Xinyu Cheng, Hyunju Kwon, Dong Li
We show the existence of nontrivial stationary weak solutions to the surface quasi-geostrophic equations on the two dimensional periodic torus.
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
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。
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。
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 $s=-1/2$。
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 continue our study of the dynamics of a nearly inviscid periodic surface quasi-geostrophic equation. Here we consider a slightly diffusive stochastic SQG equation of the form \begin{equation*} \begin{cases} dθ_t + |D|^{2δ}θ_t\,dx + (u_t \cdot \nabla)θ_t\,dx + |D|^δdW_t = 0 \\ u_t = \nabla^\perp|D|^{-1}θ_t. \end{cases} \end{equation*} We construct global energy solutions as introduced by P. Goncalves and M. Jara (2014) for any $δ> 0$, so that any small amount of diffusion permits us to construct solutions. We show moreover that pathwise uniqueness of these energy solutions holds in the prese…
This work gathers new results concerning the semi-geostrophic equations: existence and stability of measure valued solutions, existence and uniqueness of solutions under certain continuity conditions for the density, convergence to the incompressible Euler equations. Meanwhile, a general technique to prove uniqueness of sufficiently smooth solutions to non-linearly coupled system is introduced, using optimal transportation.
We show that weak solutions to the 3D quasi-geostrophic system in the class $C^ζ_{t,x}$ for $ζ<\frac{1}{5}$ are not unique and may achieve any smooth, non-negative energy profile. Our proof follows a convex integration scheme which utilizes the stratified nature of the quasi-geostrophic velocity field, providing a link with the 2D Euler equations. In fact we observe that under particular circumstances our construction coincides with the convex integration scheme for the 2D Euler equations introduced by Choffrut, De Lellis, and Szekélyhidi \cite{cdlsj12} and recovers a result which can already …
We consider the asymptotic behavior of the surface quasi-geostrophic equation, subject to a small external force. Under suitable assumptions on the forcing, we first construct the steady states and we provide a number of useful a posteriori estimates for them. Importantly, to do so, we only impose minimal cancellation conditions on the forcing function. Our main result is that all $L^1\cap L^\infty$ localized initial data produces global solutions of the forced SQG, which converge to the steady states in $L^p(\mathbf R^2), 1<p\leq 2$ as time goes to infinity. This establishes that the steady s…
Via probabilistic convex integration, we prove non-uniqueness in law of the two-dimensional surface quasi-geostrophic equations forced by random noise of additive type. In its proof we work on the equation of the momentum rather than the temperature, which is new in the study of the stochastic surface quasi-geostrophic equations. We also generalize the classical Calder$\acute{\mathrm{o}}$n commutator estimate to the case of fractional Laplacians.
Slow traveling-wave solutions for the generalized surface quasi-geostrophic equation
本文研究了广义表面准地转方程中慢传播行波解的存在性与稳定性，并通过变分方法和集中-紧致原理建立了相关稳定性定理。
Remark on the uniqueness of the mild solution of SQG equation
本文受三维Navier-Stokes方程唯一性启发，证明了二维表面准地转方程在缩放临界Lebesgue空间中温和解的唯一性。
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 study the three dimensional active scalar equation called the magneto-geostropic equation which was proposed by Moffatt and Loper as a model for the geodynamo processes in the Earth's fluid core. When the viscosity of the fluid is positive, the constitutive law that relates the drift velocity $u(x,t)$ and the scalar temperature $θ(x,t)$ produces two orders of smoothing. We study the implications of this property. For example, we prove that in the case of the non-diffusive ($\varepsilon_κ=0$) active scalar equation, initial data $θ_0\in L^3$ implies the existence of unique, global weak solut…
We develop a theory of self-similar solutions to the critical surface quasi-geostrophic equations. We construct self-similar solutions for arbitrarily large data in various regularity classes and demonstrate, in the small data regime, uniqueness and global asymptotic stability. These solutions are non-decaying as $|x| \to +\infty$, which leads to ambiguity in the velocity $\vec{R}^\perp θ$. This ambiguity is corrected by imposing $m$-fold rotational symmetry. The self-similar solutions exhibited here lie just beyond the known well-posedness theory and are expected to shed light on potential no…
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.
