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References (33)
-
R. Farwig, H. Sohr (2010)
On the existence of local strong solutions for the Navier–Stokes equations in completely general domainsNonlinear Anal., Ser. A: Theory and Methods, 73
-
K. Schumacher (2009)
The Instationary Navier–Stokes Equations in Weighted Bessel-Potential SpacesJournal of Mathematical Fluid Mechanics, 11
-
R. Farwig, H. Kozono, H. Sohr (2007)
On the Helmholtz decomposition in general unbounded domainsArchiv der Mathematik, 88
-
H. Triebel (1978)
Interpolation Theory, Function Spaces, Differential Operators -
R. Farwig, H. Kozono, H. Sohr (2007)
Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous dataJournal of The Mathematical Society of Japan, 59
-
H. Amann (2002)
Nonhomogeneous Navier—Stokes Equations with Integrable Low—Regularity Data -
G. Lumer, H. Amann (2008)
Functional analysis and evolution equations : the Gunter Lumer volume -
K. Schumacher (2008)
Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spacesANNALI DELL'UNIVERSITA' DI FERRARA, 54
-
R. Farwig, H. Sohr (2009)
Optimal initial value conditions for the existence of local strong solutions of the Navier–Stokes equationsMathematische Annalen, 345
-
R. Farwig, H. Kozono, H. Sohr (2009)
On the Stokes operator in general unbounded domainsHokkaido Mathematical Journal, 38
-
K. Schumacher (2008)
Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spacesAnn. Univ. Ferrara Sez. VII Sci. Mat., 54
-
G. Galdi, C. Simader, H. Sohr (2005)
A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W−1/q,qMathematische Annalen, 331
-
R. Farwig, Paul Riechwald (2015)
Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domainsDiscrete and Continuous Dynamical Systems - Series S, 9
-
K. Schumacher (2007)
The Navier-Stokes Equations with Low-Regularity Data in Weighted Function Spaces -
C. Amrouche, V. Girault (1994)
Decomposition of vector spaces and application to the Stokes problem in arbitrary dimensionCzechoslovak Mathematical Journal, 44
-
P.F. Riechwald (2012)
Interpolation of sum and intersection spaces of L q -type and applications to the Stokes problem in general unbounded domainsAnn. Univ. Ferrara Sez. VII Sci. Mat., 58
-
(2001)
TheNavier–Stokes Equations. AnElementary FunctionalAnalyticApproach -
K. Schumacher (2009)
The instationary Stokes equations in weighted Bessel-potential spacesJournal of Evolution Equations, 9
-
V. Maslennikova, M. Bogovskiï (1986)
Elliptic boundary value problems in unbounded domains with noncompact and nonsmooth boundariesRendiconti del Seminario Matematico e Fisico di Milano, 56
-
R. Farwig, H. Sohr, W. Varnhorn (2009)
On optimal initial value conditions for local strong solutions of the Navier–Stokes equationsAnn. Univ. Ferrara, Sez. VII, Sci. Mat., 55
-
R. Farwig, G. Galdi, H. Sohr (2006)
A New Class of Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous DataJournal of Mathematical Fluid Mechanics, 8
-
(1986)
Decomposition of L p( ; Rn) into the direct sum of subspaces of solenoidal and potential vector fields -
Journal of Nonlinear Mathematical Physics Volume 10, Supplement 1 (2003), 1--11 -
(2002)
Nonlinear Problems in Mathematical Physics and Related Problems II -
R. Farwig, H. Kozono, H. Sohr (2005)
AnLq-approach to Stokes and Navier-Stokes equations in general domainsActa Mathematica, 195
-
R. Farwig, H. Sohr (2010)
On the existence of local strong solutions for the Navier–Stokes equations in completely general domainsNonlinear Analysis-theory Methods & Applications, 73
-
P. Kunstmann (2010)
Navier-stokes equations on unbounded domains with rough initial dataCzechoslovak Mathematical Journal, 60
-
H. Amann (2000)
On the Strong Solvability of the Navier—Stokes EquationsJournal of Mathematical Fluid Mechanics, 2
-
Paul Riechwald (2012)
Interpolation of sum and intersection spaces of Lq-type and applications to the Stokes problem in general unbounded domainsANNALI DELL'UNIVERSITA' DI FERRARA, 58
-
(2009)
Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier–Stokes equations in Rn -
P. Kunstmann (2008)
H∞-calculus for the Stokes operator on unbounded domainsArchiv der Mathematik, 91
-
R. Farwig, H. Sohr, W. Varnhorn (2009)
On optimal initial value conditions for local strong solutions of the Navier–Stokes equationsANNALI DELL'UNIVERSITA' DI FERRARA, 55
-
R. Farwig, G. Galdi, H. Sohr (2005)
Very Weak Solutions of Stationary and Instationary Navier-Stokes Equations with Nonhomogeneous Data
- Publisher
- Springer Journals
- Copyright
- Copyright © 2015 by Springer Basel
- Subject
- Mathematics; Analysis
- ISSN
- 1424-3199
- eISSN
- 1424-3202
- DOI
- 10.1007/s00028-014-0258-y
- Publisher site
- See Article on Publisher Site
Abstract
We consider very weak instationary solutions u of the Navier–Stokes system in general unbounded domains $${\Omega \subset \mathbb{R}^n}$$ Ω ⊂ R n , $${n \geq 3}$$ n ≥ 3 , with smooth boundary, i.e., u solves the Navier–Stokes system in the sense of distributions and $${ u \in L^r (0,T;\tilde{L}^q(\Omega))}$$ u ∈ L r ( 0 , T ; L ~ q ( Ω ) ) where $${\frac{2}{r} + \frac{n}{q} =1}$$ 2 r + n q = 1 , 2 < r < ∞. Solutions of this class have no differentiability properties and in general are not weak solutions in the sense of Leray–Hopf. However, they lie in the so-called Serrin class $${L^r(0,T;\tilde{L}^q(\Omega))}$$ L r ( 0 , T ; L ~ q ( Ω ) ) yielding uniqueness. To deal with the unboundedness of the domain, we work in the spaces $${\tilde{L}^q(\Omega)}$$ L ~ q ( Ω ) (instead of $${L^q(\Omega)}$$ L q ( Ω ) ) defined as $${L^q \cap L^2}$$ L q ∩ L 2 when $${q \geq 2}$$ q ≥ 2 but as L q + L 2 when 1 < q < 2. The proofs are strongly based on duality arguments and the properties of the spaces $${\tilde{L}^q(\Omega)}$$ L ~ q ( Ω ) .
Journal
Journal of Evolution Equations – Springer Journals
Published: Jun 1, 2015