On the global well-posedness of the critical quasi-geostrophic equation
Hamadi Abidi, Taoufik Hmidi
在临界Besov空间$\dot B^0_{\infty,1}(\RR^2).$中，我们证明了临界耗散准地转方程在大初值条件下的整体适定性
We prove the global well-posedness of the critical dissipative quasi-geostrophic equation for large initial data belonging to the critical Besov space $\dot B^0_{\infty,1}(\RR^2).$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We derive regularized contour dynamics equations for the motion of infinite sharp fronts in the two-dimensional incompressible Euler, surface quasi-geostrophic (SQG), and generalized surface quasi-geostrophic (gSQG) equations. We derive a cubic approximation of the contour dynamics equation and prove the short-time well-posedness of the approximate equations for generalized surface quasi-geostrophic fronts and weak well-posedness for surface quasi-geostrophic fronts.
We prove the global well-posedness of the continuously stratified inviscid quasi-geostrophic equations in $\Bbb R^3$.
A regularity criterion for the dissipative quasi-geostrophic equations
在混合时空Besov空间中建立了耗散拟地转方程弱解的正则性准则。
On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations
本文利用相似变换方法证明了三维Euler方程、三维Navier-Stokes方程和二维准地转方程的新的\emph{a priori}估计。
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。"
We show the existence of nontrivial stationary weak solutions to the surface quasi-geostrophic equations on the two dimensional periodic torus.
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.
On Solutions of Three Quasi-Geostrophic Models
我们考虑二维准地转模式及其两种不同的正则化。对于具有次临界或临界指标的正则化模型，建立了全局正则性结果。将Onsager关于三维Euler方程弱解猜想的证明和Duchon和Robert的耗散解概念推广到准地转方程的弱解.
In this paper, by using Fourier splitting method and the expanded properties of decay character $r^*$, we establish the algebraic decay rate of higher order derivative of solutions to 2D dissipative quasi-geostrophic flows.
We revisit a model for three-dimensional, inviscid quasi-geostrophic flow on bounded, cylindrical domains introduced by the authors in \cite{nv18}. We prove the local-in-time existence of classical solutions.
In a recent article by Gravejat and Smets, it is built smooth solutions to the inviscid surface quasi-geostrophic equation that have the form of a traveling wave. In this article we work back on their construction to provide solution to a more general class of quasi-geostrophic equation where the half-laplacian is replaced by any fractional laplacian.
Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
给出了临界二维耗散准地转方程的整体适定性的一个初等证明。该论点是基于一个非局部极大值原则，涉及适当的连续模。
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation
受临界耗散准地转方程的启发，我们证明了具有L^2初值和关于漂移的极小假设的漂移-扩散方程是局部Holder连续的.作为应用，我们证明了具有L^2初值和临界扩散（-Δ）^{1/2}的准地转方程的解对任何空间维都是局部光滑的.
We study the existence of doubly connected rotating patches for the inviscid surface quasi- geostrophic equation left open in \cite{HHH}. By using the approach proposed by \cite{CCGS} we also prove that close to the annulus the boundaries are actually analytic curves.
In this article, the authors prove the existence of global weak solutions to the inviscid three-dimensional quasi-geostrophic equation. This equation models the evolution of the temperature on the surface of the earth. It is widely used in geophysics and meteorology.
Global well-posedness for the 2 D quasi-geostrophic equation in a critical Besov space
证明了当（大）数据属于临界尺度不变空间$\dot{B}^{2-2\al}_{2, \infty}\cap L^{2/(2\al-1)}$时，二维准地转方程存在整体唯一的强解.
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.
In this paper, we show that the global solution of the surface anisotropic two-dimensional quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion established by the author in [2] is bounded in Sobolev spaces uniformly with respect to time.
Consider the surface quasi-geostrophic equation with random diffusion, white in time. We show global existence and uniqueness in high probability for the associated Cauchy problem satisfying a Gevrey type bound. This article is inspired by recent work of Glatt-Holtz and Vicol.
We consider surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions involving conservation of a certain family of moduli of continuity.
In this paper, we prove the existence of relative equilibria with holes for quasi-geostrophic shallow-water equations. More precisely, using bifurcation techniques, we establish for any m large enough the existence of two branches of m-fold doubly-connected V-states bifurcating from any annulus of arbitrary size.
We prove that linear instability implies non-linear instability in the energy norm for the critically dissipative quasi-geostrophic equation.
Quasi-geostrophic equations with initial data in Banach spaces of local measures
这篇论文被arXiv管理员删除，因为它剽窃了Jose-A的电子文章HYKE 2005-010（http：//www.hyke.org/preprint/index.php）。Carrillo和Lucas C.f.费雷拉，题为“耗散准地转方程的自相似解和大时间渐近性”。
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.
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 compute the sharp time decay rates of the solutions of the IVP for quasi-geostrophic equation and the Boussinesq model, subject to fractional dissipation. Moreover, we explicitly identify the asymptotic profiles, the kernel of the $α$ stable processes, which are analogues of the Oseen vortices.