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.
We prove the existence of the V-states for the generalized inviscid SQG equations with $α\in ]0,1[.$ These structures are special rotating simply connected patches with $m-$ fold symmetry bifurcating from the trivial solution at some explicit values of the angular velocity. This produces, inter alia, an infinite family of non stationary global solutions with uniqueness.
Non-uniqueness in law of the surface quasi-geostrophic equations: the case of linear multiplicative noise
本文研究线性乘性噪声驱动的表面准地转方程，通过凸积分技术并利用剪切流关键观察与随机交换子误差改写，证明了其法律非唯一性。
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。
We construct a determining form for the surface quasi-geostrophic (SQG) equation with subcritical dissipation. In particular, we show that the global attractor for this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) called a determining form. Indeed, there is a one-to-one correspondence between the trajectories in the global attractor of the SQG equation and the steady state solutions of the determining form. The determining form is a true ODE in the sense that its vector field is Lipschitz. This is shown by combining De Giorgi techniques and elem…
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 a three dimensional thermocline planetary geostrophic ``horizontal" hyper--diffusion model of the gyre-scale midlatitude ocean. We show the global existence and uniqueness of the weak and strong solutions to this model. Moreover, we establish the existence of a finite dimensional global attractor to this dissipative evolution system. Preliminary computational tests indicate that our hyper--diffusion model does not exhibit any of the nonphysical instabilities near the literal boundary which are observed numerically in other models.
We provide a variational construction of special solutions to the generalized surface quasi-geostrophic equations. These solutions take the form of N vortex patches with N-fold symmetry , which are steady in a uniformly rotating frame. Moreover, we investigate their asymptotic properties when the size of the corresponding patches vanishes. In this limit, we prove these solutions to be a desingularization of N Dirac masses with the same intensity, located on the N vertices of a regular polygon rotating at a constant angular velocity.
Metastability for the dissipative quasi-geostrophic equation and the non-local enhancement
该研究针对小黏度二维线性化耗散准地转方程的亚稳态问题，证明了线性增强耗散并得到耗散率，还发现非局部项可通过剪切-扩散机制再次增强扩散效应。
This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasi-geostrophic (SQG) equation with the velocity field $u$ related to the scalar $θ$ by $u=\nabla^\perpΛ^{β-2}θ$, where $1<β\le 2$ and $Λ=(-Δ)^{1/2}$ is the Zygmund operator. The borderline case $β=1$ corresponds to the SQG equation and the situation is more singular for $β>1$. We obtain the local existence and uniqueness of classical solutions, the global existence…
We consider a class of steady solutions of the semi-geostrophic equations on $\mathbb{R}^3$ and derive the linearised dynamics around those solutions. The linear PDE which governs perturbations around those steady states is a transport equation featuring a pseudo-differential operator of order 0. We study well-posedness of this equation in $L^2(\mathbb{R}^3;\mathbb{R}^3)$ introducing a representation formula for the solutions, and extend the result to the space of tempered distributions on $\mathbb{R}^{3}$. We investigate stability of the steady solutions by looking at plane wave solutions of …
We prove the well posedness: global existence, uniqueness and regularity of the solutions, of a class of d-dimensional fractional stochastic active scalar equations. This class includes the stochastic, dD-quasi-geostrophic equation, $ d\geq 1$, fractional Burgers equation on the circle, fractional nonlocal transport equation and the 2D-fractional vorticity Navier-Stokes equation. We consider the multiplicative noise with locally Lipschitz diffusion term in both, the free and no free divergence modes. The random noise is given by an $Q-$Wiener process with the covariance $Q$ being either of fin…
We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is well-posed. In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.
