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October  2013, 6(5): 1391-1400. doi: 10.3934/dcdss.2013.6.1391

A note on local interior regularity of a suitable weak solution to the Navier--Stokes problem

Jiří Neustupa 1,

1. 

Czech Academy of Sciences, Mathematical Institute, Žitná 25, 115 67 Prague 1

Received  December 2011 Revised  January 2012 Published  March 2013

We formulate a criterion which guarantees a local regularity of a suitable weak solution $v$ to the Navier--Stokes equations (in the sense of L. Caffarelli, R. Kohn and L. Nirenberg [3]). The criterion shows that if $(x_0,t_0)$ is a singular point of solution $v$ then the $L^3$--norm of $v$ concentrates in an amount greater than or equal to some $\epsilon>0$ in an arbitrarily small neighbourhood of $x_0$ at all times $t$ in some left neighbourhood of $t_0$. As a partial result, we prove that a localized solution satisfies the strong energy inequality.
Keywords: Navier-Stokes equations, regularity., suitable weak solution.
Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D0.
Citation: 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
References:
[1]

H. Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 16-98. doi: 10.1007/s000210050018.  Google Scholar

[2]

W. Borchers and H. Sohr, On the equations rot $v=g$ and div $u=f$ with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67-87.  Google Scholar

[3]

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

[4]

L. Iskauriaza, G. Serëgin and V. Šverák, $L_{3,\infty}$-solutions of the Navier-Stokes equations and backward uniqueness, Russian Math. Surveys, 58 (2003), 211-250. doi: 10.1070/RM2003v058n02ABEH000609.  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), 21-53. doi: 10.1007/BF02588049.  Google Scholar

[6]

R. Farwig, H. Kozono and H. Sohr, Energy-based regularity criteria for the Navier-Stokes equations, J. Math. Fluid Mech., 11 (2009), 428-442. doi: 10.1007/s00021-008-0267-0.  Google Scholar

[7]

R. Farwig, H. Kozono and H. Sohr, Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition, in "Parabolic and Navier-Stokes Equations, Part 1," Banach Center Publ., 81, Polish Acad. Sci. Inst. Math., Warsaw, (2008), 175-184. doi: 10.4064/bc81-0-11.  Google Scholar

[8]

C. Foiaş and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl. (9), 58 (1979), 339-368.  Google Scholar

[9]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I. Linearized Steady Problems'," Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[10]

G. P. Galdi, An Introduction to the Navier-Stokes initial-boundary value problem, in "Fundamental Directions in Mathematical Fluid Mechanics" (eds. G. P. Galdi, J. Heywood and R. Rannacher), Advances in Mathematical Fluid Mechanics, Birkhäuser, Basel, (2000), 1-70.  Google Scholar

[11]

E. Hopf, Über die Anfangswertaufgabe für die Hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.  Google Scholar

[12]

T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbbR^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[13]

H. Kozono and H. Sohr, Remark on uniqueness of weak solutions to the Navier-Stokes equations, Analysis, 16 (1996), 255-271.  Google Scholar

[14]

H. Kozono, Uniqueness and regularity of weak solutions to the Navier-Stokes equations, in "Recent Topics on Mathematical Theory of Viscous Incompressible Fluid" (Tsukuba, 1996), Lecture Notes in Num. and Appl. Anal., 16, Kinokuniya, Tokyo, (1998), 161-208.  Google Scholar

[15]

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

[16]

F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. on Pure and 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

[17]

A. Mahalov, B. Nicolaenko and G. Seregin, New sufficient conditions of local regularity for solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 10 (2008), 106-125. doi: 10.1007/s00021-006-0220-z.  Google Scholar

[18]

J. Nečas and J. Neustupa, New conditions for local regularity of a suitable weak solution to the Navier-Stokes equations, J. Math. Fluid Mech., 4 (2002), 237-256. doi: 10.1007/s00021-002-8544-9.  Google Scholar

[19]

J. Neustupa, Partial regularity of weak solutions to the Navier-Stokes equations in the class $L^{\infty}(0,T; L^3(\Omega)^3)$, J. Math. Fluid Mech., 1 (1999), 309-325. doi: 10.1007/s000210050013.  Google Scholar

[20]

J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations, in "Mathematical Fluid Mechanics" (eds. J. Neustupa and P. Penel), Advances in Mathematical Fluid Mechanics, Birkhauser, Basel, (2001), 237-265.  Google Scholar

[21]

G. Seregin and V. Šverák, On smoothness of suitable weak solutions to the Navier-Stokes equations, J. of Math. Sci. (N. Y.), 130 (2005), 4884-4892. doi: 10.1007/s10958-005-0383-9.  Google Scholar

[22]

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

[23]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal., 9 (1962), 187-195.  Google Scholar

[24]