We provide two-sided pointwise estimates and uniform asymptotics of the solutions to the subcritical quasi-geostrophic equation with initial data in $L^{2/(α-1)}(\mathbb{R}^2)$. Furthermore, we give upper bound of similar type for any derivative of the solutions. Initial data in $L^{p}(\mathbb{R}^2)$, $p>2/(α-1)$, are also discussed.
The inviscid barotropic quasi-geostrophic equation with a free surface is considered. The free surface mandates a non-standard boundary condition. The global existence existence and uniqueness of a weak solution is established, thanks to the uniform in time bounds on the potential vorticity. The solution is also shown to satisfy the initial and boundary conditions in the classical sense.
Motivated by the recent work of Hassainia and Hmidi [Z. Hassainia, T. Hmidi - On the {V}-states for the generalized quasi-geostrophic equations,arXiv preprint arXiv:1405.0858], we close the question of the existence of convex global rotating solutions for the generalized surface quasi-geostrophic equation for $α\in [1,2)$. We also show $C^{\infty}$ regularity of their boundary for all $α\in (0,2)$.
We consider the stationary Quasi-Geostrophic equation in the whole space $\mathbb R^2$ driven by a force $f$. Under certain smallness assumptions of $f$, we establish the existence of solutions with finite $L^2$ norm. This solution is unique among all solutions with finite energy. The unique solution $Θ$ is also shown to be stable in the sense: any solution of the evolutionary Quasi-Geostrophic equation driven by $f$ and starting with finite energy, will return to $Θ$.
We construct a linear approximation of the solution to the Surface Quasi-Geostrophic Equation in $\mathbb{R}^2$, and obtain a convergence rate in $L^p$ between the solution and this approximation with respect to time. We also demonstrate that the nonlinear term of the solution is bounded sharply in $L^p$ by the same function of time.
We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^2$ or $\bT^2$. Higher regularity of mild solutions with arbitrary initial data in $H^{2-γ}$ is proved. As a corollary, we obtain a global existence result for the critical 2D quasi-geostrophic equations with periodic $\dot H^1$ data. Some decay in time estimates are also provided.
In this paper, we prove the boundedness of Riesz transforms $\partial_{j}(-Δ)^{-1/2}$ ($j=1,2,...,n$) on the Q-type spaces $Q_α^β(\mathbb{R}^{n})$. As an application, we get the well-posedness and regularity of the quasi-geostrophic equation with initial data in $Q_α^{β,-1}(\mathbb{R}^{2}).$
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.
On the global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces
本文研究超临界二维耗散准地转方程。我们得到了一些正则化效应，使得我们能够证明Lebesgue空间$L^p,$ \mbox {with $p\in[1,\infty].$上构造的临界Besov空间中的小初始数据的整体适定性结果。
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.
Well-posedness of the 2D surface quasi-geostrophic equation in variable Lebesgue spaces
本文在变指数Lebesgue空间中研究耗散型表面准地转方程的适定性，利用分数阶热核的衰减估计，建立了解的局部存在性与正则性。
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.
Remarks on the Global Regularity for the Super-Critical 2D Dissipative Quasi-Geostrophic Equation
本文将Kiselev，Nazarov和Volberg最近对临界耗散二维准地转方程适定性的优美证明方法应用到超临界情形。我们证明了如果初值是光滑的和周期的，并且$\left\| \nabla θ_0 \right\|_{L^{\infty}}^{1 - 2 s} \left\| θ_0 \right\|_{L^{\infty}}^{2 s}$很小，其中$s$是分数阶Laplacian的幂，那么超临界耗散二维准地转方程不会出现有限时间奇异性.
The purpose of this note is to study the weak solutions to the inviscid quasi-geostrophic system for initial data belonging to Lebesgue spaces. We give a global existence result as well as detail the connections between several different notions of weak solutions. In addition, we give a condition under which the energy of the system is conserved.
We study the Cauchy problem for the quasi-geostrophic equations with the critical dissipation in the two dimensional half space under the homogeneous Dirichlet boundary condition. We show the global existence, the uniqueness and the analyticity of solutions, and the real analyticity up to the boundary is obtained. We will show one of natural ways to estimate the nonlinear term for functions satisfying the Dirichlet boundary condition.
On the uniqueness of the mild solution of the critical quasi-geostrophic equation
本文在尺度临界齐次 Besov 空间中证明了带临界耗散的二维准地转方程温和解的唯一性，且无需正则性假设。
We study the behavior at infinity in time of the global solution of the anisotropic quasi-geostrophic equation $θ\in C_b(\mathbb{R}^+,H^s( \mathbb{R}^2))$. We prove that this solution decays to zero as time goes to infinity in $L^p(\mathbb{R}^2)$, $p\in [2,+\infty],$ moreover, we prove also that $\lim\limits_{t\rightarrow +\infty}\|θ(t)\|_{H^s}=0$.
In this article I study Hölder regularity for solutions of a transport equation based in the dissipative quasi-geostrophic equation. Following a recent idea of A. Kiselev and F. Nazarov, I will use the molecular characterization of local Hardy spaces in order to obtain information on Hölder regularity of such solutions. This will be done by following the evolution of molecules in a backward equation.
