Taoufik Hmidi, Sahbi Keraani
In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces $L^p,$ \mbox{with $p\in[1,\infty].$ Local results for arbitrary initial data are also given.
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We show that the the 2 D quasi-geostrophic equation has global and unique strong solution, when the (large) data belongs in the critical, scale invariant space $\dot{B}^{2-2\al}_{2, \infty}\cap L^{2/(2\al-1)}$.
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}.
We establish a regularity criterion for weak solutions of the dissipative quasi-geostrophic equations in mixed time-space Besov spaces.
We prove the global well-posedness of the critical dissipative quasi-geostrophic equation for large initial data belonging to the critical Besov space $\dot B^0_{\infty,1}(\RR^2).$
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 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.
In this paper, we consider the modified quasi-geostrophic equation \begin{gather*} \del_t θ+ (u \cdot \grad) θ+ κΛ^αθ= 0 u = Λ^{α- 1} R^{\perp}θ. \end{gather*} with $κ> 0$, $α\in (0,1]$ and $θ_0 \in \lp{2}(\R^2)$. We remark that the extra $Λ^{α- 1}$ is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence and regularity of strong solutions to this system.
Sharp well-posedness and ill-posedness of the stationary quasi-geostrophic equation
本文研究全平面稳态准地转方程，在临界 Besov 空间中确定其适定的精确 $(p,q)$ 范围，并证明其余参数范围均不适定。
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…
On the uniqueness of the quasi-geostrophic equation with the fractional Laplacian
本文研究了带分数阶拉普拉斯算子的表面准地转方程解的唯一性，证明在非齐次Besov空间中无需额外假设即可成立，并提取了方程非线性项良定性所需的最小正则性指标 $s=-1/2$。
Ill-posedness and inviscid limit of the basic equations of fluid dynamics in Besov spaces
本文针对环面上流体动力学基本方程的柯西问题，构造新初始数据证明欧拉方程和表面准地转方程$B^s_{p,\infty}$解的不适定性（覆盖已有结果），并证明纳维-斯托克斯方程与表面准地转方程无粘极限下$B^s_{p,\infty}$收敛性失效。
In this paper, we first generalize a new energy approach, developed by Y. Guo and Y. Wang \cite{GW12}, 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 supercritical and critical quasi-geostrophic equation and the critical Keller-Segel system. We show that the 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 interpolat…
We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of high-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier-Besov-Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large $L^{2}$-norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and cou…
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。
We study the Cauchy problem for the quasi-geostrophic equations in a unit ball of the two dimensional space with the homogeneous Dirichlet boundary condition. We show the existence, the uniqueness of the strong solution in the framework of Besov spaces. We establish a spectral localization technique and commutator estimates.
In this paper, we prove that if the initial data $θ_0$ and its Riesz transforms ($\mathcal{R}_1(θ_0)$ and $\mathcal{R}_2(θ_0)$) belong to the space $(\overline{S(\mathbb{R}^2))}^{B_{\infty}^{1-2α,\infty}}$, where $α\in ]1/2,1[$, then the 2D Quasi-Geostrophic equation with dissipation $α$ has a unique global in time solution $θ$. Moreover, we show that if in addition $θ_0 \in X$ for some functional space $X$ such as Lebesgue, Sobolev and Besov's spaces then the solution $θ$ belongs to the space $C([0,+\infty [,X).$
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…
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.
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.
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.
We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ($α< 1/2$) dissipation $(-Δ)^α$ : If a Leray-Hopf weak solution is Hölder continuous $θ\in C^δ({\mathbb R}^2)$ with $δ>1-2α$ on the time interval $[t_0, t]$, then it is actually a classical solution on $(t_0,t]$.
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.
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 is smooth and periodic, and $\left\| \nabla θ_0 \right\|_{L^{\infty}}^{1 - 2 s} \left\| θ_0 \right\|_{L^{\infty}}^{2 s}$ is small, 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.
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, 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.
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 $|k|\rightarrow \infty$.
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…
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.
Recently, Silvestre proved that certain weak solutions of the slightly supercritical surface quasi-geostrophic equation eventually become smooth. To prove this, he employed a De Giorgi type argument originated in the work of Caffarelli and Vasseur. Kiselev and Nazarov proved a variation of the result of Caffarelli and Vasseur by introducing a class of test functions. Motivated by the results of Silvestre, we will modify the class of test functions from the work of Kiselev and Nazarov and use this modified class to show that a solution to the supercritical SQG that is smooth up to a certain tim…
This paper was withdrawn by arXiv admin because it plagiarizes "Jiahong Wu The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation 2005 Nonlinearity 18 139-154."
Stationary solutions to the critical and super-critical quasi-geostrophic equation in the scaling critical Sobolev space
本文在尺度临界 Sobolev 空间中研究临界与超临界耗散准地转方程的稳态问题，证明小外力下小解唯一存在，并阐明数据到解映射连续但不一致连续。
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。
Unique existence of solutions to the inviscid SQG equation in a critical space
本文研究了在二维有界域内满足齐次狄利克雷边界条件的SQG方程的初值问题，证明了在临界贝塞尔空间$\dot B^2_{2,1}$中强解的唯一存在性，方法基于与狄利克雷拉普拉斯算子相关的二进制分解的谱局部化。
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.
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.
