×

Brezis-Gallouet-Wainger type inequalities and blow-up criteria for Navier-Stokes equations in unbounded domains. (English) Zbl 1397.35176

The authors consider the Navier-Stokes equations in a smooth domain \(\Omega \subset \mathbb R^3\). The initial condition is posed on the velocity \(u\) and \(u=0\) on \(\partial \Omega\). If the smooth solution \(u \in C([0, T)\) breaks down at a finite time \(t = T\), then \(\int_0^t ||rot \; u (\tau)||_{L^{\infty}} d\tau \nearrow \infty\) as \(t \nearrow T\). The authors discuss similar break down criteria in various spaces. The largest normed space where the \(\mathrm{rot }u\) can break down is found under some additional assumptions.

MSC:

35Q30 Navier-Stokes equations
35B44 Blow-up in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61-66 (1984) · Zbl 0573.76029 · doi:10.1007/BF01212349
[2] Borchers W., Miyakawa T.: L2 decay for the Navier-Stokes flow in halfspaces. Math. Ann. 282, 139-155 (1988) · Zbl 0627.35076 · doi:10.1007/BF01457017
[3] Borchers W., Sohr H.: On the semigroup of the Stokes operator for exterior domains in Lq-spaces. Math. Z. 196, 415-425 (1987) · Zbl 0636.76027 · doi:10.1007/BF01200362
[4] Bradshaw, Z., Farhat, A., Grujic, Z.: An algebraic reduction of the ’scaling gap’ in the Navier-Stokes regularity problem. arXiv:1704.05546 (preprint) · Zbl 1408.35034
[5] Brezis H., Gallouet T.: Nonlinear Schrödinger evolution equations. Nonlinear Anal. Theory Methods Appl. 4, 677-681 (1980) · Zbl 0451.35023 · doi:10.1016/0362-546X(80)90068-1
[6] Brezis H., Wainger S.: A note on limiting cases of Sobolev embeddings and convolution inequalities. Comm. Partial Differ. Equ. 5, 773-789 (1980) · Zbl 0437.35071 · doi:10.1080/03605308008820154
[7] Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771-831 (1982) · Zbl 0509.35067 · doi:10.1002/cpa.3160350604
[8] Chae D.: On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces. Commun. Pure Appl. Math. 55, 654-678 (2002) · Zbl 1025.35016 · doi:10.1002/cpa.10029
[9] Chemin J.-Y.: Perfect Incompressible Fluids. Oxford Lecture Series in Mathematics and Its Applications. Oxford Science Publications, Oxford (1998) · Zbl 0927.76002
[10] Engler H.: An alternative proof of the Brezis-Wainger inequality. Commun. Partial Differ. Equ. 14(4), 541-544 (1989) · Zbl 0688.46016
[11] Fan J., Jiang S., Nakamura G., Zhou Y.: Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations. J. Math. Fluid Mech. 13, 557-571 (2011) · Zbl 1270.35339 · doi:10.1007/s00021-010-0039-5
[12] Farwig R., Sohr H.: On the Stokes and Navier-Stokes system for domains with noncompact boundary in Lq-spaces. Math. Nachr. 170, 53-77 (1994) · Zbl 0836.35117 · doi:10.1002/mana.19941700106
[13] Farwig R., Sohr H.: Helmholtz decomposition and Stokes resolvent system for aperture domains in Lq-spaces. Analysis 16, 1-26 (1996) · Zbl 0847.35101 · doi:10.1524/anly.1996.16.1.1
[14] Ferrari A.B.: On the blow-up of solutions of the 3-D Euler equations in a bounded domain. Commun. Math. Phys. 155, 277-294 (1993) · Zbl 0787.35071 · doi:10.1007/BF02097394
[15] Fujiwara D., Morimoto H.: An Lr-theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24, 685-700 (1977) · Zbl 0386.35038
[16] Giga Y.: Analyticity of the semigroup generated by the Stokes operator in Lr spaces. Math. Z. 178, 297-329 (1981) · Zbl 0473.35064 · doi:10.1007/BF01214869
[17] Giga Y., Miyakawa T.: Solutions in Lr of the Navier-Stokes initial value problem. Arch. Ration. Mech. Anal. 89, 267-281 (1985) · Zbl 0587.35078 · doi:10.1007/BF00276875
[18] Grujic Z., Guberovic R.: A regularity criterion for the 3D NSE in a local version of the space of functions of bounded mean oscillations. Ann. I. H. Poincare 27, 773-778 (2010) · Zbl 1187.35153 · doi:10.1016/j.anihpc.2009.11.009
[19] Gustafson S., Kang K., Tsai T.-P.: Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations. Commun. Math. Phys. 273, 161-176 (2007) · Zbl 1126.35042 · doi:10.1007/s00220-007-0214-6
[20] Ibrahim S., Majdoub M., Masmoudi N.: Double logarithmic inequality with a sharp constant. Proc. AMS. 135, 87-97 (2007) · Zbl 1130.46018 · doi:10.1090/S0002-9939-06-08240-2
[21] Iwashita H.: Lq−Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces. Math. Ann. 285, 265-288 (1989) · Zbl 0659.35081 · doi:10.1007/BF01443518
[22] Kato T.: Strong Lp-solutions of the Navier-Stokes equation in Rm, with applications to weak solutions. Math. Z. 187, 471-480 (1984) · Zbl 0545.35073 · doi:10.1007/BF01174182