Existence and stability of smooth traveling circular pairs for the generalized surface quasi-geostrophic equation
本文针对广义表面准地转方程，通过能量最大化方法构造了光滑行进的反向旋转圆形涡对，证明了极大元的唯一性与序列紧性，并进一步确立了这些涡对的轨道稳定性。
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。
The semi-geostrophic equations have attracted the attention of the physical and mathematical communities since the work of Hoskins in the 1970s owing to their ability to model the formation of fronts in rotation-dominated flows, and also to their connection with optimal transport theory. In this paper, we study an active scalar equation, whose activity is determined by way of a Neumann-to-Dirichlet map associated to a fully nonlinear second-order Neumann boundary value problem on the infinite strip $\mathbb{R}^{2}\times(0,1)$, that models a semi-geostrophic flow in regime of constant potential…
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…
Non-uniqueness up to the Onsager threshold for the forced SQG equation
本文通过凸积分技术的交替形式，证明了受迫无粘表面准地转方程在特定弱解类中直至Onsager阈值的非唯一性。
This paper is devoted to the study of the dynamical behavior of the critically dissipative quasi-geostrophic equation in $\textbf{T}^2$. We prove that this system possesses time-dependent periodic solutions, bifurcating from a smooth steady solution, i.e. a Hopf-Bifurcation theorem.
The quasi-geostrophic equation or the Euler equation with dissipation studied in the present paper is a simplified form of the atmospheric circulation model introduced by Charney and DeVore [J. Atmos. Sci. 36(1979), 1205-1216] on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking. The fluid motion defined by the equation is driven by a zonal thermal forcing and an Ekman friction forcing measured by $κ>0$. It is proved that the steady-state solution is unique for $κ>1$ while multiple steady-state solutions exist for $κ<κ_{crit}$ with respect …
Surface quasi-geostrophic equations forced by random noise: prescribed energy and non-unique Markov selections
本文研究受加性与线性乘性随机噪声驱动的二维表面准地转方程动量形式，通过构造能量为任意指定确定性函数的解，证明相应弱解的几乎必然马尔可夫选取具有非唯一性。
Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular that the active scalar. In \cite{Friedlander-Vicol_3}, the authors prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for periodic perturbations with initial data localized …
Well-posedness and convergence of entropic approximation of semi-geostrophic equations
本文证明了半地转方程熵逼近解的存在性、唯一性，以及当熵参数消失时向弱解的收敛性，并验证了数值离散化的收敛性。
This paper is about the evolution of a temperature front governed by the Surface quasi-geostrophic equation. The existence part of that program within the scale of Sobolev spaces was obtained by one of the authors [10]. Here we revisit that proof introducing some new tools and points of view which allow us to conclude the also needed uniqueness result.
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。
We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô's formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Lévy noise is established.
It is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction, coupled to the heat diffusion and transport, is mathematically ill-posed. This is because the no-normal flow physical boundary condition implicitly produces an additional boundary condition for the temperature at the literal boundary. This additional boundary condition is different, because of the Coriolis forcing term, than the no heat flux physical boundary condition. Consequently, the second order parabolic heat equation is over de…
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 investigate the uniqueness of solutions of the steady planar Navier-Stokes equations with different boundary conditions in the exterior domain. For a class of incompressible flow with constant vorticity, we prove the uniqueness of the solution under the enhanced Navier boundary conditions. At the same time, some counterexamples are given to show that the uniqueness of the solution fails under the Navier boundary conditions. For the general incompressible flow with Dirichlet boundary condition, we prove various sufficient conditions for the uniqueness of the solution.
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。
In this paper we study a higher order viscous quasi-geostrophic type equation. This equation was derived in [11] as the limit dynamics of a singularly perturbed Navier-Stokes-Korteweg system with Coriolis force, when the Mach, Rossby and Weber numbers go to zero at the same rate. The scope of the present paper is twofold. First of all, we investigate well-posedness of such a model on the whole space $\R^2$: we prove that it is well-posed in $H^s$ for any $s\geq3$, globally in time. Interestingly enough, we show that this equation owns two levels of energy estimates, for which one gets existenc…
Global Uniqueness of Subsonic Flows for the Steady Euler-Poisson System
研究证明了在受限喷嘴中稳态欧拉-泊松系统亚音速流的全局唯一性，基于亚音速状态集合的凸性及能量估计。
On the local uniqueness of steady states for the Vlasov-Poisson system
本文受Vlasov-Poisson系统稳态定量稳定性研究及二维Euler方程类似结果的启发，研究了特定稳态附近的局部唯一性问题。
Under the condition of small external forces, we obtain existence of a weak solution of the steady Hall-MHD system with Hölder continuous magnetic field. We also established regularity of weak solutions provided that magnetic fields are bounded. For sufficiently small external forces, uniqueness result is also established.
