# American Institute of Mathematical Sciences

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

 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.

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

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##### 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
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