[23] Kato T., Ponce G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. 41, 891-907 (1988) · Zbl 0671.35066 · doi:10.1002/cpa.3160410704
[24] Kozono H., Ogawa T., Taniuchi Y.: The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations. Math. Z. 242, 251-278 (2002) · Zbl 1055.35087 · doi:10.1007/s002090100332
[25] Kozono H., Taniuchi Y.: Limiting case of the Sobolev inequality in BMO, with application to the Euler equations. Commun. Math. Phys. 214, 191-200 (2000) · Zbl 0985.46015 · doi:10.1007/s002200000267
[26] Kozono H., Wadade H.: Remarks on Gagliardo-Nirenberg type inequality with critical Sobolev space and BMO. Math. Z. 259, 935-950 (2008) · Zbl 1151.46019 · doi:10.1007/s00209-007-0258-5
[27] Kubo, T., Shibata, Y.: On some properties of solutions to the Stokes equation in the half-space and perturbed half-space. In: D’Ancona, P., Georgev, V. (eds.) Dispersive Nonlinear Problems in Mathematical Physics. Quad. Mat., vol. 15, pp. 149-220. Dept. Math., Seconda Univ. Napoli, Caserta (2004) · Zbl 1129.35443
[28] Kukavica I.: On the dissipative scale for the Navier-Stokes equation. Indiana Univ. Math. J. 48, 1057-1081 (1999) · Zbl 0937.35126 · doi:10.1512/iumj.1999.48.1748
[29] Matsumoto T., Ogawa T.: Interpolation inequality of logarithmic type in abstract Besov spaces and an application to semilinear evolution equations. Math. Nachr. 283, 1810-1828 (2010) · Zbl 1215.46025 · doi:10.1002/mana.200710165
[30] Miyakawa T.: On nonstationary solutions of the Navier-Stokes equations in an exterior domain. Hiroshima Math. J. 12, 115-140 (1982) · Zbl 0486.35067
[31] Morii K., Sato T., Wadade H.: Brézis-Gallouët-Wainger type inequality with a double logarithmic term in the Hölder space: Its sharp constants and extremal functions. Nonlinear Anal. 73, 1747-1766 (2010) · Zbl 1208.46034 · doi:10.1016/j.na.2010.05.012
[32] Ogawa T.: Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow. SIAM J. Math. Anal. 34, 1318-1330 (2003) · Zbl 1036.35082 · doi:10.1137/S0036141001395868
[33] Ogawa T., Taniuchi Y.: A note on blow-up criterion to the 3-D Euler equations in a bounded domain. J. Math. Fluid Mech. 5, 17-23 (2003) · Zbl 1044.35046 · doi:10.1007/s000210300001
[34] Ogawa T., Taniuchi Y.: On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain. J. Differ. Equ. 190, 39-63 (2003) · Zbl 1038.76014 · doi:10.1016/S0022-0396(03)00013-5
[35] Ozawa T.: On critical cases of Sobolev’s inequalities. J. Funct. Anal. 127, 259-269 (1995) · Zbl 0846.46025 · doi:10.1006/jfan.1995.1012
[36] Planchon F.: An extension of the Beale-Kato-Majda criterion for the Euler equations. Commun. Math. Phys. 232, 319-326 (2003) · Zbl 1022.35048 · doi:10.1007/s00220-002-0744-x
[37] Shirota T., Yanagisawa T.: A continuation principle for the 3-D Euler equations for incompressible fluids in a bounded domain. Proc. Jpn. Acad. Ser. A 69, 77-82 (1993) · Zbl 0790.35086 · doi:10.3792/pjaa.69.77
[38] Simader, C.G., Sohr, H.: A new approach to the Helmholtz decomposition and the Neumann problem in Lq-spaces for bounded and exterior domains. In: Galdi, G.P. (ed.) Mathematical Problems Relating to the Navier-Stokes Equation, Series Advances in Mathematical Sciences and Applications, vol. 11, pp. 1-35. World Scientific, River Edge (1992) · Zbl 0791.35096
[39] Stein E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993) · Zbl 0821.42001
[40] Stein E.M., Shakarchi R.: Functional Analysis. Introduction to Further Topics in Analysis. Princeton Lectures in Analysis. Princeton University Press, Princeton (2011) · Zbl 1235.46001
[41] Sohr H.: A regularity class for the Navier-Stokes equations in Lorentz spaces. J. Evol. Equ. 1, 441-467 (2001) · Zbl 1007.35051 · doi:10.1007/PL00001382
[42] Taylor M.E.: Pseudodifferential Operators and Nonlinear PDE. Progress in Mathematics. Birkhäuser, Boston (1991) · Zbl 0746.35062 · doi:10.1007/978-1-4612-0431-2
[43] Vishik M.: Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. École Norm. Sup. 32, 769-812 (1999) · Zbl 0938.35128 · doi:10.1016/S0012-9593(00)87718-6
[44] Weissler F.B.: The Navier-Stokes initial value problem in Lp. Arch. Ration. Mech. Anal. 74, 219-230 (1980) · Zbl 0454.35072 · doi:10.1007/BF00280539
[45] Yudovich V.I.: Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2, 27-38 (1995) · Zbl 0841.35092 · doi:10.4310/MRL.1995.v2.n1.a4
[46] Zajaczkowski W.M.: Remarks on the breakdown of smooth solutions for the 3-d Euler equations in a bounded domain. Bull. Pol. Acad. Sci. Math. 37, 169-181 (1989) · Zbl 0755.76027
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.