February  2016, 9(1): 157-172. doi: 10.3934/dcdss.2016.9.157

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

1. 

Department of Mathematics and Center of Smart Interfaces (CSI), Technische Universität Darmstadt, 64289 Darmstadt

2. 

Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, Germany

Received  September 2014 Revised  February 2015 Published  December 2015

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 $\frac{2}{r} + \frac{3}{q} =1$.
Citation: Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157
References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I: Abstract Linear Theory,, Monographs in Mathematics, (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar

[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. doi: 10.1007/BF02925141. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00021-005-0182-6. Google Scholar

[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. doi: 10.1007/BF02588049. Google Scholar

[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. doi: 10.2969/jmsj/1180135504. Google Scholar

[7]

R. Farwig, H. Kozono and H. Sohr, On the Helmholtz decomposition in general unbounded domains,, Arch. Math., 88 (2007), 239. doi: 10.1007/s00013-006-1910-8. Google Scholar

[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, (2007), 79. Google Scholar

[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, (2008), 257. doi: 10.1007/978-3-7643-7794-6_17. Google Scholar

[10]

R. Farwig, H. Kozono and H. Sohr, On the Stokes operator in general unbounded domains,, Hokkaido Math. J., 38 (2009), 111. doi: 10.14492/hokmj/1248787007. Google Scholar

[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. doi: 10.1007/s00028-014-0258-y. Google Scholar

[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. doi: 10.1007/s00021-011-0078-6. Google Scholar

[13]

H. Kozono and H. Sohr, Remark on uniqueness of weak solutions to the Navier-Stokes equations,, Analysis, 16 (1996), 255. doi: 10.1524/anly.1996.16.3.255. Google Scholar

[14]

P. C. Kunstmann, $H^{\infty}$-calculus for the Stokes operator on unbounded domains,, Arch. Math., 91 (2008), 178. doi: 10.1007/s00013-008-2621-0. Google Scholar

[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. doi: 10.1007/s11565-011-0140-6. Google Scholar

[16]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach,, Birkhäuser Verlag, (2001). Google Scholar

[17]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland Publ., (1978). Google Scholar

show all references

References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I: Abstract Linear Theory,, Monographs in Mathematics, (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar

[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. doi: 10.1007/BF02925141. Google Scholar

[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. Google Scholar

[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. doi: 10.1007/s00021-005-0182-6. Google Scholar

[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. doi: 10.1007/BF02588049. Google Scholar

[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. doi: 10.2969/jmsj/1180135504. Google Scholar

[7]

R. Farwig, H. Kozono and H. Sohr, On the Helmholtz decomposition in general unbounded domains,, Arch. Math., 88 (2007), 239. doi: 10.1007/s00013-006-1910-8. Google Scholar

[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, (2007), 79. Google Scholar

[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, (2008), 257. doi: 10.1007/978-3-7643-7794-6_17. Google Scholar

[10]

R. Farwig, H. Kozono and H. Sohr, On the Stokes operator in general unbounded domains,, Hokkaido Math. J., 38 (2009), 111. doi: 10.14492/hokmj/1248787007. Google Scholar

[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. doi: 10.1007/s00028-014-0258-y. Google Scholar

[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. doi: 10.1007/s00021-011-0078-6. Google Scholar

[13]

H. Kozono and H. Sohr, Remark on uniqueness of weak solutions to the Navier-Stokes equations,, Analysis, 16 (1996), 255. doi: 10.1524/anly.1996.16.3.255. Google Scholar

[14]

P. C. Kunstmann, $H^{\infty}$-calculus for the Stokes operator on unbounded domains,, Arch. Math., 91 (2008), 178. doi: 10.1007/s00013-008-2621-0. Google Scholar

[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. doi: 10.1007/s11565-011-0140-6. Google Scholar

[16]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach,, Birkhäuser Verlag, (2001). Google Scholar

[17]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland Publ., (1978). Google Scholar

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