Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains

Reinhard Farwig; Paul Felix  Riechwald

doi:10.3934/dcdss.2016.9.157

Article Contents

2016, Volume 9, Issue 1: 157-172. Doi: 10.3934/dcdss.2016.9.157

This issue Previous Article Solidification and separation in saline water Next Article On weak solutions to a diffuse interface model of a binary mixture of compressible fluids

Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains

Reinhard Farwig^1, and
Paul Felix Riechwald^2,

1.
Department of Mathematics and Center of Smart Interfaces (CSI), Technische Universität Darmstadt, 64289 Darmstadt
2.
Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt

Received: August 31, 2014

Revised: January 31, 2015

Published: January 2016

Abstract Related Papers Cited by

Abstract

We consider weak solutions of the instationary Navier-Stokes system in general unbounded smooth domains $\Omega\subset \mathbb{R}^3$ and discuss several criteria to prove that the weak solution is locally or globally in time a strong solution in the sense of Serrin. Since the usual Stokes operator cannot be defined on all types of unbounded domains we have to replace the space $L^q(\Omega)$, $q>2$, by $\tilde L^q(\Omega) = L^q(\Omega) \cap L^2(\Omega)$ and Serrin's class $L^r(0,T;L^q(\Omega))$ by $L^r(0,T;\tilde L^q(\Omega))$ where $2< r <\infty$, $3< q <\infty$ and $\frac{2}{r} + \frac{3}{q} =1$.

Keywords:

Mathematics Subject Classification: Primary: 35B65, 35Q30; Secondary: 76D05.

Citation:

Related Papers

Cited by

References

[1]	H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I: Abstract Linear Theory, Monographs in Mathematics, 89, Birkhäuser, Basel-Boston-Berlin, 1995.doi: 10.1007/978-3-0348-9221-6.
[2]	M. E. Bogovskij and V. N. Maslennikova, Elliptic boundary value problems in unbounded domains with noncompact and nonsmooth boundaries, Sem. Mat. Fis. Milano, 56 (1986), 125-138.doi: 10.1007/BF02925141.
[3]	M. E. Bogovskij, Decomposition of $L_p(\Omega;R^n)$ into the direct sum of subspaces of solenoidal and potential vector fields, Math. Dokl., 33 (1986), 161-165.
[4]	R. Farwig, G. P. Galdi and H. Sohr, A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data, J. Math. Fluid Mech., 8 (2006), 423-444.doi: 10.1007/s00021-005-0182-6.
[5]	R. Farwig, H. Kozono and H. Sohr, An $L^q$-approach to Stokes and Navier-Stokes equations in general domains, Acta Math., 195 (2005), 21-53.doi: 10.1007/BF02588049.
[6]	R. Farwig, H. Kozono and H. Sohr, Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data, J. Math. Soc. Japan, 59 (2007), 127-150.doi: 10.2969/jmsj/1180135504.
[7]	R. Farwig, H. Kozono and H. Sohr, On the Helmholtz decomposition in general unbounded domains, Arch. Math., 88 (2007), 239-248.doi: 10.1007/s00013-006-1910-8.
[8]	R. Farwig, H. Kozono and H. Sohr, The Stokes resolvent problem in general unbounded domains, in Kyoto Conference on the Navier-Stokes Equations and their Applications, RIMS Kôkyûroku Bessatsu, Res. Inst. Math. Sci., B1, Kyoto, 2007, 79-91.
[9]	R. Farwig, H. Kozono and H. Sohr, Maximal regularity of the Stokes operator in general unbounded domains, in Functional Analysis and Evolution Equations. The Günter Lumer Volume (eds. H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise and J. von Below), Birkhäuser Verlag, Basel, 2008, 257-272.doi: 10.1007/978-3-7643-7794-6_17.
[10]	R. Farwig, H. Kozono and H. Sohr, On the Stokes operator in general unbounded domains, Hokkaido Math. J., 38 (2009), 111-136.doi: 10.14492/hokmj/1248787007.
[11]	R. Farwig and P. F. Riechwald, Very weak solutions to the Navier-Stokes system in general unbounded domains, J. Evol. Equ., 15 (2015), 253-279.doi: 10.1007/s00028-014-0258-y.
[12]	R. Farwig, H. Sohr and W. Varnhorn, Extensions of Serrin's uniqueness and regularity conditions for the Navier-Stokes equations, J. Math. Fluid Mech., 14 (2012), 529-540.doi: 10.1007/s00021-011-0078-6.
[13]	H. Kozono and H. Sohr, Remark on uniqueness of weak solutions to the Navier-Stokes equations, Analysis, 16 (1996), 255-271.doi: 10.1524/anly.1996.16.3.255.
[14]	P. C. Kunstmann, $H^{\infty}$-calculus for the Stokes operator on unbounded domains, Arch. Math., 91 (2008), 178-186.doi: 10.1007/s00013-008-2621-0.
[15]	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 (2012), 167-181.doi: 10.1007/s11565-011-0140-6.
[16]	H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Verlag, Basel, 2001.
[17]	H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publ., Amsterdam, 1978.