. Farwig, P.F. Riechwald, Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discr. Contin. Dyn. Syst. Ser. S 9, 157–172 (2015)MathSciNetCrossRef
29.
R. Farwig, V. Rosteck, Resolvent estimates of the Stokes system with Navier boundary conditions in general unbounded domains. Adv. Differ. Equs. 21(5–6), 401–428 (2016)MathSciNetMATH
30.
R. Farwig, V. Rosteck, Note on Friedrichs’ inequality in N-star-shaped domains. J. Math. Anal. Appl. 435, 1514–1524 (2016)MathSciNetCrossRef
31.
R. Farwig, H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Jpn. 46, 607–643 (1994)MathSciNetCrossRef
32.
R. Farwig, H. Sohr, W. Varnhorn, On optimal initial value conditions for local strong solutions of the Navier-Stokes equations. Ann. Univ. Ferrara 55, 89–110 (2009)MathSciNetCrossRef
33.
R. Farwig, H. Sohr, W. Varnhorn, Besov space regularity conditions for weak solutions of the Navier-Stokes equations. J. Math. Fluid Mech. 16, 307–320 (2014)MathSciNetCrossRef
34.
35.
G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Linearized Steady Problems, vol. I (Springer, New York, 1994)
36.
G.P. Galdi, C.G. Simader, H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in \(W^{-\frac{1} {q},q}\). Math. Ann. 331, 41–74 (2005)MathSciNetCrossRef
37.
M. Geissert, H. Heck, M. Hieber, O. Sawada, Weak Neumann implies Stokes. J. Reine Angew. Math. 669, 75–100 (2012)MathSciNetMATH
38.
Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L r -spaces. Math. Z. 178, 297–265 (1981)MathSciNetCrossRef
39.
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 (1992)CrossRef
40.
41.
J. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980)MathSciNetCrossRef
42.
43.
P.C. Kunstmann, Navier-Stokes equations on unbounded domains with rough initial data. Czech. Math. J. 60, 297–313 (2010)MathSciNetCrossRef
44.
O.A. Ladyzhenskaya, G.A. Seregin, On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J. Math. Fluid Mech. 1, 356–387 (1999)MathSciNetCrossRef
45.
O.A. Ladyzehnskaya, V.A. Solonnikov, The solvability of boundary value and initial boundary value problem for the Navier-Stokes equation in domains with non-compact boundaries. Vestnik Leningrad Univ. 13, 39–47 (1977, in Russian)
46.
47.
P. Maremonti, V.A. Solonnikov, On nonstationary Stokes problem in exterior domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24, 395–449 (1997)
48.
V.N. Maslennikova, M.E. Bogovskiĭ, Elliptic boundary value problems in unbounded domains with noncompact and nonsmooth boundaries. Rend. Sem. Mat. Fis. Milano LVI, 125–138 (1986)MathSciNetCrossRef
49.
50.
P.F. Riechwald, Very weak solutions to the Navier-Stokes equations in general unbounded domains. PhD thesis, Technische Universität Darmstadt (2011) (Logos-Verlag Berlin, 2011)
51.
P.F. Riechwald, Interpolation of sum and intersection spaces of L q -type and applications to the Stokes problem in general unbounded domains. Ann. Univ. Ferrara Sez. VII Sci. Mat. 58, 167–181 (2012)MathSciNetCrossRef
52.
V. Rosteck, The Stokes system with the Navier boundary condition in general unbounded domains. PhD thesis, Technische Universität Darmstadt (Verlag Dr. Hut, München, 2013)
53.
J. Serrin, The initial value problem for the Navier-Stokes equations. in Nonlinear Problems, ed. by R.E. Langer (University Wisconsin Press, Madison, 1963), pp. 69–98
54.
Y. Shibata, R. Shimada, On a generalized resolvent estimate for the Stokes system with Robin boundary condition. J. Math. Soc. Jpn. 59, 469–519 (2007)MathSciNetCrossRef
55.
R. Shimada, On the L p − L q maximal regularity for Stokes equations with Robin boundary condition in a bounded domain. Math. Methods Appl. Sci. 30, 257–289 (2007)MathSciNetCrossRef
56.
H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach (Birkhäuser, Basel, 2001)MATH
57.
V.A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations. J. Sov. Math. 8, 467–529 (1977)CrossRef
58.
V.A. Solonnikov, Estimates of solutions of the Stokes equations in Sobolev spaces with mixed norms. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288, 204–231 (2002, Russian); J. Math. Sci. 123, 4637–4653 (2004, English)
59.
R. Temam, Navier-Stokes equations. An elementary functional analytic approach (North-Holland Publication Company, Amsterdam/New York/Oxford, 1979)
60.
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam/New York, 1978)MATH
Metadata
Title
Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains
Authors
Reinhard Farwig
Hideo Kozono
Hermann Sohr
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_8

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