A New Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation

We show a new Bernstein’s inequality which generalizes the results of Cannone-Planchon, Danchin and Lemarié-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and super-critical dissipation for the small initial data in the critical Besov space, and local well-posedness for the large initial data.

[1]  Charles Fefferman,et al.  Growth of solutions for QG and 2D Euler equations , 2001 .

[2]  Hideyuki Miura,et al.  Dissipative Quasi-Geostrophic Equation for Large Initial Data in the Critical Sobolev Space , 2006 .

[3]  Raphaël Danchin,et al.  Global existence in critical spaces for compressible Navier-Stokes equations , 2000 .

[4]  Diego Cordoba,et al.  Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation , 1998, math/9811184.

[5]  R. Danchin.  Density-dependent incompressible viscous fluids in critical spaces , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  Dongho Chae,et al.  The quasi-geostrophic equation in the Triebel–Lizorkin spaces , 2003 .

[7]  J. Pedlosky.  Geophysical Fluid Dynamics , 1979 .

[8]  P. Constantin,et al.  On the critical dissipative quasi-geostrophic equation , 2001 .

[9]  Pierre Gilles Lemarié-Rieusset,et al.  Recent Developments in the Navier-Stokes Problem , 2002 .

[10]  Antonio Córdoba,et al.  Communications in Mathematical Physics A Maximum Principle Applied to Quasi-Geostrophic Equations , 2004 .

[11]  N. Ju.  Existence and Uniqueness of the Solution to the Dissipative 2D Quasi-Geostrophic Equations in the Sobolev Space , 2004 .

[12]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[13]  Jiahong Wu,et al.  Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces , 2006 .

[14]  Jiahong Wu,et al.  The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation , 2005 .

[15]  Ning Ju,et al.  The Maximum Principle and the Global Attractor for the Dissipative 2D Quasi-Geostrophic Equations , 2005 .

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

[17]  Raphaël Danchin,et al.  Poches de tourbillon visqueuses , 1997 .

[18]  N. Lerner,et al.  Flow of Non-Lipschitz Vector-Fields and Navier-Stokes Equations , 1995 .

[19]  Peter Constantin,et al.  Behavior of solutions of 2D quasi-geostrophic equations , 1999 .

[20]  Jiahong Wu,et al.  Global Solutions of the 2D Dissipative Quasi-Geostrophic Equation in Besov Spaces , 2005, SIAM J. Math. Anal..

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

[22]  J. Bony,et al.  Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires , 1980 .

[23]  Andrew J. Majda,et al.  Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar , 1994 .

[24]  Dongho Chae,et al.  Global Well-Posedness in the Super-Critical Dissipative Quasi-Geostrophic Equations , 2003 .