Uniqueness of the dissipative SQG without time-continuity assumption
研究耗散表面准地转方程在不假设时间连续性和解小性条件下的唯一性，通过能量法和分数阶热半群平滑效应证明在临界Lebesgue空间和非齐次Besov空间中的唯一性。
We address the well-posedness theory for the magento-geostrophic equation, namely an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In the presence of supercritical fractional diffusion given by (-Δ)^γ, where 0<γ<1, we discover that for γ>1/2 the equations are locally well-posed, while for γ<1/2 they are ill-posed, in the sense that there is no Lipschitz solution map. The main reason for the striking loss of regularity when γgoes below 1/2 is that the constitutive law used to obtain the velocity from the active scalar …
In this paper, we prove the global regularity of smooth solutions to 2D surface quasi-geostrophic (SQG) equations with super-critical dissipation for a class of large initial data, where the velocity and temperature can be arbitrarily large in spaces $L^\infty(\mathbb{R}^2)$ and $H^3(\mathbb{R}^2)$. This result could be seen as an improvement work of Liu-Pan-Wu \cite{Liu}, for it's without any smallness hypothesis of the $L^\infty(\mathbb{R}^2)$ norm of the initial data.
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.
This article is devoted to the study of the critical dissipative surface quasi-geostrophic $(SQG)$ equation in $\mathbb{R}^2$. For any initial data $θ_{0}$ belonging to the space $Λ^{s} ( H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)$, we show that the critical (SQG) equation has at least one global weak solution in time for all $1/4\leq s \leq 1/2$ and at least one local weak solution in time for all $0<s<1/4$. The proof for the global existence is based on a new energy inequality which improves the one obtain in \cite{Laz} whereas the local existence uses more refined energy estima…
In this paper, we first establish the local well-posedness (existence, uniqueness and continuous dependence) for the Fornberg-Whitham equation in both supercritical Besov spaces $B^s_{p,r},\ s>1+\frac{1}{p},\ 1\leq p,r\leq+\infty$ and critical Besov spaces $B^{1+\frac{1}{p}}_{p,1},\ 1\leq p<+\infty$, which improves the previous work \cite{y2,ho,ht}. Then, we prove the solution is not uniformly continuous dependence on the initial data in supercritical Besov spaces $B^s_{p,r},\ s>1+\frac{1}{p},\ 1\leq p\leq+\infty,\ 1\leq r<+\infty$ and critical Besov spaces $B^{1+\frac{1}{p}}_{p,1},\ 1\leq p<+…
The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model then solutions always exist if the dissipation's wave number dependence is super-linear. Below this critical power the dissipation appears to be insufficient. For instance, it is not known if the critical dissipative QG equation has global smooth solutions for arbitrary large initial data. In this paper we prove existence and uniqueness of global classical solutions of the critical dissipative QG equation for initial data that have small $L^\inft…
Strong ill-posedness of the 2D Boussinesq equations in supercritical Besov spaces
通过构造任意小范数的初始数据导致解瞬间范数膨胀，证明二维Boussinesq方程在超临界Besov空间和Sobolev空间中强不适定，并进一步证明静水平衡附近扰动的失稳。
In this paper, we first establish the local well-posedness and continuous dependence for the rotation Camassa-Holm equation modelling the equatorial water waves with the weak Coriolis effect in nonhomogeneous Besov spaces $B^s_{p,r}$ with $s>1+1/p$ or $s=1+1/p,\ p\in[1,+\infty),\ r=1$ by a new way: the compactness argument and Lagrangian coordinate transformation, which removes the index constraint $s>3/2$ and improves our previous work \cite{guoy1}. Then, we prove the solution is not uniformly continuous dependence on the initial data in both supercritical and critical Besov spaces.
We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to $\infty$. Both results rely on the Strichartz-type estimates for the corresponding linear equation.
The local well-posedness, blow-up and global solution of a new integrable system in Besov spaces
该研究在临界与超临界Besov空间中建立一类新可积N-peakon系统柯西问题的局部适定性，推导爆破准则与全局解存在条件，并通过一阶差分法给出相关仿真算例。
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。
We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($α<1/2$) dissipation $(-Δ)^α$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ($α= 1/2$) QG equation \cite{CV}. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from $L^2$ to $L^\infty$, from $L^\infty$ to Hölder ($C^δ$, $δ>0$), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be $L^\infty$, but it do…
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…
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…
The nonlinear estimates on quantum Besov spaces
本文研究了非光滑符号的超位置算子在量子贝塞尔空间上的有界性估计，扩展了McDonald的结果，并在非交换PDE的适定性理论中具有广泛应用。
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。
The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasi-geostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.
Global unique solutions with instantaneous loss of regularity for SQG with fractional diffusion
本文构造了在超临界分数阶扩散下耗散表面准地转方程（$α$-SQG）的全局唯一解，该解会瞬时失去正则性。
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。
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。
Norm Inflation for Inviscid and Fully Dissipative Boussinesq Systems in Supercritical Spaces
研究二维Boussinesq系统在超临界Besov空间的范数膨胀，通过分析无粘和全耗散系统在特定条件下的情况，得出范数膨胀发生在密度分量\(\rho\)且速度分量\(u\)有界等结论，覆盖多数满足条件的超临界空间。
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。
On rough Calderón solutions to the Navier-Stokes equations and applications to the singular set
本文采用类Calderón拆分方法，证明了超临界Besov空间初始数据下Navier-Stokes方程全局弱解的存在性，填补了Besov空间中Leray理论与温和解理论的空白，并研究了更粗糙框架下I型爆破假设的奇异集结构。
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