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A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law
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 |
Starting from the partial regularity results for suitable weak solutions to the Navier-Stokes Cauchy problem by Caffarelli, Kohn and Nirenberg [
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. |
[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. |
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. |
[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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