J. Wolf, A direct proof of the Caffarelli-Kohn-Nirenberg theorem, in "Parabolic and Navier-Stokes Equations, Part 2," Banach Center Publ., 81, Polish Acad. Sci. Inst. Math., Warsaw, (2008), 533-552. doi: 10.4064/bc81-0-34.  Google Scholar

[25]

J. Wolf, A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations, in "Advances in Mathematical Fluid Mechanics" (eds. R. Rannacher and A. Sequeira), Springer, Berlin, (2010), 613-630.  Google Scholar

show all references

References:
[1]

H. Amann, On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech., 2 (2000), 16-98. doi: 10.1007/s000210050018.  Google Scholar

[2]

W. Borchers and H. Sohr, On the equations rot $v=g$ and div $u=f$ with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67-87.  Google Scholar

[3]

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

[4]

L. Iskauriaza, G. Serëgin and V. Šverák, $L_{3,\infty}$-solutions of the Navier-Stokes equations and backward uniqueness, Russian Math. Surveys, 58 (2003), 211-250. doi: 10.1070/RM2003v058n02ABEH000609.  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), 21-53. doi: 10.1007/BF02588049.  Google Scholar

[6]

R. Farwig, H. Kozono and H. Sohr, Energy-based regularity criteria for the Navier-Stokes equations, J. Math. Fluid Mech., 11 (2009), 428-442. doi: 10.1007/s00021-008-0267-0.  Google Scholar

[7]

R. Farwig, H. Kozono and H. Sohr, Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition, in "Parabolic and Navier-Stokes Equations, Part 1," Banach Center Publ., 81, Polish Acad. Sci. Inst. Math., Warsaw, (2008), 175-184. doi: 10.4064/bc81-0-11.  Google Scholar

[8]

C. Foiaş and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl. (9), 58 (1979), 339-368.  Google Scholar

[9]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I. Linearized Steady Problems'," Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[10]

G. P. Galdi, An Introduction to the Navier-Stokes initial-boundary value problem, in "Fundamental Directions in Mathematical Fluid Mechanics" (eds. G. P. Galdi, J. Heywood and R. Rannacher), Advances in Mathematical Fluid Mechanics, Birkhäuser, Basel, (2000), 1-70.  Google Scholar

[11]

E. Hopf, Über die Anfangswertaufgabe für die Hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.  Google Scholar

[12]

T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbbR^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.  Google Scholar

[13]

H. Kozono and H. Sohr, Remark on uniqueness of weak solutions to the Navier-Stokes equations, Analysis, 16 (1996), 255-271.  Google Scholar

[14]

H. Kozono, Uniqueness and regularity of weak solutions to the Navier-Stokes equations, in "Recent Topics on Mathematical Theory of Viscous Incompressible Fluid" (Tsukuba, 1996), Lecture Notes in Num. and Appl. Anal., 16, Kinokuniya, Tokyo, (1998), 161-208.  Google Scholar

[15]

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

[16]

F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. on Pure and 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

[17]

A. Mahalov, B. Nicolaenko and G. Seregin, New sufficient conditions of local regularity for solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 10 (2008), 106-125. doi: 10.1007/s00021-006-0220-z.  Google Scholar

[18]

J. Nečas and J. Neustupa, New conditions for local regularity of a suitable weak solution to the Navier-Stokes equations, J. Math. Fluid Mech., 4 (2002), 237-256. doi: 10.1007/s00021-002-8544-9.  Google Scholar

[19]

J. Neustupa, Partial regularity of weak solutions to the Navier-Stokes equations in the class $L^{\infty}(0,T; L^3(\Omega)^3)$, J. Math. Fluid Mech., 1 (1999), 309-325. doi: 10.1007/s000210050013.  Google Scholar

[20]

J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations, in "Mathematical Fluid Mechanics" (eds. J. Neustupa and P. Penel), Advances in Mathematical Fluid Mechanics, Birkhauser, Basel, (2001), 237-265.  Google Scholar

[21]

G. Seregin and V. Šverák, On smoothness of suitable weak solutions to the Navier-Stokes equations, J. of Math. Sci. (N. Y.), 130 (2005), 4884-4892. doi: 10.1007/s10958-005-0383-9.  Google Scholar

[22]

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

[23]

J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal., 9 (1962), 187-195.  Google Scholar

[24]

J. Wolf, A direct proof of the Caffarelli-Kohn-Nirenberg theorem, in "Parabolic and Navier-Stokes Equations, Part 2," Banach Center Publ., 81, Polish Acad. Sci. Inst. Math., Warsaw, (2008), 533-552. doi: 10.4064/bc81-0-34.  Google Scholar

[25]

J. Wolf, A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations, in "Advances in Mathematical Fluid Mechanics" (eds. R. Rannacher and A. Sequeira), Springer, Berlin, (2010), 613-630.  Google Scholar

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