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。
A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation
我们证明了一个新的Bernstein不等式，它推广了Cannone-Planchon，Danchin和Lemarié-Rieusset的结果.作为该不等式的应用，我们证明了临界Besov空间中具有临界和超临界耗散的二维准地转方程在小初值下的整体适定性，以及在大初值下的局部适定性.
In this paper, we prove the nonlinear orbital stability of vortex dipoles for the quasi-geostrophic shallow-water (QGSW) equations. The vortex dipoles are explicit travelling wave solutions to the QGSW equations, which are analogues of the classical circular vortex of Lamb and Chaplygin for the steady planar Euler equations. We establish a variational characterization of these vortex poles, which provides a basis for the stability result.
In this paper, we study the super-critical Quasi-Geostrophic equation in Gevrey-Sobolev space. We prove the local existence of $(QG)$ for any large initial data and we give an exponential type of Blow-up to the solution. Moreover, we establish the existence global for a small initial data and we show that $\|θ\|_{H^s_{a, α^{-1}}}$ decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.
In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion which represents a general case of the classical surface quasi-geostrophic equation. On the one hand, we will show the local existence and uniqueness of the solution in Sobolev space $H^{2-2α}(\mathbb{R}^2)\cap H^{2-2β}(\mathbb{R}^2)$, which is the critical space in the classical case. Furthermore, we will demonstrate that the solution is global even when the initial data is very small. Finally, we will study the asymptotic representation of ou…
We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation $ε$ from a radial stationary solution $θ=|x|$. We use a modified energy method to prove the existence time of classical solutions from $\frac{1}ε$ to a time scale of $\frac{1}{ε^4}$. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space.
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.
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…
Global regularity for a modified critical dissipative quasi-geostrophic equation
本文考虑修正的准地转方程\begin{gather*} \del_t θ+ (u \cdot \grad) θ+ κΛ^αθ= 0 u = Λ^{α- 1} R^{\perp}θ. \end{gather*}，其中包含$κ> 0$，$α\in (0,1]$和$θ_0 \in \lp{2}(\R^2)$。我们注意到，额外的$Λ^{α- 1}$的引入是为了使该系统的标度不变性类似于临界准地转方程的标度不变性。本文利用Besov空间技巧证明了该系统强解的整体存在性和正则性。
In this paper, we consider the following modified quasi-geostrophic equations $\partial_tθ+Λ^αθ+u\vec\nablaθ=0$, $u=Λ^{α-1}\mathcal{R}^\perp(θ)$ where $α\in ]0,1[$ is a fixed parameter. This equation was recently introduced by P. Constantin, G. Iyer and J. Wu in \cite{CIW} as a modification of the classical quasi-geostrophic equation. In this paper, we prove that for any initial data $θ_\ast$ in the Sobolev space $H^1(\mathbb{R}^2),$ the equation (MQG) has a global and smooth solution $θ$ in $C(\mathbb{R}^{+},H^1(\mathbb{R}^2)) .$
Existence of Non-Decaying Solutions to the Generalized Surface Quasi-Geostrophic Equations
本文证明了广义表面准地转方程在 Hölder-Zygmund 空间和一致局部 Sobolev 空间中非衰减解的短时存在性与唯一性。
In this paper we show the global existence for critical dissipative quasi-geostrophic equations if $\|\widehat{θ^0}\|_{L^1}$ is small enough; among others we prove the analyticity of such a solution. If in addition the initial condition verifies $|D|^{-δ}θ^0\in L^1(\mathbb R^2)$ with $0<δ<1$, then the solution remains regular and $\lim_{t\rightarrow\infty}t^δ\|\widehatθ(t)\|_{L^1}=0$. Fourier analysis and standard techniques are used.
We use a nonlocal 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 $i k^\perp |k|^{-1} m(k|)$, where $m$ is a smooth increasing function that grows slower than $\log \log |k|$ as $|k|\rightarrow \infty$.
The role of the dimension in uniqueness results for the stationary quasi-geostrophic system
本文针对二维、三维和四维稳态分数阶准地转方程，在特定Lebesgue可积性条件下证明了弱解的唯一性，并阐明了维数与Laplace算子分数幂次对唯一性的影响。
We establish analyticity of the subcritical and critical quasi-geostrophic equations in critical Besov spaces. The main method is so-called Gevrey estimates, which is motivated by the work of Foias and Temam. We show that mild solutions θ(t), are Gevrey regular, i.e. they satisfy the estimate \sup_{t>0}\|e^{αt^{1/γ}Λ_1}θ(t)\|_{\cap L}<\infty. for a suitably chosen α>0 and a scaling invariant Besov space {\cap L}.
In this paper, the authors show the existence of global in time classical solutions to the 3D quasi-geostrophic system with Ekman pumping for any smooth initial value (possibly large). This system couples an inviscid transport equation in $\mathbb{R}^3_+$ with an equation on the boundary satisfied by the trace. The proof combines the De Giorgi regularization effect on the boundary $z=0$ -similar to the so called surface quasi-geostrophic equation- with Beale-Kato-Majda techniques to propagate regularity for $z>0$. A potential theory argument is used to strengthen the regularization effect on t…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。