# American Institute of Mathematical Sciences

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
 [1] Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215 [2] Lukáš Poul. Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains. Conference Publications, 2007, 2007 (Special) : 834-843. doi: 10.3934/proc.2007.2007.834 [3] Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783 [4] Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic & Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545 [5] Zijin Li, Xinghong Pan. Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1333-1350. doi: 10.3934/cpaa.2019064 [6] Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 [7] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [8] Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 171-183. doi: 10.3934/dcds.2010.27.171 [9] Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161 [10] Wendong Wang, Liqun Zhang, Zhifei Zhang. On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2609-2627. doi: 10.3934/dcds.2018110 [11] Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033 [12] Francesca Crispo, Paolo Maremonti. A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1283-1294. doi: 10.3934/dcds.2017053 [13] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [14] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717 [15] Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715 [16] Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397 [17] Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079 [18] Giovanni P. Galdi. Existence and uniqueness of time-periodic solutions to the Navier-Stokes equations in the whole plane. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1237-1257. doi: 10.3934/dcdss.2013.6.1237 [19] Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613 [20] Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045

2018 Impact Factor: 0.545