Regularity Criterion and Energy Conservation for the Supercritical Quasi-geostrophic Equation

  title={Regularity Criterion and Energy Conservation for the Supercritical Quasi-geostrophic Equation},
  author={Mimi Dai},
  journal={Journal of Mathematical Fluid Mechanics},
  • Mimi Dai
  • Published 9 May 2015
  • Mathematics, Environmental Science
  • Journal of Mathematical Fluid Mechanics
This paper studies the regularity and energy conservation problems for the 2D supercritical quasi-geostrophic (SQG) equation. We apply an approach of splitting the dissipation wavenumber to obtain a new regularity condition which is weaker than all the Prodi–Serrin type regularity conditions. Moreover, we prove that any viscosity solution of the supercritical SQG in $$L^2(0,T; B^{1/2}_{2,c(\mathbb N)})$$L2(0,T;B2,c(N)1/2) satisfies energy equality. 

Determining modes for the surface quasi-geostrophic equation

Regularity criterion for the 3D Hall-magneto-hydrodynamics

Stability of Fractionally Dissipative 2D Quasi-geostrophic Equation with Infinite Delay

In this paper, fractionally dissipative 2D quasi-geostrophic equations with an external force containing infinite delay is considered in the space $$H^s$$ H s with $$s\ge 2-2\alpha $$ s ≥ 2 - 2 α and

Energy and helicity conservation for the generalized quasi-geostrophic equation

In this paper, we consider the 2-D generalized surface quasi-geostrophic equation with the velocity v determined by v = R ⊥ Λ γ − 1 θ . It is shown that the L p type energy norm of weak solutions is

Lipschitz Continuity of Solutions to Drift-Diffusion Equations in the Presence of Nonlocal Terms

  • H. Ibdah
  • Mathematics
    Journal of Mathematical Fluid Mechanics
  • 2022
We analyze the propagation of Lipschitz continuity of solutions to various linear and nonlinear drift-diffusion systems, with and without incompressibility constraints. Diffusion is assumed to be

Regularity criteria for the 3D Navier-Stokes and MHD equations

We prove that a solution to the 3D Navier-Stokes or MHD equations does not blow up at $t=T$ provided $\displaystyle \limsup_{q \to \infty} \int_{\mathcal{T}_q}^T \|\Delta_q(\nabla \times u)\|_\infty

Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics.

Using Littlewood-Paley theory, one formulates the determining wavenumbers for the Hall-MHD system, defined for each individual solution $(u,b)$. It is shown that the long time behaviour of strong

Regularity problem for the nematic LCD system with Q-tensor in $\mathbb R^3$

We study the regularity problem of a nematic liquid crystal model with local configuration represented by Q-tensor in three dimensions. It was an open question whether the classical Prodi-Serrin


The above system describes the evolution of a system consisting of a magnetic field b, electrons and ions, whose collective motion under b can be approximated as an electrically conducting fluid with

Regularity Problem for the Nematic LCD System with Q-tensor in ℝ3

  • Mimi Dai
  • Mathematics
    SIAM J. Math. Anal.
  • 2017
Applying a wavenumber splitting method, it is shown that a solution does not blow-up under certain extended Beale-Kato-Majda condition solely imposed on velocity.



On the Regularity Conditions for the Dissipative Quasi-geostrophic Equations

  • D. Chae
  • Mathematics
    SIAM J. Math. Anal.
  • 2006
We obtain regularity conditions for solutions of the dissipative quasi-geostrophic equation. The first one imposes on the integrability of the magnitude of the temperature gradient, and corresponds

A regularity criterion for the dissipative quasi-geostrophic equations

Regularity criteria of supercritical beta-generalized quasi-geostrophic equations in terms of partial derivatives

We study the two-dimensional beta-generalized supercritical quasi-geostrophic equation and in particular show that in order to obtain global regularity results, one only needs to bound its partial

Global well-posedness of slightly supercritical active scalar equations

The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasigeostrophic (SQG) and Burgers equations, when the

Energy conservation and Onsager's conjecture for the Euler equations

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in conserve energy only if they have a certain minimal smoothness (of the order of 1/3 fractional derivatives)

Behavior of solutions of 2D quasi-geostrophic equations

We study solutions to the 2D quasi-geostrophic (QGS) equation $$ \frac{\partial \theta}{\partial t}+u\cdot\nabla\theta + \kappa (-\Delta)^{\alpha}\theta=f $$ and prove global existence and uniqueness

The quasi-geostrophic equation in the Triebel–Lizorkin spaces

We prove the local-in-time well-posedness in the Triebel–Lizorkin spaces for the two-dimensional quasi-geostrophic equation. We also obtain a sharp finite time blow-up criterion of solutions both in

A Maximum Principle Applied to Quasi-Geostrophic Equations

We study the initial value problem for dissipative 2D Quasi-geostrophic equations proving local existence, global results for small initial data in the super-critical case, decay of Lp-norms and