American Institute of Mathematical Sciences

Advanced
March  2017, 37(3): 1283-1294. doi: 10.3934/dcds.2017053

A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem

Francesca Crispo , and  Paolo Maremonti

Dipartimento di Matematica e Fisica, Seconda Università degli Studi di Napoli, via Vivaldi, 43 -Caserta, I 81100, Italy

* Corresponding author: P. Maremonti

Received  March 2016 Revised  October 2016 Published  December 2016

Fund Project: This research was partly supported by GNFM-INdAM, and by MIUR via the PRIN 2012 "Nonlinear Hyperbolic Partial Differential Equations, Dispersive and Transport Equations: Theoretical and Applicative Aspects".

Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg [1], as a corollary, under suitable assumptions of local character on the initial data, we investigate the behavior in time of the $L_{loc}^\infty$-norm of the solution in a neighborhood of $t=0$. The behavior is the same as for the resolvent operator associated to the Stokes operator.

Keywords: Navier-Stokes equations, suitable weak solutions, partial regularity.
Mathematics Subject Classification: Primary:35Q30, 35B65;Secondary:76D03.
Citation: 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, 2017, 37 (3) : 1283-1294. doi: 10.3934/dcds.2017053
References:
[1]

L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.  doi: 10.1002/cpa.3160350604.  Google Scholar

[2]

F. Crispo and P. Maremonti, On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem, Nonlinearity, 29 (2016), 1355-1383.  doi: 10.1088/0951-7715/29/4/1355.  Google Scholar

[3]

R.Farwig,Partial regularity and weighted energy estimates of global weak solutions of the Navier-Stokes system, Progress in partial differential equations: The Metz surveys, 4 (1996), 205–215, Pitman Res. Notes Math. Ser.,345,Longman, Harlow.  Google Scholar

[4]

O. A. Ladyzhenskaya and G. A. Seregin, On partial regularity of suitable weaks olutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 356-387.  doi: 10.1007/s000210050015.  Google Scholar

[5]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[6]

F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm.Pure Appl. Math., 51 (1998), 241-257.  doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.  Google Scholar

[7]

P. Maremonti, Partial regularity of a generalized solution to the Navier-Stokes equations in exterior domain, Comm. Math. Phys., 110 (1987), 75-87.  doi: 10.1007/BF01209017.  Google Scholar

[8]

P. Maremonti, On the asymptotic behavior of the $L^2$-norm of suitable weak solutions to the Navier-Stokes equations in three-dimensional exterior domains, Comm. Math. Phys., 118 (1988), 385-400.  doi: 10.1007/BF01466723.  Google Scholar

[9]

P. Maremonti, Weak solutions to the Navier-Stokes equations with data in $\mathbb L(3, \infty )$, to appear in the Proceedings "Mathematical Nonlinear Phenomena: Analysis and Computation" (2015) Springer. Google Scholar

[10]

P. Maremonti and V. A. Solonnikov, An estimate for the solutions of Stokes equations in exterior domains, Zap. Nauch. Sem. LOMI, 180 (1990), 105-120, trasl.  doi: 10.1007/BF01249337.  Google Scholar

[11]

P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains, Ann. Sc. Norm. Sup. Pisa, 24 (1997), 395-449.   Google Scholar

[12]

J. A. Mauro, Some analytic questions in mathematical physic problems, Pliska Stud. Math. Bulgar., 23 (2014), 95-118.   Google Scholar

[13]

V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys., 55 (1977), 97-112.  doi: 10.1007/BF01626512.  Google Scholar

[14]

G. A. Seregin, Local regularity for suitable weak solutions of the Navier-Stokes equations, Russian Math. Surveys, 62 (2007), 595-614.  doi: 10.1070/RM2007v062n03ABEH004415.  Google Scholar

[15]

E. A. Stein, Note on singular integrals, Proc. Amer. Math. Soc., 8 (1957), 250-254.  doi: 10.1090/S0002-9939-1957-0088606-8.  Google Scholar

[16]

A. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations, Nonlin. Diff. Eq. Appl., 14 (2007), 753-785.  doi: 10.1007/s00030-007-6001-4.  Google Scholar

show all references

References:
[1]

L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.  doi: 10.1002/cpa.3160350604.  Google Scholar

[2]

F. Crispo and P. Maremonti, On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem, Nonlinearity, 29 (2016), 1355-1383.  doi: 10.1088/0951-7715/29/4/1355.  Google Scholar

[3]

R.Farwig,Partial regularity and weighted energy estimates of global weak solutions of the Navier-Stokes system, Progress in partial differential equations: The Metz surveys, 4 (1996), 205–215, Pitman Res. Notes Math. Ser.,345,Longman, Harlow.  Google Scholar

[4]

O. A. Ladyzhenskaya and G. A. Seregin, On partial regularity of suitable weaks olutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 356-387.  doi: 10.1007/s000210050015.  Google Scholar

[5]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[6]

F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm.Pure Appl. Math., 51 (1998), 241-257.  doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A.  Google Scholar

[7]

P. Maremonti, Partial regularity of a generalized solution to the Navier-Stokes equations in exterior domain, Comm. Math. Phys., 110 (1987), 75-87.  doi: 10.1007/BF01209017.  Google Scholar

[8]

P. Maremonti, On the asymptotic behavior of the $L^2$-norm of suitable weak solutions to the Navier-Stokes equations in three-dimensional exterior domains, Comm. Math. Phys., 118 (1988), 385-400.  doi: 10.1007/BF01466723.  Google Scholar

[9]

P. Maremonti, Weak solutions to the Navier-Stokes equations with data in $\mathbb L(3, \infty )$, to appear in the Proceedings "Mathematical Nonlinear Phenomena: Analysis and Computation" (2015) Springer. Google Scholar

[10]

P. Maremonti and V. A. Solonnikov, An estimate for the solutions of Stokes equations in exterior domains, Zap. Nauch. Sem. LOMI, 180 (1990), 105-120, trasl.  doi: 10.1007/BF01249337.  Google Scholar

[11]

P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains, Ann. Sc. Norm. Sup. Pisa, 24 (1997), 395-449.   Google Scholar

[12]

J. A. Mauro, Some analytic questions in mathematical physic problems, Pliska Stud. Math. Bulgar., 23 (2014), 95-118.   Google Scholar

[13]

V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys., 55 (1977), 97-112.  doi: 10.1007/BF01626512.  Google Scholar

[14]

G. A. Seregin, Local regularity for suitable weak solutions of the Navier-Stokes equations, Russian Math. Surveys, 62 (2007), 595-614.  doi: 10.1070/RM2007v062n03ABEH004415.  Google Scholar

[15]

E. A. Stein, Note on singular integrals, Proc. Amer. Math. Soc., 8 (1957), 250-254.  doi: 10.1090/S0002-9939-1957-0088606-8.  Google Scholar

[16]

A. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations, Nonlin. Diff. Eq. Appl., 14 (2007), 753-785.  doi: 10.1007/s00030-007-6001-4.  Google Scholar

[1]

Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033
[2]

Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717
[3]

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
[4]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161
[5]

Jiří Neustupa. A note on local interior regularity of a suitable weak solution to the Navier--Stokes problem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1391-1400. doi: 10.3934/dcdss.2013.6.1391
[6]

Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4397-4419. doi: 10.3934/dcds.2021041
[7]

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
[8]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319
[9]

Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185
[10]

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
[11]

Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021051
[12]

Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279
[13]

Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137
[14]

Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837
[15]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159
[16]

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
[17]

Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141
[18]

Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228
[19]

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
[20]

Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (73)
  • HTML views (54)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences