Global weak solutions for SQG in bounded domains

Nonuniqueness of weak solutions to the SQG equation

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 in the survey arXiv:1111.2700 by De Lellis and Székelyhidi Jr. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian.

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。

Estimates near the boundary for critical SQG

We obtain estimates near the boundary for the critical dissipative SQG equation in bounded domains, with the square root of the Dirichlet Laplacian dissipation. We prove that global regularity up to the boundary holds if and only if a certain quantitative vanishing of the scalar at the boundary is maintained.

A direct approach to nonuniqueness and failure of compactness for the SQG equation

We give an alternative proof of the nonuniqueness of weak solutions to the surface quasigeostrophic equation (SQG) first shown in [Buckmaster-Shkoller-Vicol, '16]. Our approach proceeds directly at the level of the scalar field. Furthermore, we prove that every smooth scalar field with compact support that conserves the integral can be realized as a weak limit of solutions to SQG.

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。

Smooth attractors for weak solutions of the SQG equation with critical dissipation

We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Hölder norms, which complement the existing estimates based on commutator analysis.

On nonexistence of splash singularities for the $α$-SQG patches

In this paper, we consider patch solutions to the $α$-SQG equation and derive new criteria for the absence of splash singularity where different patches or parts of the same patch collide in finite time. Our criterion refines a result due to Gancedo and Strain \cite{GS}, providing a condition on the growth of curvature of the patch necessary for the splash and an exponential in time lower bound on the distance between patches with bounded curvature.

On the determining wavenumber for the nonautonomous subcritical SQG equation

A time-dependent determining wavenumber was introduced in [5] to estimate the number of determining modes for the surface quasi-geostrophic (SQG) equation. In this paper we continue this investigation focusing on the subcritical case and study trajectories inside an absorbing set bounded in $L^\infty$. Utilizing this bound we find a time-independent determining wavenumber that improves the estimate obtained in [5]. This classical approach is more direct, but it is contingent on the existence of the $L^\infty$ absorbing set.

Uniformly attracting limit sets for the critically dissipative SQG equation

We consider the global attractor of the critical SQG semigroup $S(t)$ on the scale-invariant space $H^1(\mathbb{T}^2)$. It was shown in~\cite{CTV13} that this attractor is finite dimensional, and that it attracts uniformly bounded sets in $H^{1+δ}(\mathbb{T}^2)$ for any $δ>0$, leaving open the question of uniform attraction in $H^1(\mathbb{T}^2)$. In this paper we prove the uniform attraction in $H^1(\mathbb{T}^2)$, by combining ideas from DeGiorgi iteration and nonlinear maximum principles.

Global and local existence for the dissipative critical SQG equation with small oscillations

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…

Sobolev空间中广义SQG方程弱解的存在性

Existence of weak solutions to the generalized SQG equations in Sobolev spaces

该研究针对二维耗散型广义表面准地转方程，在初始数据属于Ḣ^{β/2 -1}(ℝ^2)的条件下证明了弱解的整体存在性，将Marchand的解框架推广到更大的主动标量方程族。

SQG方程在临界空间中解的唯一存在性

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。

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…

Non-unique stationary solutions of forced SQG

We show the existence of non-unique stationary weak solutions for forced surface quasi-geostrophic (SQG) equation via a convex integration scheme. The scheme is implemented for the sum-difference system of two distinct solutions. Through this scheme, one observes the external forcing is naturally generated accompanying the flexibility in means of lack of uniqueness. It thus provides a transparent way to reveal the flexibility of the system with the presence of a forcing.

关于SQG及其推广的守恒律与非线性项结构

On the conservation laws and the structure of the nonlinearity for SQG and its generalizations

该研究通过对非线性项的新定义，证明SQG及mSQG方程弱解的角动量守恒，在更优假设下改进哈密顿量守恒判据，且新非线性估计最优并唯一刻画mSQG非线性项特征。

SQG 的 Onsager 型定理

An Onsager-type theorem for SQG

本文利用 Nash 迭代与线性解耦方法构造时间紧支集的 SQG 非平凡弱解，证明其违反 Hamiltonian 守恒，从而在临界正则性意义上解决 SQG 的 Onsager 猜想。

涡度在点涡附近为常数时SQG涡-波系统的存在性与唯一性

Existence and Uniqueness for the SQG Vortex-Wave System when the Vorticity is Constant near the Point-Vortex

本文研究带参数0 < s < 1的Surface Quasi-Geostrophic方程的涡-波系统，得到经典解的局部存在性、稳定性与爆破准则，在亚临界和临界情形下分别建立了弱解和V-弱解的全局存在性。

临界SQG方程的范数膨胀

Norm Inflation For The Critical SQG Equation

本文研究临界耗散表面准地转（SQG）方程，证明在临界空间H^1中解映射不一致有界，并构造了从光滑紧支初值出发的H^1范数膨胀解，同时在超临界Sobolev空间W^{β,p}中展示了小数据范数膨胀。

