Eventual regularization for the slightly supercritical quasi-geostrophic equation
Luis Silvestre
本文证明了一类略超临界准地转方程的弱解在长时间内是光滑的。我们证明它使用德乔治型参数使用的想法从最近的一篇论文由Caffarelli和Vasseur。
We prove that weak solutions of a slightly supercritical quasi-geostrophic equation become smooth for large time. We prove it using a De Giorgi type argument using ideas from a recent paper by Caffarelli and Vasseur.
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Motivated by the De Giorgi type argument used in a recent paper by Caffarelli and Vasseur, we prove Hölder-regularity for weak solutions of the supercritical quasi-geostrophic equation with minimal assumptions on the initial datum.
Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation
本文给出了具有超临界耗散的二维准地转方程（$α< 1/2$）弱解的一个正则性结果：如果Leray-Hopf弱解在时间间隔$[t_0, t]$上是Hölder连续的$θ\in C^δ({\mathbb R}^2)$，且具有$δ>1-2α$，则它实际上是$(t_0,t]$上的经典解.
In this paper we prove, if $θ\in C([0,\infty),H^{2-2α}(\mathbb R^2))$ is a global solution of supercritical surface Quasi-Geostrophic equation with small initial data, then $\|θ(t)\|_{H^{2-2α}}$ decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.
This paper studies the regularity and energy conservation problems for the 2D supercritical quasi-geostrophic (SQG) equation. We apply an approach of splitting the dissipation wavenumber to obtain a new regularity condition which is weaker than all the Prodi-Serrin type regularity conditions. Moreover, we prove that any weak solution of the supercritical SQG in $L^2(0,T; B^{1/2}_{2,c(\mathbb N)})$ satisfies energy equality.
Global well-posedness for a slightly supercritical surface quasi-geostrophic equation
利用非局部最大值原理证明了一个略超临界地表准地转方程整体光滑解的存在性。这意味着速度场$u$是通过具有符号$i k^\perp |k|^{-1} m(k|)$的傅立叶乘法器从有源标量$θ$获得的，其中$m$是比$\log \log |k|$增长得慢的平滑增函数，作为$|k|\rightarrow \infty$。
We consider the $2$D 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 correponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.
Eventual Regularity of the Solutions to the Supercritical Dissipative Quasi-Geostrophic Equation
最近，Silvestre证明了轻微超临界地面准地转方程的某些弱解最终变得光滑。为了证明这一点，他采用了起源于卡法雷利和瓦瑟尔著作中的德乔治式论证。Kiselev和Nazarov通过引入一类检验函数证明了Caffarelli和Vasseur结果的一个变化。受Silvestre结果的启发，我们将修改Kiselev和Nazarov工作中的检验函数类，并使用此修改后的类来表明在一定时间内光滑的超临界SQG的解必须永远保持光滑。
In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in "critical" Sobolev spaces. Moreover, the Gevrey class that we obtain is "near optimal" and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier-Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the…
Nonlinear instability for the surface quasi-geostrophic equation in the supercritical regime
该研究针对超临界耗散的受迫表面准地转方程，证明稳态解的线性不稳定性会引发非线性不稳定性，还得到对数超临界情形下的全局适定性结果，且该结果本身具有独立研究价值。
On the supercritically diffusive magneto-geostrophic equations
我们解决的适定性理论的磁地转方程，即一个积极的标量方程，其中的发散自由漂移速度是一个衍生物比积极的标量奇异。当存在由（-Δ）^γ给出的超临界分数阶扩散时，其中0<γ<1，我们发现当γ>1/2时方程是局部适定的，而当γ<1/2时方程是不适定的，即不存在Lipschitz解映射.当γ低于1/2时，规律性显著丧失的主要原因是，从有效标量中获得速度的本构关系是由一个无界的Fourier乘子给出的，它既是均匀的又是各向异性的。最后，我们注意到，可以探索速度的本构律的各向异性，以便在初始数据和力具有薄傅立叶支持（即，它们在频率空间中的平面上被支持）时获得解的正则性的改进。特别地，对于这样的充分准备的数据，可以证明对于γ \in（0，1）的所有值的解的局部存在性和唯一性。
Dissipative quasi-geostrophic equations with initial data
这篇论文被arXiv管理员撤回，因为它抄袭了“吴家宏具有临界或超临界耗散的二维准地转方程2005非线性18 139-154。"
Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing
本文研究了具有超临界耗散和全空间色散强迫的二维准地转方程。当色散振幅参数足够大时，我们证明了具有大初值的方程的强解的整体适定性。当振幅参数趋于$\infty$时，我们还证明了强收敛性.这两个结果依赖于相应的线性方程的Escherichartz型估计。
Optimal Gevrey regularity for supercritical quasi-geostrophic equations
针对带超临界耗散的二维地表类地转方程，本文证明临界 Sobolev 与 Besov 空间中大初值具有和线性部分同衰减指数的最优 Gevrey 正则性，解决了相关公开问题。
The initial value problem for the two dimensional dissipative quasi-geostrophic equation derived from geophisical fluid dynamics is studied. The dissipation of this equation is given by the fractional Laplacian. It is known that the half Laplacian is a critical dissipation for the quasi-geostrophic equation. In this paper, far field asymptotics of solutions are given in the critical and the supercritical cases.
