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

Dai, Mimi

doi:10.1007/s00021-017-0320-y

Cited by 12 publications

(8 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[13, 16-18, 30,32,34, 36] and the references therein. The existence, uniqueness and regularity of the quasi-geostrophic equation have been considered in [12,14,15,19,20]. The asymptotic analysis of the systems, as a key method to explore the evolution of the systems in the future, has also been investigated in [11,34], where the large time behavior of solutions has been discussed by several decay estimates.…”

Section: Introductionmentioning

confidence: 99%

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

2020

View full text Add to dashboard Cite

In this paper, fractionally dissipative 2D quasi-geostrophic equations with an external force containing infinite delay is considered in the space H s with s ≥ 2 − 2α and α ∈ ( 1 2 , 1). First, we investigate the existence and regularity of solutions by Galerkin approximation and the energy method. The continuity of solutions with respect to initial data and the uniqueness of solutions are also established. Then we prove the existence and uniqueness of a stationary solution by the Lax-Milgram theorem and the Schauder fixed point theorem. Using the classical Lyapunov method, the construction method of Lyapunov functionals and the Razumikhin-Lyapunov technique, we analyze the local stability of stationary solutions. Finally, the polynomial stability of stationary solutions is verified in a particular case of unbounded variable delay.

show abstract

Section: Introductionmentioning

confidence: 99%

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

2020

View full text Add to dashboard Cite

show abstract

“…More precisely, it is enough to control a weak solution of the 3D Navier-Stokes equations in the inertial range, i.e., bellow the dissipation wavenumber, in order to ensure regularity. The dissipation wavenumber was also adapted to the supercritical SQG by Dai in [19], where the smallest critical norm was used as well.…”

Section: Introductionmentioning

confidence: 99%

Determining modes for the surface quasi-geostrophic equation

Cheskidov

Dai

2018

Physica D: Nonlinear Phenomena

Self Cite

View full text Add to dashboard Cite

We introduce a determining wavenumber for the surface quasi-geostrophic (SQG) equation defined for each individual trajectory and then study its dependence on the force. While in the subcritical and critical cases this wavenumber has a uniform upper bound, it may blow up when the equation is supercritical. A bound on the determining wavenumber provides determining modes, which in some sense measure the number of degrees of freedom of the flow, or resolution needed to describe a solution to the SQG equation.

show abstract

“…We mention that the wavenumber splitting method has also been successfully applied to other disipative equations, for instance, the supercritical surface quasigeostrophic equation in [10], the magneto-hydrodynamics system in [8], and the Hall magneto-hydrodynamics system in [11].…”

Section: Introductionmentioning

confidence: 99%

Regularity Problem for the Nematic LCD System with Q-tensor in $\mathbb R^3$

Dai¹

2017

SIAM J. Math. Anal.

Self Cite

View full text Add to dashboard Cite

Abstract. 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 condition implies regularity for this model. Applying a wavenumber splitting method, we show that a solution does not blow-up under certain extended Beale-Kato-Majda condition solely imposed on velocity. This regularity criterion automatically implies that the classical Prodi-Serrin or Beale-Kato-Majda condition prevents blow-up of solutions.

show abstract

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

Cited by 12 publications

References 41 publications

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

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

Determining modes for the surface quasi-geostrophic equation

Regularity Problem for the Nematic LCD System with Q-tensor in $\mathbb R^3$

Contact Info

Product

Resources

About