## On a parabolic logarithmic Sobolev inequality.(English)Zbl 1180.46024

Authors’ abstract: In order to extend the blow-up criterion of solutions to the Euler equations, H. Kozono and Y. Taniuchi [Commun. Math. Phys. 214, No. 1, 191–200 (2000; Zbl 0985.46015)] proved a logarithmic Sobolev inequality by means of the isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of the anisotropic (parabolic) BMO norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time existence result for a class of nonlinear parabolic problems.
Reviewer: Liu Zheng (Anshan)

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26D10 Inequalities involving derivatives and differential and integral operators 26D15 Inequalities for sums, series and integrals 35K55 Nonlinear parabolic equations

Zbl 0985.46015
Full Text:

### References:

 [1] Beale, J.T.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. math. phys., 94, 61-66, (1984) · Zbl 0573.76029 [2] Bownik, M., Anisotropic triebel – lizorkin spaces with doubling measures, J. geom. anal., 17, 387-424, (2007) · Zbl 1147.42006 [3] Brézis, H.; Gallouët, T., Nonlinear Schrödinger evolution equations, Nonlinear anal., 4, 677-681, (1980) · Zbl 0451.35023 [4] Brézis, H.; Wainger, S., A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. partial differential equations, 5, 773-789, (1980) · Zbl 0437.35071 [5] Evans, L.C., Partial differential equations, Grad. stud. math., vol. 19, (1998), American Mathematical Society Providence, RI [6] Frazier, M.; Jawerth, B., A discrete transform and decompositions of distribution spaces, J. funct. anal., 93, 34-170, (1990) · Zbl 0716.46031 [7] Hayashi, N.; von Wahl, W., On the global strong solutions of coupled klein – gordon – schrödinger equations, J. math. soc. Japan, 39, 489-497, (1987) · Zbl 0657.35034 [8] Ibrahim, H.; Jazar, M.; Monneau, R., Dynamics of dislocation densities in a bounded channel. part I: smooth solutions to a singular coupled parabolic system, preprint · Zbl 1175.70020 [9] Ibrahim, H.; Jazar, M.; Monneau, R., Global existence of solutions to a singular parabolic/hamilton – jacobi coupled system with Dirichlet conditions, C. R. math. acad. sci. Paris, ser. I, 346, 945-950, (2008) · Zbl 1166.35313 [10] Johnsen, J.; Sickel, W., A direct proof of Sobolev embeddings for quasi-homogeneous lizorkin – triebel spaces with mixed norms, J. funct. spaces appl., 5, 183-198, (2007) · Zbl 1140.46014 [11] Kozono, H.; Ogawa, T.; Taniuchi, Y., Navier – stokes equations in the Besov space near $$L^\infty$$ and BMO, Kyushu J. math., 57, 303-324, (2003) · Zbl 1067.35064 [12] Kozono, H.; Taniuchi, Y., Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. math. phys., 214, 191-200, (2000) · Zbl 0985.46015 [13] Ladyženskaja, O.A.; Solonnikov, V.A.; Uraĺceva, N.N., Linear and quasilinear equations of parabolic type, Transl. math. monogr., vol. 23, (1967), American Mathematical Society Providence, RI, Translated from the Russian by S. Smith [14] 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), (electronic) · Zbl 1036.35082 [15] Stöckert, B., Remarks on the interpolation of anisotropic spaces of besov – hardy – sobolev type, Czechoslovak math. J., 32, 107, 233-244, (1982) [16] Triebel, H., Theory of function spaces. II, Monogr. math., vol. 84, (1992), Birkhäuser-Verlag Basel · Zbl 0778.46022
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.