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

By Qionglei Chen, Changxing Miao, Zhifei Zhang. 2006

240 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6bb756c80ed00debbd919276efe274ae0b385dbf

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

On a maximum principle and its application to logarithmically critical Boussinesq system (2009, background): “The first estimate is a particular case of a general result in [Chen et al. 2007].”
Global Regularity for a Class of Generalized Magnetohydrodynamic Equations (2011, background): “The following proposition provides a lower bound for an integral originated from the dissipative term in the process of L p estimates (see [ 7 ,24]).”
On the global regularity for the supercritical SQG equation (2014, result): “Similar results w ere obtained in [2, 4, 20, 24, 25] for the critical Besov spaces X = B p,1 .”
Regularity Criteria for the Generalized MHD Equations (2008, background): “The following proposition provides a lower bound for an integral originated from the dissipative term in the process of Lp estimates (see Chen et al., 2007; Wu, 2006).”
Global regularity for a modified critical dissipative quasi-geostrophic equation (2008, unknown): “4) (see [18], or Chen, Miao, Zhang [2]).”
On the regularity of a class of generalized quasi-geostrophic equations (2010, methodology): “3) by |∆̇qθ|∆̇qθ, integrating over the spacial variable and using the divergence-free property and the following generalized Bernstein inequality in [3] ∫”
On the global well-posedness of a class of Boussinesq–Navier–Stokes systems (2009, background): “Then the a priori estimate (4.2) with σ = 1 ensures that the solution (u(n), θ(n)) is globally defined.”
Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces (2008, methodology): “For the Q-G equation, please refer to [3,5,28–31,33].”
Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations (2007, background): “this problem has attracted a significant amount of research ([2],[3],[4],[ 5 ],[6],[7],[11],[13] [14],[15],[16],[19],[20],[21],[22], [23],[24]).”