We prove the global regularity of smooth solutions for a dissipative surface quasi-geostrophic 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 a uniform-in-time gr…
Optimal estimates for far field asymptotics of solutions to the quasi-geostrophic equation
研究准地转方程临界和超临界初始值问题，通过异常扩散现象揭示解在空间参数趋于无穷时的缓慢衰减行为，并给出远场渐进行为的统一估计。
In the present study, we find that the surface quasi-geostrophic equation admits exact solutions, which evolve with time in quasi-stationary states. The solutions presented are available for any dissipation effect $κ(-Δ)^α$ ($κ>0$, $0\le α<1$), involved in the equation. When the equation is supercritical ($0\le α<\frac12$), the problem on the existence of large global regular solutions remains open. This study, however, provides explicit sample solutions for the understanding of the uncertain problem.
We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high freq…
Eventual regularization of the slightly supercritical fractional Burgers equation
我们证明了一个略超临界分数Burgers方程的弱解在长时间内是Holder连续的.
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.
Global well-posedness of slightly supercritical active scalar equations
本文研究了具有非局域扩散的微超临界活性标量。我们证明了全局正则性的表面准地转（SQG）和Burgers方程，当扩散项是超临界的符号与大致对数行为在无穷远。我们证明了Burgers方程的结果是尖锐的。我们还证明了一个略超临界二维欧拉方程的整体正则性。我们的主要工具是一个非局部的最大值原理，它控制一定的模的连续性的解决方案。
We construct families of smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation (SQG). These solutions can be viewed as the equivalents for this equation of the vortex anti-vortex pairs in the context of the incompressible Euler equation. Our argument relies on the stream function formulation and eventually amounts to solving a fractional nonlinear elliptic equation by variational methods.
Global well-posedness for supercritical SQG with perturbations of radially symmetric data
本文针对超临界耗散SQG方程大初始数据整体适定性难题，利用径向解的增强衰减，证明大径向数据的小扰动存在唯一整体解。
We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation \[ \partial_t θ+ \mathcal{R}^\perp θ\cdot \nabla θ+ Λ^γθ= 0, \qquad θ(\cdot,0) =θ_0 \] on $\mathbb{T}^2 = [0,1]^2$, with $γ\in (0,1)$. The coefficient in front of the dissipative term $Λ^γ= (-Δ)^{γ/2}$ is normalized to $1$. We show that given a smooth initial datum with $\|θ_0\|_{L^2}^{γ/2} \|θ_0\|_{\dot{H}^2}^{1-γ/2}\leq R$, where {\em $R$ is arbitrarily large}, there exists $γ_1 = γ_1(R) \in (0,1)$ such that for $γ\geq γ_1$, the solution of the supercritical SQG equation with dissipation $Λ^γ$ d…
Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation
针对表面准地转方程，证明了任意初始数据下全局弱解的存在性，该解保持哈密顿量（$\dot{H}^{-1/2}$范数）守恒，并通过消失粘性极限获得。
Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations
研究了具有超临界耗散的准地转（QG）型方程（$α<1/2$）弱解的正则性$(-Δ)^α$。本研究的动机是最近的工作Caffarelli和Vasseur，其中他们研究的关键（$α= 1/2$）QG方程\cite{CV}的整体正则性问题。他们的方法依次增加了Leray-Hopf弱解的正则性水平：从$L^2$到$L^\infty$，从$L^\infty$到Hölder（$C^δ$，$δ>0$），从Hölder到经典解。在超临界情况下，Leray-Hopf弱解仍然可以被证明是$L^\infty$，但似乎他们的方法不能很容易地扩展到建立$L^\infty$解的Hölder连续性。为了使他们的方法起作用，我们要求速度在赫尔德空间$C^{1-2α}$中。从$C^δ$和$δ>1-2α$开始，可以通过Besov空间技术建立更高的正则性，并将在其他地方介绍\cite{CW6}。
On the regularity of a class of generalized quasi-geostrophic equations
本文考虑广义准地转方程\partial_t θ+ uPDMATHABMN \nabla θ+ v Λ^βθ=0，\quad u= Λ^α \mathcal{R}^\ botθ，\quad x \in \mathbb{R}^2，其中$ν>0$，$Λ:=\sqrt{-Δ}$，$α\in ]0,1[$和$β\in ]0,2[$ .首先给出了一个产生全空间活动标量非局部最大值原理的一般判据，然后主要应用这个一般判据，对$α\in]0,1[$和$β\in ]α+1,2]$情形，在光滑初值下证明了系统的整体适定性;对$α\in ]0,1[$和$β\in ]2α,α+1]$情形，在适当的初值下证明了系统弱解的局部光滑性和最终正则性.
