On the critical one component regularity for 3-D Navier-Stokes system

Chemin, Jean-Yves; Zhang, Ping

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 1, pp. 131-167.

On considère une donnée initiale $v_0$ dont la vorticité ${\Omega }_0=\nabla \times v_0$ appartient à $L^{\frac{3}{2}}$ (ce qui implique que $v_0$ appartient à l'espace de Sobolev $H^{\frac{1}{2}}$). Nous démontrons que si la solution $v$ de l'équation de Navier-Stokes tridimensionnelle associée à $v_0$ par le théorème de Fujita-Kato développe une singularité à l'instant $T^{☆}$ (fini) alors, pour tout $p$ dans l'intervalle $]4,6[$ et tout vecteur unitaire $e$ de ${ℝ}^3,$ on a ${\int }_0^{T^{☆}}{\parallel v\left(t\right)·e\parallel }_{H^{\frac{1}{2}+\frac{2}{p}}}^{p}dt=\infty .$ Remarquons que toutes ses quantités sont invariantes par les changements d'échelle de l'équation de Navier-Stokes.

Given an initial data $v_0$ with vorticity ${\Omega }_0=\nabla \times v_0$ in $L^{\frac{3}{2}}$ (which implies that $v_0$ belongs to the Sobolev space $H^{\frac{1}{2}}$), we prove that the solution $v$ given by the classical Fujita-Kato theorem blows up in a finite time $T^{☆}$ only if, for any $p$ in $]4,6[$ and any unit vector $e$ in ${ℝ}^3,$ there holds ${\int }_0^{T^{☆}}{\parallel v\left(t\right)·e\parallel }_{H^{\frac{1}{2}+\frac{2}{p}}}^{p}dt=\infty .$ We remark that all these quantities are scaling invariant under the scaling transformation of Navier-Stokes system.

Classification : 35Q30, 76D03
Keywords: Incompressible Navier-Stokes Equations, Blow-up criteria, Anisotropic, Littlewood-Paley Theory.
Mot clés : Équations de Navier-Stokes incompressibles, critère de l'explosion, théorie de Littlewood-Paley anisotropique.

Beirão da Veiga, H. A new regularity class for the Navier-Stokes equations in ${𝐑}^n$ , Chinese Ann. Math. Ser. B, Volume 16 (1995), pp. 407-412 ; a Chinese summary appears in Chinese Ann. Math. Ser. A 16 (1995), 797 (ISSN: 0252-9599) | MR | Zbl

Bahouri, H.; Chemin, J.-Y.; Danchin, R., Grundl. math. Wiss., 343, Springer, Heidelberg, 2011, 523 pages (ISBN: 978-3-642-16829-1) | DOI | MR | Zbl

Beale, J. T.; Kato, T.; Majda, A. Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys., Volume 94 (1984), pp. 61-66 http://projecteuclid.org/euclid.cmp/1103941230 (ISSN: 0010-3616) | DOI | MR | Zbl

Bony, J.-M. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., Volume 14 (1981), pp. 209-246 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Chemin, J.-Y.; Desjardins, B.; Gallagher, I.; Grenier, E. Fluids with anisotropic viscosity, M2AN Math. Model. Numer. Anal., Volume 34 (2000), pp. 315-335 (ISSN: 0764-583X) | DOI | Numdam | MR | Zbl

Chemin, J.-Y.; Gallagher, I.; Zhang, P. Sums of large global solutions to the incompressible Navier-Stokes equations, J. reine angew. Math., Volume 681 (2013), pp. 65-82 (ISSN: 0075-4102) | MR | Zbl

Chemin, J.-Y. Remarques sur l'existence globale pour le système de Navier-Stokes incompressible, SIAM J. Math. Anal., Volume 23 (1992), pp. 20-28 (ISSN: 0036-1410) | DOI | MR | Zbl

Chemin, J.-Y.; Planchon, F. Self-improving bounds for the Navier-Stokes equations, Bull. Soc. Math. France, Volume 140 (2012), pp. 583-597 (ISSN: 0037-9484) | DOI | Numdam | MR | Zbl

Cao, C.; Titi, E. S. Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., Volume 57 (2008), pp. 2643-2661 (ISSN: 0022-2518) | DOI | MR | Zbl

Cao, C.; Titi, E. S. Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., Volume 202 (2011), pp. 919-932 (ISSN: 0003-9527) | DOI | MR | Zbl

Chemin, J.-Y.; Zhang, P. On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Comm. Math. Phys., Volume 272 (2007), pp. 529-566 (ISSN: 0010-3616) | DOI | MR | Zbl

Fujita, H.; Kato, T. On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., Volume 16 (1964), pp. 269-315 (ISSN: 0003-9527) | DOI | MR | Zbl

Fang, D.; Qian, C. Some new regularity criteria for the 3D Navier-Stokes Equations (preprint arXiv:1212.2335 )

Gallagher, I.; Koch, G. S.; Planchon, F. A profile decomposition approach to the $L_t^{\infty }\left(L_x^3\right)$ Navier-Stokes regularity criterion, Math. Ann., Volume 355 (2013), pp. 1527-1559 (ISSN: 0025-5831) | DOI | MR | Zbl

He, C. Regularity for solutions to the Navier-Stokes equations with one velocity component regular, Electron. J. Differential Equations (2002) (ISSN: 1072-6691) | MR | Zbl

Iskauriaza, L.; Serëgin, G. A.; Shverak, V. $L_{3,\infty }$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, Volume 58 (2003), pp. 3-44 (ISSN: 0042-1316) | DOI | MR | Zbl

Kukavica, I.; Ziane, M. One component regularity for the Navier-Stokes equations, Nonlinearity, Volume 19 (2006), pp. 453-469 (ISSN: 0951-7715) | DOI | MR | Zbl

Kukavica, I.; Ziane, M. Navier-Stokes equations with regularity in one direction, J. Math. Phys., Volume 48 (2007) (ISSN: 0022-2488) | DOI | MR | Zbl

Leray, J. Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., Volume 63 (1934), pp. 193-248 (ISSN: 0001-5962) | DOI | JFM | MR

Neustupa, J.; Novotný, A.; Penel, P., Topics in mathematical fluid mechanics (Quad. Mat.), Volume 10, Dept. Math., Seconda Univ. Napoli, Caserta, 2002, pp. 163-183 | MR | Zbl

Neustupa, J.; Penel, P., Applied nonlinear analysis, Kluwer/Plenum, New York, 1999, pp. 391-402 | MR | Zbl

Paicu, M. Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana, Volume 21 (2005), pp. 179-235 (ISSN: 0213-2230) | DOI | MR | Zbl

Pokorný, M. On the result of He concerning the smoothness of solutions to the Navier-Stokes equations, Electron. J. Differential Equations (2003) (ISSN: 1072-6691) | MR | Zbl

Penel, P.; Pokorný, M. Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity, Appl. Math., Volume 49 (2004), pp. 483-493 (ISSN: 0862-7940) | DOI | MR | Zbl

Skalák, Z.; Kučera, P. A note on coupling of velocity components in the Navier-Stokes equations, ZAMM Z. Angew. Math. Mech., Volume 84 (2004), pp. 124-127 (ISSN: 0044-2267) | DOI | MR | Zbl

Zhou, Y.; Pokorný, M. On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, Volume 23 (2010), pp. 1097-1107 (ISSN: 0951-7715) | DOI | MR | Zbl