2D Dissipative Quasi-geostrophic Equation Research Articles

Uniform sparse domination and quantitative weighted boundedness for singular integrals and application to the dissipative quasi-geostrophic equation

In this paper, we consider a kind of singular integralsTλf(x)= p.v. ∫RnΩ(y)|y|n−λf(x−y)dy for any f∈Lq(Rn),1<q<∞ and 0<λ<n, which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation(0.1)∂tθ+u⋅∇θ+κΛ2βθ=0,(x,t)∈R2×R+,κ>0, where u=−∇⊥Λ−2+2αθ, α∈[0,12] and β∈(0,1]. Firstly, we give a uniform sparse domination for this kind of singular integral operators. Secondly, we obtain the uniform quantitative weighted bounds for the operator Tλ with rough kernel. As an application, we obtain the uniform quantitative weighted bounds for the commutator [b,Tλ] with rough kernel and study solutions to the generalized 2D dissipative quasi-geostrophic (QG) equation.

Metastability for the Dissipative Quasi-Geostrophic Equation and the Non-local Enhancement

In this paper, we study the linear metastability for the linearized 2D dissipative surface quasi-geostrophic equation with small viscosity $$\nu $$ around the quasi-steady state $$\Theta _{\sin }=-e^{-\nu t}\sin y$$ . We proved the linear enhanced dissipation and obtained the dissipation rate. Moreover, the new non-local enhancement phenomenon was discovered and discussed. Precisely we showed that the non-local term $$\cos y\partial _x(-\Delta )^{-\frac{1}{2}}\theta $$ re-enhances the enhanced diffusion effect by the shear-diffusion mechanism.

Time periodic solutions to the 2D quasi-geostrophic equation with the supercritical dissipation

We consider the 2D dissipative quasi-geostrophic equation with the time periodic external force and prove the existence of a unique time periodic solution in the case of the supercritical dissipation. In this case, the smoothing effect of the semigroup generated by the dissipation term is too weak to control the nonlinearity in the Duhamel term of the corresponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.

Well-posedness and blow-up of solutions for the 2D dissipative quasi-geostrophic equation in critical Fourier-Besov-Morrey spaces.

Well-posedness and blow-up of solutions for the 2D dissipative quasi-geostrophic equation in critical Fourier-Besov-Morrey spaces.

Global smoothness for a 1D supercritical transport model with nonlocal velocity

We are concerned with a nonlocal transport 1D-model with supercritical dissipation γ ∈ ( 0 , 1 ) \gamma \in (0,1) in which the velocity is coupled via the Hilbert transform, namely the so-called CCF model. This model arises as a lower dimensional model for the well-known 2D dissipative quasi-geostrophic equation and in connection with vortex-sheet problems. It is known that its solutions can blow up in finite time when γ ∈ ( 0 , 1 / 2 ) \gamma \in (0,1/2) . On the other hand, as stated by Kiselev (2010), in the supercritical subrange γ ∈ [ 1 / 2 , 1 ) \gamma \in \lbrack 1/2,1) it is an open problem to know whether its solutions are globally regular. We show global existence of nonnegative H 3 / 2 H^{3/2} -strong solutions in a supercritical subrange (close to 1) that depends on the initial data norm. Then, for each arbitrary smooth nonnegative initial data, the model has a unique global smooth solution provided that γ ∈ [ γ 1 , 1 ) \gamma \in \lbrack \gamma _{1},1) where γ 1 \gamma _{1} depends on the H 3 / 2 H^{3/2} -initial data norm. Our approach is inspired by that of Coti Zelati and Vicol (IUMJ, 2016).

Remarks on the weak–strong uniqueness for the 2D quasi-geostrophic equation in BMO space

This paper is concerned with the weak–strong uniqueness of the Cauchy problem for the 2D dissipative quasi-geostrophic equation. We proved the weak–strong uniqueness when one of the two weak solutions lies in the regular class ∇ θ ∈ L 1 ( ( 0 , T ) ; B M O ) .

Long-Time Decay to the Global Solution of the 2D Dissipative Quasigeostrophic Equation

We study the behavior at infinity in time of any global solutionθ∈C(R+,Ḣ2-2α(R2))of the surface quasigeostrophic equation with subcritical exponent2/3≤α≤1. We prove thatlim⁡t→∞∥θ(t)∥Ḣ2-2α=0. Moreover, we prove also the nonhomogeneous version of the previous result, and we prove that ifθ∈C(R+,Ḣ2-2α(R2))is a global solution, thenlim⁡t→∞∥θ(t)∥H2-2α=0.

Decay of Weak Solutions to the 2D Dissipative Quasi-Geostrophic Equation

We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data θ0 is in L 2 only, we prove that the L 2 norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For θ0 in L p ∩ L 2, with 1 ≤ p < 2, we are able to obtain a uniform decay rate in L 2. We also prove that when the \(L^{\frac{2}{2\alpha-1}}\) norm of θ0 is small enough, the L q norms, for \(q > {\frac{2}{2\alpha-1}}\) , have uniform decay rates. This result allows us to prove decay for the L q norms, for \(q \geq {\frac{2}{2\alpha-1}}\) , when θ0 is in \(L^2 \cap L^{\frac{2}{2\alpha-1}}\) .

Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation

In this article we apply the method used in the recent elegant proof by Kiselev, Nazarov and Volberg of the well-posedness of critically dissipative 2D quasi-geostrophic equation to the super-critical case. We prove that if the initial value satisfies ‖ ∇ θ 0 ‖ L ∞ 1 − 2 s ‖ θ 0 ‖ L ∞ 2 s < c s for some small number c s > 0 , where s is the power of the fractional Laplacian, then no finite time singularity will occur for the super-critically dissipative 2D quasi-geostrophic equation.

Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces

In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove a global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces L p , with p ∈ [ 1 , ∞ ] . Local results for arbitrary initial data are also given.

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

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.

Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity.

Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation

This paper concerns itself with Besov space solutions of the 2-D quasi-geostrophic (QG) equation with dissipation induced by a fractional Laplacian ( − Δ ) α . The goal is threefold: first, to extend a previous result on solutions in the inhomogeneous Besov space B 2 , q r [J. Wu, Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces, SIAM J. Math. Anal. 36 (2004–2005) 1014–1030] to cover the case when r = 2 − 2 α ; second, to establish the global existence of solutions in the homogeneous Besov space B ̊ p , q r with general indices p and q ; and third, to determine the uniqueness of solutions in any one of the four spaces: B 2 , q s , B ̊ p , q r , L q ( ( 0 , T ) ; B 2 , q s + 2 α q ) and L q ( ( 0 , T ) ; B ̊ p , q r + 2 α q ) , where s ≥ 2 − 2 α and r = 1 − 2 α + 2 p .

The 2D dissipative quasi-geostrophic equation

In this paper, we consider the initial value problem for the 2D critical dissipative quasigeostrophic equation and present results concerning global existence and uniqueness of its solutions in L q ([0, T]; L p ) and Sobolev spaces.