[Critères de régularité des solutions faibles des équations de Navier–Stokes fondés sur les projections spectrales de la vorticité]
Soit la résolution spectrale de lʼidentité associée à lʼopérateur auto-adjoint curl dans lʼespace , et soit une fonction à valeurs réelles définie sur . On note , puis, lorsque v est solution faible du problème de condition initiale de Navier–Stokes dans , . On établit alors la régularité de v sous certaines conditions imposées à a et, ou bien à , ou bien à sa troisième composante seulement.
We denote , where is the spectral resolution of identity associated with the self-adjoint operator curl in the space . Further, we denote , where v is a weak solution to the Navier–Stokes initial value problem in . We assume that is a real function in . We show that certain conditions imposed on function a and , or only on the third component of , imply regularity of solution v.
Accepté le :
Publié le :
Jiří Neustupa 1 ; Patrick Penel 2
@article{CRMATH_2012__350_11-12_597_0, author = {Ji\v{r}{\'\i} Neustupa and Patrick Penel}, title = {Regularity criteria for weak solutions to the {Navier{\textendash}Stokes} equations based on spectral projections of vorticity}, journal = {Comptes Rendus. Math\'ematique}, pages = {597--602}, publisher = {Elsevier}, volume = {350}, number = {11-12}, year = {2012}, doi = {10.1016/j.crma.2012.06.008}, language = {en}, }
TY - JOUR AU - Jiří Neustupa AU - Patrick Penel TI - Regularity criteria for weak solutions to the Navier–Stokes equations based on spectral projections of vorticity JO - Comptes Rendus. Mathématique PY - 2012 SP - 597 EP - 602 VL - 350 IS - 11-12 PB - Elsevier DO - 10.1016/j.crma.2012.06.008 LA - en ID - CRMATH_2012__350_11-12_597_0 ER -
%0 Journal Article %A Jiří Neustupa %A Patrick Penel %T Regularity criteria for weak solutions to the Navier–Stokes equations based on spectral projections of vorticity %J Comptes Rendus. Mathématique %D 2012 %P 597-602 %V 350 %N 11-12 %I Elsevier %R 10.1016/j.crma.2012.06.008 %G en %F CRMATH_2012__350_11-12_597_0
Jiří Neustupa; Patrick Penel. Regularity criteria for weak solutions to the Navier–Stokes equations based on spectral projections of vorticity. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 597-602. doi : 10.1016/j.crma.2012.06.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.06.008/
[1] A new regularity class for the Navier–Stokes equations in , Chin. Ann. Math. Ser. B, Volume 16 (1995), pp. 407-412
[2] Regularity criteria for the three dimensional Navier–Stokes equations, Indiana Univ. Math. J., Volume 57 (2008) no. 6, pp. 2643-2661
[3] Regularity of solutions to the Navier–Stokes equation, Electron. J. Differential Equations, Volume 5 (1999), pp. 1-7
[4] Spectral analysis of a Stokes-type operator arising from flow around a rotating body, J. Math. Soc. Japan, Volume 63 (2011) no. 1, pp. 163-194
[5] An introduction to the Navier–Stokes initial–boundary value problem (G.P. Galdi; J. Heywood; R. Rannacher, eds.), Fundamental Directions in Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics, Birkhäuser Verlag, Basel, 2000, pp. 1-98
[6] One component regularity for the Navier–Stokes equations, Nonlinearity, Volume 19 (2006) no. 2, pp. 453-469
[7] Navier–Stokes equations with regularity in one direction, J. Math. Phys., Volume 48 (2007) no. 6, p. 065203 (10 pp)
[8] Regularity of a suitable weak solution of the Navier–Stokes equations as a consequence of regularity of one velocity component (A. Sequeira; H. Beirao da Veiga; J.H. Videman, eds.), Applied Nonlinear Analysis, Kluwer Academic/Plenum Publishers, New York, 1999, pp. 391-402
[9] An interior regularity of a weak solution to the Navier–Stokes equations in dependence on one component of velocity (G.P. Galdi; R. Rannacher, eds.), Topics in Mathematical Fluid Mechanics, Quaderni di Matematica, vol. 10, Dipartimento di Matematica della SUN, Napoli, 2003, pp. 163-183
[10] Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier–Stokes equations (J. Neustupa; P. Penel, eds.), Mathematical Fluid Mechanics, Recent Results and Open Questions, Birkhäuser Verlag, Basel–Boston–Berlin, 2001, pp. 237-268
[11] J. Neustupa, P. Penel, Regularity of a weak solution to the Navier–Stokes equation via one component of a spectral projection of vorticity, preprint, 2012.
[12] Some new regularity criteria for the Navier–Stokes equations containing the gradient of velocity, Appl. Math., Volume 49 (2004) no. 5, pp. 483-493
[13] P. Penel, M. Pokorný, Improvement of some anisotropic regularity criteria for the Navier–Stokes equations, Discrete Contin. Dynam. Systems, Ser. S, in press.
[14] The Navier–Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel–Boston–Berlin, 2001
[15] On the regularity of the solutions of the Navier–Stokes equations via one velocity component, Nonlinearity, Volume 23 (2010) no. 5, pp. 1097-1107
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