×

Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations. (English) Zbl 1126.35047

The authors consider a Leray-Hopf weak solution \((u,p)\) of the Navier-Stokes equations in \(\mathbb{R}^3\times(0,T)\). They prove that this solution is regular provided that the initial velocity \(u_0\) belongs to \(L^2 (\mathbb{R}^3)\cap L^q (\mathbb{R}^3)\) for some \(q>3\) and that the pressure satisfies \[ \int^T_0\|p(t) \|_{\dot B^0_{\infty,\infty}}\,dt<\infty. \] The main tool in obtaining this result is an a-priori-estimate of the form \[ \sup_{0\leq t\leq T}\|u(t)\|_{L^s}\leq C(\| u_0\|_{L^s}+(CT)^{\frac{1} {s}}+e)^{\exp(C\int^T_0\|p(t)\|_{\dot B^0_{\infty, \infty}}\,dt)},\quad 3<s\leq 4, \] which is derived with the help of the Paley-Littlewood decomposition.

MSC:

35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071
[2] Luigi C. Berselli and Giovanni P. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc. 130 (2002), no. 12, 3585 – 3595. · Zbl 1075.35031
[3] Dongho Chae and Jihoon Lee, Regularity criterion in terms of pressure for the Navier-Stokes equations, Nonlinear Anal. 46 (2001), no. 5, Ser. A: Theory Methods, 727 – 735. · Zbl 1007.35064 · doi:10.1016/S0362-546X(00)00163-2
[4] E. B. Fabes, B. F. Jones, and N. M. Rivière, The initial value problem for the Navier-Stokes equations with data in \?^{\?}, Arch. Rational Mech. Anal. 45 (1972), 222 – 240. · Zbl 0254.35097 · doi:10.1007/BF00281533
[5] Yoshikazu Giga, Solutions for semilinear parabolic equations in \?^{\?} and regularity of weak solutions of the Navier-Stokes system, J. Differential Equations 62 (1986), no. 2, 186 – 212. · Zbl 0577.35058 · doi:10.1016/0022-0396(86)90096-3
[6] Eberhard Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4 (1951), 213 – 231 (German). · Zbl 0042.10604 · doi:10.1002/mana.3210040121
[7] Hideo Kozono and Yasushi Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations, Math. Z. 235 (2000), no. 1, 173 – 194. · Zbl 0970.35099 · doi:10.1007/s002090000130
[8] Hideo Kozono, Takayoshi Ogawa, and Yasushi Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z. 242 (2002), no. 2, 251 – 278. · Zbl 1055.35087 · doi:10.1007/s002090100332
[9] J. Leray, Sur le mouvement d’un liquids visqeux emplissant l’espace, Acta Math. 63(1934), 193-248. · JFM 60.0726.05
[10] James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), 187 – 195. · Zbl 0106.18302 · doi:10.1007/BF00253344
[11] James Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962) Univ. of Wisconsin Press, Madison, Wis., 1963, pp. 69 – 98.
[12] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[13] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. · Zbl 0821.42001
[14] Hermann Sohr, Zur Regularitätstheorie der instationären Gleichungen von Navier-Stokes, Math. Z. 184 (1983), no. 3, 359 – 375 (German). · Zbl 0506.35084 · doi:10.1007/BF01163510
[15] Michael Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 4, 437 – 458. · Zbl 0632.76034 · doi:10.1002/cpa.3160410404
[16] Shuji Takahashi, On interior regularity criteria for weak solutions of the Navier-Stokes equations, Manuscripta Math. 69 (1990), no. 3, 237 – 254. · Zbl 0718.35022 · doi:10.1007/BF02567922
[17] Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. · Zbl 0546.46028
[18] Yong Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in \Bbb R&sup3;, Proc. Amer. Math. Soc. 134 (2006), no. 1, 149 – 156. · Zbl 1075.35044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.