Sharp Sobolev Inequality of Logarithmic Type and the Limiting Regularity Condition to the Harmonic Heat Flow

We show a sharp version of the Sobolev inequality of the Beale--Kato--Majda and the Kozono--Taniuchi type in Lizorkin--Triebel space. As an application of this inequality, the regularity problem un...

[1]  J. Serrin.  On the interior regularity of weak solutions of the Navier-Stokes equations , 1962 .

[2]  Thierry Gallouët,et al.  Nonlinear Schrödinger evolution equations , 1980 .

[3]  H.BeirāodaVeiga.  A New Regularity Class for the Navier-Stokes Equations in IR^n , 1995 .

[4]  M. Vishik,et al.  Hydrodynamics in Besov Spaces , 1998 .

[5]  Rugang Ye,et al.  Finite-time blow-up of the heat flow of harmonic maps from surfaces , 1992 .

[6]  T. Ogawa,et al.  On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain , 2003 .

[7]  J. Eells,et al.  Harmonic Mappings of Riemannian Manifolds , 1964 .

[8]  Michael Struwe,et al.  Existence and partial regularity results for the heat flow for harmonic maps , 1989 .

[9]  Takayoshi Ogawa,et al.  The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations , 2002 .

[10]  Y. Meyer,et al.  Compensated compactness and Hardy spaces , 1993 .

[11]  Mikhail Feldman,et al.  Partial regularity for harmonic maps of evolution into spheres , 1994 .

[12]  Tosio Kato,et al.  Commutator estimates and the euler and navier‐stokes equations , 1988 .

[13]  F. Hélein.  Régularité des applications faiblement harmoniques entre une surface et une sphère , 1990 .

[14]  V. I. Yudovich,et al.  Uniqueness Theorem for the Basic Nonstationary Problem in the Dynamics of an Ideal Incompressible Fluid , 1995 .

[15]  H. Kozono,et al.  Limiting Case of the Sobolev Inequality in BMO,¶with Application to the Euler Equations , 2000 .

[16]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[17]  Takehiko Ohyama,et al.  Interior regularity of weak solutions of the time-dependent Navier-Stokes equation , 1960 .

[18]  Wei Ding,et al.  Blow-up and global existence for heat flows of harmonic maps , 1990 .

[19]  Jean-Yves Chemin,et al.  Perfect Incompressible Fluids , 1998 .

[20]  H. Triebel.  Theory Of Function Spaces , 1983 .

[21]  J. Coron,et al.  Explosion en temps fini pour le flot des applications harmoniques , 1989 .

[22]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[23]  Michael Struwe,et al.  Geometric evolution problems , 1995 .

[24]  Lawrence C. Evans,et al.  Partial regularity for stationary harmonic maps into spheres , 1991 .

[25]  Haim Brezis,et al.  A note on limiting cases of sobolev embeddings and convolution inequalities , 1980 .

[26]  Yoshikazu Giga,et al.  Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system , 1986 .

[27]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[28]  Tosio Kato,et al.  Remarks on the breakdown of smooth solutions for the 3-D Euler equations , 1984 .

[29]  G. Ponce.  Remarks on a paper by J. T. Beale, T. Kato, and A. Majda , 1985 .