Supercritical Surface Quasi-geostrophic Equation Research Articles

Global well-posedness of slightly supercritical SQG equations and gradient estimate

We prove the global regularity of smooth solutions for a dissipative surface quasi-geostrophic (SQG) equation with both velocity and dissipation logarithmically supercritical compared to the critical equation. By this, we mean that a symbol defined as a power of logarithm is added to both velocity and dissipation terms to penalize the equation’s criticality. Our primary tool is the nonlinear maximum principle which provides transparent proofs of global regularity for nonlinear dissipative equations. Combining the nonlinear maximum principle with a modulus of continuity, we prove an uniform-in-time gradient estimate for the critical and slightly supercritical SQG equation. It improves the previous double exponential bound by Kiselev–Nazarov–Volberg to the single exponential. In addition, we prove the eventual exponential decay of the solutions.

Asymptotic study of supercritical surface Quasi-Geostrophic equation in critical space

In this paper we prove, if θ∈C([0,∞),H2−2α(R2)) is a global solution of supercritical surface Quasi-Geostrophic equation with small initial data, then ‖θ(t)‖H2−2α decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.

Estimate on the Dimension of the Singular Set of the Supercritical Surface Quasigeostrophic Equation

We consider the SQG equation with dissipation given by a fractional Laplacian of order alpha <frac{1}{2}. We introduce a notion of suitable weak solution, which exists for every L^2 initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most frac{1}{2alpha } left( frac{1+alpha }{alpha } (1-2alpha ) + 2right) .

A class of large solutions to the supercritical surface quasi-geostrophic equation

Whether or not classical solutions to the surface quasi-geostrophic (SQG) equation can develop finite time singularities remains an outstanding open problem. This paper constructs a class of large global-in-time classical solutions to the SQG equation with supercritical dissipation. The construction process presented here implies that any solution of the supercritical SQG equation must be globally regular if its initial data is sufficiently close to a function (measured in a Sobolev norm) whose Fourier transform is supported in a suitable region away from the origin.

Optimal convergence rates of the supercritical surface quasi-geostrophic equation

This study is concerned with the optimal convergence rates of the supercritical surface quasi-geostrophic equation. When the global weak solution θ of the supercritical surface quasi-geostrophic equation satisfies ∇θ∈Lr(0,∞;Ḃp,∞0(R2))for2p+αr=α,2α<p<∞,then even for large initial data perturbation, every weak solution θ̃(x,t) of the perturbed quasi-geostrophic equation converges to θ(x,t) with the optimal algebraic convergence rate ‖θ̃(t)−θ(t)‖L2≤C(1+t)−1αt>0.The findings are mainly based on some new estimates for the trilinear form in Besov spaces and the generalized Fourier splitting method.

The optimal temporal decay estimates for the fractional power dissipative equation in negative Besov spaces

In this paper, we first generalize a new energy approach, developed by Guo and Wang [Commun. Partial Differ. Equations 37, 2165–2208 (2012)] in the framework of homogeneous Besov spaces for proving the optimal temporal decay rates of solutions to the fractional power dissipative equation, then we apply this approach to the critical and supercritical surface quasi-geostrophic equation and the critical Keller-Segel system. We show that certain weighted negative Besov norm of solutions is preserved along time evolution and obtain the optimal temporal decay rates of the spatial derivatives of solutions by the Fourier splitting approach and the interpolation techniques.

On Gevrey regularity of the supercritical SQG equation in critical Besov spaces

In this paper, we show that the solution of the supercritical surface quasi-geostrophic (SQG) equation, with initial data in a critical Besov space, belongs to a subanalytic Gevrey class. In order to prove this, a suitable estimate on the nonlinear term, in the form of a commutator, is required. We express the commutator as a bilinear multiplier operator and obtain single-scale estimates for its symbol. In particular, we show that the localized symbol is of Marcinkiewicz type, and show that due to the localizations inherited from working in the Besov spaces, this condition implies the requisite boundedness of the corresponding operator. This result strengthens previous ones which showed that solutions starting from initial data in critical Besov spaces are classical. As a direct consequence of our method, decay estimates of higher-order derivatives are easily deduced.