We study a 1D transport equation with nonlocal velocity. First, we prove eventual regularization of the viscous regularization when dissipation is in the supercritical range with non-negative initial data. Next, we will prove global regularity for solutions when dissipation is slightly supercritical. Both results utilize a nonlocal maximum principle.
Global existence of solutions of a loglog energy-supercritical Klein-Gordon equation
该研究针对n=3、4、5维的loglog能量超临界Klein-Gordon方程，通过反证法结合爆破时刻附近的Strichartz型范数有界分析，证明了解的全局存在性。
Global Large Solution to Navier-Stokes and SQG Equations with Time Oscillation
该研究针对非线性项含时间振荡的三维不可压缩Navier-Stokes方程与二维超临界SQG方程，证明了任意$H^2$初值下全局光滑解的存在性，丰富了两类流体方程的大解全局存在性结论。
Harnack inequalities for nonlocal operators with supercritical drifts and their applications
本文研究带临界/超临界无散度漂移的非局部方程弱解的哈纳克估计，应用该结果建立了三类方程（临界随机准地转、二维分数阶纳维-斯托克斯、临界随机微分方程广义鞅问题）的适定性或弱解存在性。
Homogeneous steady states for the generalized surface quasi-geostrophic equations
本文研究了广义表面准地转方程的均匀稳态解，通过参数化分析确定了不同情形下的存在性和非存在性，并提供了自相似解的例子。
In this paper we show that the solution of the supercrti- cal surface quasi-geostrophic (SQG) equation, starting from initial data in homogeneous critical Besov spaces belong to a subanalytic Gevrey class. In particular, we improve upon the result of Dong and Li in [26], where they showed that the solutions of Chen-Miao-Zhang (cf. [11]) are classical solutions. We extend the approach of Biswas (cf. [7]) to critical, L^p -based Besov spaces, and adapt the point of view of Lemarie- Rieusset (cf. [36]), who treated the operator arising from applying the analytic Gevrey operator to a product of an…
In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the sup…
Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise
该研究针对受时空白噪声导数扰动的奇异表面准地转方程，通过改进的Da Prato--Debussche技巧与凸积分迭代，首次证明了次临界、临界及超临界情形下无穷多非高斯概率强解和遍历平稳解的存在性。
We are motivated by the study of the Microcanonical Variational Principle within the Onsager's description of two-dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails. We obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding Mean Field Equation on convex and "thin" enough domains in the supercritical (with respect to the Moser-Trudinger inequality) regime. This is a brand new achievement since existence results in the supercritical region were previously known \un{only} on multiply connected domains. T…
Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation
该研究证明了超临界区域带强迫的α-SQG方程弱解的非唯一性，构造了非线性不稳定的光滑紧支涡旋，并作为副产品得到了无强迫α-SQG方程既不旋转也不移动的整体光滑解。
Hamiltonian compactness and dissipation for the generalized SQG equation in the inviscid limit
研究无黏极限下广义SQG方程的耗散机制，通过强紧致性防止哈密顿量异常耗散，揭示更一般的机制，证明在临界积分条件下全局解的存在性。
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