Supercritical Dissipative Quasi-geostrophic Equation Research Articles

Existence of a unique global solution, and its decay at infinity, for the modified supercritical dissipative quasi-geostrophic equation

Existence of a unique global solution, and its decay at infinity, for the modified supercritical dissipative quasi-geostrophic equation

Long time stability for the dispersive SQG equation and Boussinesq equations in Sobolev space Hs

Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity 23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys. 67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684], where the initial condition [Formula: see text] [Formula: see text] plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like [Formula: see text]. As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium.

Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing

We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to $\infty$. Both results rely on the Strichartz-type estimates for the corresponding linear equation.

Eventual Regularity of the Solutions to the Supercritical Dissipative Quasi-Geostrophic Equation

We show that a solution to the supercritical surface quasi-geostrophic equation that is smooth up to a certain time must remain smooth forever.

A regularity criterion for the critical and supercritical dissipative quasi-geostrophic equations

In this note, by using a well-known commutator estimate, we give a new regularity criterion for the 2D dissipative quasi-geostrophic equations in the critical and supercritical cases.

On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces

We prove the local smoothing effect of the 2D critical and supercritical dissipative quasi-geostrophic equations in critical Besov spaces. As an application, a global well-posedness result is established by adapting a method in Kiselev, Nazarov, and Volberg (2007) [16] and an idea in Dong and Du (2008) [15] with suitable modifications. Moreover, we show that the unique solution obtained in Chen, Miao, and Zhang (2007) [11] is a classical solution. These generalize some previous results in Dong (2010) [13], Dong and Du (2008) [15]. The main ingredients of the proofs are two commutator estimates and the preservation of suitable modulus of continuity of the solutions.

Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness

We study the critical and super-critical dissipativequasi-geostrophic equations in $\R^2$ or $\T^2$. An optimal local smoothing effect of solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a main application, we establish theglobal well-posedness for the critical 2D quasi-geostrophicequations with periodic $H^1$ data. Some decay in time estimates arealso provided.

Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation

The asymptotic stability for the weak solution θ of the critical and supercritical dissipative quasi-geostrophic equation in the Serrin-type class ∇θ ∊ Lr(0, ∞;Lp(R2)) is examined. This equation is perturbed by large initial data and external functions. It is shown that every weak perturbed solution has the same asymptotic behaviour as that of θ. More precisely, the difference decays in the norm of L2(R2).

Dissipative Quasi-Geostrophic Equation for Large Initial Data in the Critical Sobolev Space

The critical and super-critical dissipative quasi-geostrophic equations are investigated in \(\mathbb{R}^2\). We prove local existence of a unique regular solution for arbitrary initial data in H2-2α which corresponds to the scaling invariant space of the equation. We also consider the behavior of the solution near t = 0 in the Sobolev space.