Weak strong uniqueness criteria for the critical quasi-geostrophic equation

Abstract We give two weak–strong uniqueness results for the weak solutions to the critical dissipative quasi-geostrophic equation when the initial data belongs to H − 1 / 2 . The first one shows that we can construct a unique H − 1 / 2 -solution when the initial data belongs moreover to L ∞ with a small L ∞ norm. The other one gives the uniqueness of a H − 1 / 2 -solution which belongs to C ( [ 0 , T ) , CMO ) .

[1]  Jiahong Wu,et al.  The 2D dissipative quasi-geostrophic equation , 2002, Appl. Math. Lett..

[2]  Ronald R. Coifman,et al.  Factorization theorems for Hardy spaces in several variables , 1976 .

[3]  S. Gala,et al.  Remark on uniqueness of weak solutions to the Navier–Stokes equations , 2008 .

[4]  Akihito Uchiyama.  On the compactness of operators of Hankel type , 1978 .

[5]  Wolf von Wahl,et al.  The equations of Navier-Stokes and abstract parabolic equations , 1985 .

[6]  H. Kozono,et al.  REMARK ON UNIQUENESS OF WEAK SOLUTIONS TO THE NAVIER-STOKES EQUATIONS , 1996 .

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

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

[9]  L. Caffarelli,et al.  Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation , 2006, math/0608447.

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

[11]  Taoufik Hmidi,et al.  On the Global Well-Posedness of the Critical Quasi-Geostrophic Equation , 2008, SIAM J. Math. Anal..

[12]  F. Marchand.  Existence and Regularity of Weak Solutions to the Quasi-Geostrophic Equations in the Spaces Lp or $$\dot{H}^{-1/2}$$ , 2007 .

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

[14]  Fabien Marchand,et al.  Propagation of Sobolev regularity for the critical dissipative quasi-geostrophic equation , 2006, Asymptot. Anal..

[15]  E. M. STEIN,et al.  THE CHARACTERIZATION OF FUNCTIONS ARISING AS POTENTIALS , 2007 .

[16]  Pierre Gilles Lemarié-Rieusset,et al.  Solutions auto-similaires non radiales pour l'équation quasi-géostrophique dissipative critique , 2005 .

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

[18]  F. Beaufils,et al.  FRANCE , 1979, The Lancet.

[19]  E. Stein,et al.  The characterization of functions arising as potentials. II , 1961 .

[20]  A. Volberg,et al.  Global well-posedness for the critical 2D dissipative quasi-geostrophic equation , 2007 .