Asymptotic convergence rate of supercritical surface quasi-geostrophic equation in Morrey space

In this paper the asymptotic convergence rate of the supercritical surface quasi-geostrophic equation in Morrey space is investigated. It is shown that if the global solution θ(x,t) of the supercritical surface quasi-geostrophic equation satisfies the growth condition in critical Morrey space∫0∞‖∇θ(t)‖M˙p,q(R2)rdt<∞for αr+2p=α,2α<p<∞, then every perturbed weak solution θ˜(x,t) of the perturbed quasi-geostrophic equation converges asymptotically to θ(x,t) with the optimal rate‖θ˜(t)−θ(t)‖L2=O((1+t)−1α),as t→∞. In particular, the initial perturbation is allowed large. The findings are based on the spectral decomposition methods of fractional Laplace and Morrey space technique.

Remarks on the logarithmical regularity criterion of the supercritical surface quasi-geostrophic equation in Morrey spaces

This paper is concerned with the regularity criterion of the weak solutions of the two-dimensional supercritical surface quasi-geostrophic equation in a critical Morrey space. It is proved that if the weak solution θ(x,t) of the supercritical quasi-geostrophic equation satisfies the growth condition∫0T‖∇θ(t)‖Ṁp,qr1+ln(e+‖∇θ‖Lp)dt<∞with2p+αr=α,2α<p<∞, then the solution θ(x,t) is regular on R2×(0,T]. This improves the earlier results by Xiang (2010).

Eventual Regularity of the Two-Dimensional Boussinesq Equations with Supercritical Dissipation

This paper studies solutions of the two-dimensional incompressible Boussinesq equations with fractional dissipation. The spatial domain is a periodic box. The Boussinesq equations concerned here govern the coupled evolution of the fluid velocity and the temperature and have applications in fluid mechanics and geophysics. When the dissipation is in the supercritical regime (the sum of the fractional powers of the Laplacians in the velocity and the temperature equations is less than 1), the classical solutions of the Boussinesq equations are not known to be global in time. Leray–Hopf type weak solutions do exist. This paper proves that such weak solutions become eventually regular (smooth after some time \(T>0\)) when the fractional Laplacian powers are in a suitable supercritical range. This eventual regularity is established by exploiting the regularity of a combined quantity of the vorticity and the temperature as well as the eventual regularity of a generalized supercritical surface quasi-geostrophic equation.

Stability Analysis of the Supercritical Surface Quasi-Geostrophic Equation

This paper is devoted to the study of the stability issue of the supercritical dissipative surface quasi-geostrophic equation with nondecay low-regular external force. Supposing that the weak solution of the surface quasi-geostrophic equation with the force satisfies the growth condition in the critical BMO space , it is proved that every perturbed weak solution converges asymptotically to solution of the original surface quasi-geostrophic equation. The initial and external forcing perturbations are allowed to be large.

Global well-posedness for a slightly supercritical surface quasi-geostrophic equation

We use a non-local maximum principle to prove the global existence of smooth solutions for a slightly supercritical surface quasi-geostrophic equation. By this we mean that the velocity field u is obtained from the active scalar θ by a Fourier multiplier with symbol iζ⊥|ζ|−1m(|ζ|), where m is a smooth increasing function that grows slower than log log|ζ| as |ζ| → ∞.

Note on the well-posedness of a slightly supercritical surface quasi-geostrophic equation

For the following slightly supercritical surface quasi-geostrophic equation∂tθ+u⋅∇θ+|D|βθ=0,u=∇⊥|D|β−2m(D)θ,β∈]0,1], where m∈C∞(R2∖{0}) is a radial non-decreasing positive function which roughly has a logarithmic growth near infinity, we apply the method of nonlocal maximum principle to show the global well-posedness of smooth solutions.

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.