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

  • * Corresponding author: P. Maremonti

    * Corresponding author: P. Maremonti 
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".
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  • 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.

    Mathematics Subject Classification: Primary:35Q30, 35B65;Secondary:76D03.


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  • [1] L. CaffarelliR. 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.
    [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.
    [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.
    [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.
    [5] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.
    [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.
    [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.
    [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.
    [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.
    [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.
    [11] P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains, Ann. Sc. Norm. Sup. Pisa, 24 (1997), 395-449. 
    [12] J. A. Mauro, Some analytic questions in mathematical physic problems, Pliska Stud. Math. Bulgar., 23 (2014), 95-118. 
    [13] V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys., 55 (1977), 97-112.  doi: 10.1007/BF01626512.
    [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.
    [15] E. A. Stein, Note on singular integrals, Proc. Amer. Math. Soc., 8 (1957), 250-254.  doi: 10.1090/S0002-9939-1957-0088606-8.
    [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.
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