In this paper we study the uniqueness property of solutions to the steady incompressible Euler equations with perturbations in $\Bbb R^N$. Our perturbations include as special cases the Euler equations with a `single signed' nonlinear term, the self-similar Euler equations, and the steady Navier-Stokes equations. For these equations show that suitable decay assumptions at infinity on the solution or its derivatives, imposed by the $L^q$ conditions imply that the only possible solution is zero.
The paper is concerned with the steady-state Burgers equation of fractional dissipation on the real line. We first prove the global existence of viscosity weak solutions to the fractal Burgers equation driven by the external force. Then the existence and uniqueness of solution with finite $H^{\fracα{2}}$ energy to the steady-state equation are established by estimating the decay of fractal Burgers' solutions. Furthermore, we show that the unique steady-state solution is nonlinearly stable, which means any viscosity weak solution of fractal Burgers equation, starting close to the steady-state s…
The Derivative Structure for a Quadratic Nonlinearity and Uniqueness for SQG
本文研究有界光滑区域上的二维 SQG 方程，通过处理分数阶 Laplace 算子并推导非线性项的二阶导数结构，证明了 $L^2$ 温和解的唯一性。
Probabilistic Approaches to The Energy Equality in Forced Surface Quasi-Geostrophic Equations
本文通过概率方法研究了强制性表面准地转方程的能量相等性，证明了大偏差的上下界匹配，并探讨了非高斯大偏差与确定性方程唯一性问题的联系。
We consider a nonlocal aggregation equation with degenerate diffusion, which describes the mean-field limit of interacting particles driven by nonlocal interactions and localized repulsion. When the interaction potential is attractive, it is previously known that all steady states must be radially decreasing up to a translation, but uniqueness (for a given mass) within the radial class was open, except for some special interaction potentials. For general attractive potentials, we show that the uniqueness/non-uniqueness criteria are determined by the power of the degenerate diffusion, with the …
Regularity, uniqueness and the relative size of small and large scales in SQG flows
本文针对超临界耗散SQG方程的正则性与唯一性开放问题，利用能量方法和Littlewood-Paley理论，研究了假设爆破与非唯一性情形中小尺度和大尺度的必然活跃程度，扩展了先前关于三维Navier-Stokes方程的结果。
We consider the inverse boundary value problem for the steady state convection diffusion equation. We prove that a velocity field $V$, is uniquely determined by the Dirichlet-to-Neumann map, when $V \in C^{0,γ} (Ω)$, $2/3< γ\leq 1$, i.e. when $V$ is a Hölder continuous vector field with $2/3< γ\leq 1$.
A global well-posedness result for the three-dimensional inviscid quasi-geostrophic equation over a cylindrical domain
该研究在柱形域上证明了三维无粘quasi-地转方程在特定边界条件下的全局良好设定性，针对初始位势涡度场的有界性和可微性给出了解的存在性和唯一性结果。
Dynamic Behavior of a Multi-Layer Quasi-Geostrophic Model: Weak and Time-Periodic Solutions
本文研究准地转两层系统的全局弱解存在唯一性，并证明从两个同角速度旋转的初始圆盘可分支产生m重时间周期解，同时克服了多层耦合给Crandall-Rabinowitz定理应用带来的困难。
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。
We establish the local uniqueness of steady transonic shock solutions with spherical symmetry for the three-dimensional full Euler equations. These transonic shock-fronts are important for understanding transonic shock phenomena in divergent nozzles. From mathematical point of view, we show the uniqueness of solutions of a free boundary problem for a multidimensional quasilinear system of mixed-composite elliptic--hyperbolic type. To this end, we develop a decomposition of the Euler system which works in a general Riemannian manifold, a method to study a Venttsel problem of nonclassical nonloc…
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。