A regularity criterion for positive part of radial component in the case of axially symmetric Navier-Stokes equations

• Autorzy: Kubica A.
• Języki publikacji: EN

Abstrakty

EN

We examine the conditional regularity of the solutions of Navier–Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of velocity satisfies a weighted Serrin type condition and in addition angular component satisfies some condition, then the solution is regular.

Słowa kluczowe

EN

Navier-Stokes equations; axial symmetry; regularity criteria; weighted spaces

PL

równania Naviera-Stokesa; symetria osiowa; kryteria regularności

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2015
• Tom: Vol. 48, nr 1
• Strony: 62--72
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Kubica A.

• Faculty of Mathematics and Information Science, Warsaw University af Technology, Pl. Politechniki 1 00-661 Warsaw

Bibliografia

• [1] D. Chae, J. Lee, On the regularity of the axisymmetric solutions of the Navier–Stokes equations, Math. Z. 239(4) (2002), 645–671.
• [2] C. C. Chen, R. M. Strain, T. P. Tsai, H. T. Yau, Lower bounds on the blow-up rate of the axisymmetric Navier–Stokes equations. II, Comm. Partial Differential Equations 34(1–3) (2009), 203–232.
• [3] O. Kreml, M. Pokorný, A regularity criterion for the angular velocity component in axisymmetric Navier–Stokes equations, Electron. J. Differential Equations 2007, No. 08, 10 pp.
• [4] A. Kubica, M. Pokorný, W. Zajączkowski, Remarks on regularity criteria for axially symmetric weak solutions to the Navier–Stokes equations, Math. Methods Appl. Sci. 35(3) (2012), 360–371.
• [5] O. A. Ladyzhenskaya, Unique global solvability of the three-dimensional Cauchy problem for the Navier–Stokes equations in the presence of axial symmetry, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 155–177.
• [6] S. Leonardi, J. Málek, J. Nečas, M. Pokorný, On axially symmetric flows in R3, Z. Anal. Anwendungen 18 (1999), 639–649.
• [7] S. Montgomery-Smith, Conditions implying regularity of the three dimensional Navier–Stokes equation, Appl. Math. 50(5) (2005), 451–464.
• [8] J. Neustupa, M. Pokorný, An interior regularity criterion for an axially symmetric suitable weak solution to the Navier–Stokes equations, J. Math. Fluid Mech. 2(4) (2000), 381–399.
• [9] M. Pokorný, A regularity criterion for the angular velocity component in the case of axisymmetric Navier–Stokes equations, Elliptic and Parabolic Problems (Rolduc/Gaeta, 2001), 233–242, World Sci. Publ., River Edge, NJ, 2002.
• [10] B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, 1736. Springer-Verlag, Berlin, 2000.
• [11] M. R. Uchovskii, B. I. Yudovich, Axially symmetric flows of an ideal and viscous fluid in the whole space (in Russian, also J. Appl. Math. Mech. 32 (1968), 52–61), Prikl. Mat. Mekh. 32 (1968), 59–69.