Dissipative Quasi-geostrophic Equation Research Articles

Global Unique Solutions with Instantaneous Loss of Regularity for SQG with Fractional Diffusion

In this work we construct global unique solutions of the dissipative Surface quasi-geostrophic equation (α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-SQG) that lose regularity instantly when there is super-critical fractional diffusion.

Long time dynamics of stochastic fractionally dissipative quasi-geostrophic equations with stochastic damping

Long time dynamics of stochastic fractionally dissipative quasi-geostrophic equations with stochastic damping

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

Stochastic Quasi-Geostrophic Equation with Jump Noise in Lp Spaces

In this paper, we consider a 2D stochastic quasi-geostrophic equation driven by jump noise in a smooth bounded domain. We prove the local existence and uniqueness of mild Lp(D)-solutions for the dissipative quasi-geostrophic equation with a full range of subcritical powers α∈(12,1] by using the semigroup theory and fixed point theorem. Our approach, based on the Yosida approximation argument and Itô formula for the Banach space valued processes, allows for establishing some uniform bounds for the mild solutions and we prove the global existence of mild solutions in L∞(0,T;Lp(D)) space for all p>22α−1, which is consistent with the deterministic case.

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.

A uniform Besov boundedness and the well-posedness of the generalized dissipative quasi-geostrophic equation in the critical Besov space

Abstract In this paper, we consider a kind of singular integrals which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation ∂ t ⁡ θ + u ⋅ ∇ ⁡ θ + κ ⁢ Λ 2 ⁢ β ⁢ θ = 0 , ( x , t ) ∈ ℝ 2 × ℝ + , κ &gt; 0 , \partial_{t}\theta+u\cdot\nabla\theta+\kappa\Lambda^{2\beta}\theta=0,\quad(x,t% )\in\mathbb{R}^{2}\times\mathbb{R}^{+},\;\kappa&gt;0, where u = - ∇ ⊥ ⁡ Λ - 2 + 2 ⁢ α ⁢ θ {u=-\nabla^{\bot}\Lambda^{-2+2\alpha}\theta} , α ∈ [ 0 , 1 2 ] {\alpha\in[0,\frac{1}{2}]} and β ∈ ( 0 , 1 ] {\beta\in(0,1]} . First, we give a relationship between this kind of singular integrals and Calderón–Zygmund singular integral operators and obtain a uniform Besov estimates. As an application, we give the well-posedness of the generalized 2D dissipative quasi-geostrophic (QG) in the critical Besov space.

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.

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.

Global solutions to the dissipative quasi-geostrophic equation with dispersive forcing

We consider the initial value problem for the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing and prove the global existence of a unique solution in the scaling subcritical Sobolev spaces $H^{s}(\mathbb{R}^{2})$ ($s > 2 - \alpha$) and the scaling critical space $H^{2-\alpha}(\mathbb{R}^2)$. More precisely, for the scaling subcritical case, we establish a unique global solution for a given initial data $\theta_{0} \in H^{s}(\mathbb{R}^{2})$ ($s > 2 - \alpha$) if the size of dispersion parameter is sufficiently large and also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. For the scaling critical case, we find that the size of dispersion parameter to ensure the global existence is determined by each subset $K \subset H^{2-\alpha}(\mathbb{R}^{2})$, which is precompact in some homogeneous Sobolev spaces.

Global Well-Posedness of the Dissipative Quasi-Geostrophic Equation with Dispersive Forcing

The dissipative quasi-geostrophic equation with dispersive forcing is considered. By striking new balances between the dispersive effects of the dispersive forcing and the smoothing effects of the viscous dissipation, we obtain the global well-posedness for Cauchy problem of the dissipative quasi-geostrophic equation with dispersive forcing for arbitrary initial data, provided that the dispersive parameter is large enough.

Global well-posedness, Gevrey class regularity and large time asymptotics for the dissipative quasi-geostrophic equation in Fourier–Besov spaces

Global well-posedness, Gevrey class regularity and large time asymptotics for the dissipative quasi-geostrophic equation in Fourier–Besov spaces

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.

Optimal estimates for far field asymptotics of solutions to the quasi-geostrophic equation

The initial value problem for the two dimensional dissipative quasi-geostrophic equation of the critical and the supercritical cases is considered. Anomalous diffusion on this equation provides slow decay of solutions as the spatial parameter tends to infinity. In this paper, uniform estimates for far field asymptotics of solutions are given.

Strong solutions to a modified Michelson-Sivashinsky equation

We prove a global well-posedness and regularity result of strong solutions to a slightly modified Michelson-Sivashinsky equation in any spatial dimension and in the absence of physical boundaries. Local-in-time well-posedness (and regularity) in the space $W^{1,\infty}(\mathbb{R}^d)$ is established and is shown to be global if in addition the initial data is either periodic or vanishes at infinity. The proof of the latter result utilizes ideas previously introduced by Kiselev, Nazarov, Volberg and Shterenberg to handle the critically dissipative surface quasi-geostrophic equation and the critically dissipative fractional Burgers equation. Namely, the global regularity result is achieved by constructing a time-dependent modulus of continuity that must be obeyed by the solution of the initial-value problem for all time, preventing blowup of the gradient of the solution. This work provides an example where regularity is shown to persist even when a-priori bounds are not available.

Asymptotic behavior of critical dissipative quasi-geostrophic equation in Fourier space

In this paper we show the global existence for critical dissipative quasi-geostrophic equations if ‖θ0ˆ‖L1 is small enough; among others we prove the analyticity of such a solution. If in addition the initial condition verifies |D|−δθ0∈L1(R2) with 0<δ<1, then the solution remains regular and limt→∞⁡tδ‖θˆ(t)‖L1=0. Fourier analysis and standard techniques are used.

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 $$H^s$$ with $$s\ge 2-2\alpha $$ and $$\alpha \in (\frac{1}{2},1)$$ . 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.

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).

Paracontrolled quasi-geostrophic equation with space-time white noise

We study the stochastic dissipative quasi-geostrophic equation with space-time white noise on the two-dimensional torus. This equation is highly singular and basically ill-posed in its original form. The main objective of the present paper is to formulate and solve this equation locally in time in the framework of paracontrolled calculus when the differential order of the main term, the fractional Laplacian, is larger than $7/4$. No renormalization has to be done for this model.

On the global regularity for the anisotropic dissipative surface quasi-geostrophic equation

In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. The global existence of classical solutions is established when the dissipation powers are restricted to a suitable range. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques no longer apply. To overcome this difficulty, we establish several anisotropic embedding and interpolation inequalities involving fractional derivatives. In addition, in order to bypass the unavailability of the classical Gronwall inequality, we establish a new logarithmic-type Gronwall inequality, which may be of independent interest and have potential applications.