强制2D纳维-斯托克斯方程和耗散SQG方程的非唯一性

强迫SQG的非唯一弱解

Non-unique weak solutions of forced SQG

本文通过适用于两个解的和差系统的convex integration方案，成功构造了强迫SQG方程的非唯一弱解。

Vishik非唯一性定理的证明：强迫二维欧拉方程在涡度类L¹∩L^p中的情况

A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation

本文给出强迫二维欧拉方程在涡度类L¹∩L^p中Vishik非唯一性定理的更简单证明，通过构造稳定的涡旋来简化证明过程。

三维受迫分数阶Navier-Stokes方程Leray-Hopf解的非唯一性，直至J. L. Lions指数

Nonuniqueness of Leray-Hopf solutions to the forced fractional Navier-Stokes Equations in three dimensions, up to the J. L. Lions exponent

本文证明在三维受迫分数阶Navier-Stokes方程中，当指数$\alpha\in(1/2,5/4)$时存在非唯一Leray-Hopf解，从而表明J. L. Lions指数的尖锐性。

Long time dynamics of forced critical SQG

We prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi-geostrophic equation (SQG) and prove that it has finite fractal (box-counting) dimension. In order to do so we give a new proof of global regularity for critical SQG. The main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional Laplacian, which is used to bootstrap the regularity directly from $L^\infty$ to $C^α$, without the use of De Giorgi techniques. We prove that for large time, the norm of the solution measured in a sufficiently str…

概率方法在强制性表面准地转方程中的能量相等性

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。

Absence of anomalous dissipation of energy in forced two dimensional fluid equations

We prove absence of anomalous dissipation of energy for forced critical SQG in two space dimensions.

Stationary solutions and nonuniqueness of weak solutions for the Navier-Stokes equations in high dimensions

Consider the unforced incompressible homogeneous Navier-Stokes equations on the $d$-torus $\mathbb{T}^d$ where $d\geq 4$ is the space dimension. It is shown that there exist nontrivial steady-state weak solutions $u\in L^{2}(\mathbb{T}^d)$. The result implies the nonuniqueness of finite energy weak solutions for the Navier-Stokes equations in dimensions $d \geq 4$. And it also suggests that the uniqueness of forced stationary problem is likely to fail however smooth the given force is.

不稳定涡旋、带强迫的尖锐非唯一性及SQG方程的整体光滑解

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。

关于 Dirichlet 分数阶 Laplace 算子及其在有界区域上 SQG 方程中的应用

On the Dirichlet Fractional Laplacian and Applications to the SQG Equation on Bounded Domains

本文研究有界区域上的分数阶 Dirichlet Laplace 算子，建立乘积估计与非线性 Poincaré 不等式，并用于证明受稳态外力驱动的 SQG 方程存在有限维全局吸引子。

哈密顿量紧致性与耗散在无黏极限下的广义SQG方程中

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。

随机SQG方程中双级联的混合三阶通量定律

Mixed Third-Order Flux Laws for Dual Cascade in the Stochastic SQG Equation

研究随机强迫耗散表面准地转方程的双级联通量定律，推导了直接级联和逆级联的混合三阶结构函数定律，并证明了Onsager型障碍结果。

具有$L^{4/3+}$主动标量的SQG方程的灵活性

Flexibility for the SQG Equation with an $L^{4/3+}$ Active Scalar

该研究针对二维环面上的无粘表面准地转方程，利用凸积分格式在$p=4/3+10^{-5$的主动标量下，证明了弱解的灵活性与非唯一性，拓展了浓度临界指数相关结论。

On the forced surface quasi-geostrophic equation: Existence of steady states and sharp relaxation rates

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…

Sobolev适定性区域内广义SQG方程的有限时间爆破

Blow-up at finite time for the generalized SQG equations in the Sobolev well-posedness regime

该研究针对γ∈(0,1)的受迫广义SQG方程，构造了初始数据与力，证明其经典有限能量解在Sobolev适定性区域内于t=1时发生有限时间爆破，为该领域首个此类结果。