Regularity criteria for weak solutions to the Navier–Stokes equations based on spectral projections of vorticity

Jiří Neustupa; Patrick Penel

doi:10.1016/j.crma.2012.06.008

Partial Differential Equations/Mathematical Problems in Mechanics

Regularity criteria for weak solutions to the Navier–Stokes equations based on spectral projections of vorticity
[Critères de régularité des solutions faibles des équations de Navier–Stokes fondés sur les projections spectrales de la vorticité]

Présenté par : Gérard Iooss

Jiří Neustupa ¹ ; Patrick Penel ²

¹ Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 115 67 Praha 1, Czech Republic
² Université du Sud, Toulon-Var, Mathématique, BP 20132, 83957 La Garde cedex, France

Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 597-602.

Résumé
Abstract

Soit ${E_{λ}}$ la résolution spectrale de lʼidentité associée à lʼopérateur auto-adjoint curl dans lʼespace $L_{σ}^{2} (R^{3})$ , et soit $a = a (t)$ une fonction à valeurs réelles définie sur $(0, T)$ . On note $P_{a}^{+} : = \int_{a}^{\infty} d E_{λ}$ , puis, lorsque v est solution faible du problème de condition initiale de Navier–Stokes dans $R^{3} \times (0, T)$ , $ω_{a}^{+} : = P_{a}^{+} curl v$ . On établit alors la régularité de v sous certaines conditions imposées à a et, ou bien à $ω_{a}^{+}$ , ou bien à sa troisième composante $ω_{a 3}^{+}$ seulement.

We denote $P_{a}^{+} : = \int_{a}^{\infty} d E_{λ}$ , where ${E_{λ}}$ is the spectral resolution of identity associated with the self-adjoint operator curl in the space $L_{σ}^{2} (R^{3})$ . Further, we denote $ω_{a}^{+} : = P_{a}^{+} curl v$ , where v is a weak solution to the Navier–Stokes initial value problem in $R^{3} \times (0, T)$ . We assume that $a = a (t)$ is a real function in $(0, T)$ . We show that certain conditions imposed on function a and $ω_{a}^{+}$ , or only on the third component $ω_{a 3}^{+}$ of $ω_{a}^{+}$ , imply regularity of solution v.

Reçu le : 2012-03-27
Accepté le : 2012-06-21
Publié le : 2012-06-01

DOI : 10.1016/j.crma.2012.06.008

Affiliations des auteurs :

Jiří Neustupa ¹ ; Patrick Penel ²

¹ Czech Academy of Sciences, Institute of Mathematics, Žitná 25, 115 67 Praha 1, Czech Republic
² Université du Sud, Toulon-Var, Mathématique, BP 20132, 83957 La Garde cedex, France

@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/

Bibliographie
Cité par

[1] H. Beirao da Veiga A new regularity class for the Navier–Stokes equations in $R^{n}$ , Chin. Ann. Math. Ser. B, Volume 16 (1995), pp. 407-412

[2] C. Cao; E.S. Titi Regularity criteria for the three dimensional Navier–Stokes equations, Indiana Univ. Math. J., Volume 57 (2008) no. 6, pp. 2643-2661

[3] D. Chae; H.J. Choe Regularity of solutions to the Navier–Stokes equation, Electron. J. Differential Equations, Volume 5 (1999), pp. 1-7

[4] R. Farwig; Š. Nečasová; J. Neustupa 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] G.P. Galdi 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] I. Kukavica; M. Ziane One component regularity for the Navier–Stokes equations, Nonlinearity, Volume 19 (2006) no. 2, pp. 453-469

[7] I. Kukavica; M. Ziane Navier–Stokes equations with regularity in one direction, J. Math. Phys., Volume 48 (2007) no. 6, p. 065203 (10 pp)

[8] J. Neustupa; P. Penel 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] J. Neustupa; A. Novotný; P. Penel 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] J. Neustupa; P. Penel 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] P. Penel; M. Pokorný 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] H. Sohr The Navier–Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel–Boston–Berlin, 2001

[15] Y. Zhou; M. Pokorný On the regularity of the solutions of the Navier–Stokes equations via one velocity component, Nonlinearity, Volume 23 (2010) no. 5, pp. 1097-1107

Myong-Hwan Ri Helicity and regularity of weak solutions to 3D Navier-Stokes equations, Annali dell'Università di Ferrara. Sezione VII. Scienze Matematiche, Volume 67 (2021) no. 2, pp. 435-445 | DOI:10.1007/s11565-021-00370-w | Zbl:1477.35134
Myong-Hwan Ri A regularity criterion for 3D Navier-Stokes equations via one component of velocity and vorticity, Annali dell'Università di Ferrara. Sezione VII. Scienze Matematiche, Volume 63 (2017) no. 2, pp. 353-363 | DOI:10.1007/s11565-017-0274-2 | Zbl:1386.35330
Zhengguang Guo; Matteo Caggio; Zdeněk Skalák Regularity criteria for the Navier-Stokes equations based on one component of velocity, Nonlinear Analysis. Real World Applications, Volume 35 (2017), pp. 379-396 | DOI:10.1016/j.nonrwa.2016.11.005 | Zbl:1360.35146
Zdeněk Skalák A regularity criterion for the Navier-Stokes equations based on the gradient of one velocity component, Journal of Mathematical Analysis and Applications, Volume 437 (2016) no. 1, pp. 474-484 | DOI:10.1016/j.jmaa.2016.01.023 | Zbl:1334.35210
Zdeněk Skalák Criteria for the regularity of the solutions to the Navier-Stokes equations based on the velocity gradient, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 118 (2015), pp. 1-21 | DOI:10.1016/j.na.2015.01.011 | Zbl:1322.35109
Jiří Neustupa; Patrick Penel Regularity of a Weak Solution to the Navier–Stokes Equations via One Component of a Spectral Projection of Vorticity, SIAM Journal on Mathematical Analysis, Volume 46 (2014) no. 2, p. 1681 | DOI:10.1137/120874874

Cité par 6 documents. Sources : Crossref, zbMATH

Commentaires - Politique