直至Onsager阈值的受迫SQG方程的非唯一性

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。

二次非线性项的导数结构与 SQG 的唯一性

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。

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。

广义SQG方程的适定性与极限行为

On wellposedness and limiting behavior of generalized SQG equations

本文研究二维广义SQG方程，证明当参数β接近β₀时，光滑解的存在区间保持一致，并发展了守恒律估计和一致有界的交换子估计，揭示了方程的适定性与极限行为。

半平面上广义SQG方程的局部正则性与有限时间奇异性

Local regularity and finite time singularity for the generalized SQG equation on the half-plane

该研究针对半平面上的广义SQG方程，明确了α∈(0,1/4]时的局部适定性与有限时间爆破解的存在性，同时证明α>1/4时方程不适定，完善了该类方程的正则性理论。

具有洛伦兹数据的无粘表面准地转方程的弱解

Weak Solutions for Inviscid SQG with Lorentz Data

研究在\(\mathbb{R}^2\)及光滑有界域中无粘表面准地转方程的全局弱解，针对临界洛伦兹空间\(L^{4/3,2}\)的任意初始数据。核心方法是用为洛伦兹空间定制的近似方案，主要贡献是构造出守恒哈密顿量的弱解。

Long time behavior and critical limit of subcritical SQG equations in scale-invariant Sobolev spaces

We consider the subcritical SQG equation in its natural scale invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional laplacian. This estimate appears to be new in the literature, and allows a sharp use of the subcritical nature of the $L^\infty$ bounds for this problem. As a byproduct, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attracto…

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。

Regularity criterion and energy conservation for the supercritical 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.

Global existence for the critical dissipative surface quasi-geostrophic equation

In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in $ \mathbb{R}^2$. Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data $θ_{0}$ liying in the space $Λ^{s} (\dot H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)$ the critical (SQG) has a global weak solution in time for all $1/2< s<1$. Our proof is based on an energy inequality verified by the truncated $(SQG)_{R,\ep}$ equation. By classical compactness arguments, we show that we are able to pass to the limit ($R \rightarrow \…

水平周期广义表面准地转斑块与层

Horizontally periodic generalized surface quasigeostrophic patches and layers

本研究扩展了$\alpha$-SQG方程中有界斑块的存在性结果，证明了水平周期斑块边界$H^k$正则性的有限时间传播，并重点分析了介于欧拉与SQG之间的参数范围。

g-SQG Patches的一个改进的正则性判据与Splash-like奇性的不存在性

An Improved Regularity Criterion and Absence of Splash-like Singularities for g-SQG Patches

本文证明了参数 $α\le 1/4$ 的广义表面准地转方程的充分正则patch解不会发生splash-like奇性，并得到了 $H^3$ patch解的改进全局正则性判据。

Global regularity and time decay for the SQG equation with anisotropic fractional dissipation

In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. On the one hand, when the dissipation powers are restricted to a suitable range, the global regularity of the surface quasi-geostrophic equation is obtained by some anisotropic embedding and interpolation inequalities involving fractional derivatives. One the one hand, we obtain the optimal large time decay estimates for global weak solutions by an anisotropic interpolation inequality. Moreover, based on the argument adopted in esta…

On Gevrey Regularity of the Supercritical SQG equation in Critical Besov Spaces

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…

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。

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。

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。

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。

A simple proof of the generalized Leibniz rule on bounded Euclidean domains

This note is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have been lately considered by Constantin and Ignatova in the framework of the SQG equation \cite{CI} in bounded domains and by two of the authors \cite{HungJuan} in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of Porous Medium Equations with pressure given by Ries…

Estimate on the dimension of the singular set of the supercritical surface quasigeostrophic equation

We consider the SQG equation with dissipation given by a fractional Laplacian of order $α<\frac{1}{2}$. We introduce a notion of suitable weak solution, which exists for every $L^2$ initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most $\frac{1}{2α} \left( \frac{1+α}α (1-2α) + 2\right)$.

临界空间中Navier-Stokes方程与SQG方程的新正则性准则

New Regularity Criteria for Navier-Stokes and SQG Equations in Critical Spaces

本文针对三维不可压缩Navier-Stokes方程与二维超临界SQG方程，在Besov空间、Lorentz空间及临界空间和等多种临界空间框架下推导了新的正则性准则，拓展了两类方程正则性条件的适用范围。

Generalized surface quasi-geostrophic equations with singular velocities

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…

粗糙Kraichnan噪声对广义SQG方程的正则化

Regularization by rough Kraichnan noise for the generalised SQG equations

本文证明在广义表面准地转方程中加入粗糙Kraichnan噪声可正则化该方程，为初值在L^1∩L^p中的解提供强存在性与路径唯一性，并涵盖任意β∈(0,1)，同时结果也适用于带外力及二维线性输运方程。

Bounded Mean Oscillation and the Uniqueness of Active Scalar Equations

We consider a number of uniqueness questions for several wide classes of active scalar equations, unifying and generalizing the techniques of several authors. As special cases of our results, we provide a significantly simplified proof to the known uniqueness result for the 2D Euler equations in $L^1 \cap BMO$ and provide a mild improvement to the recent results of Rusin for the 2D inviscid surface quasi-geostrophic (SQG) equations, which are now to our knowledge, the best results known for this model. We also obtain what are (to our knowledge) the strongest known uniqueness results for the Pa…

On the construction of suitable weak solutions to the 3D Navier-Stokes equations in a bounded domain by an artificial compressibility method

In this paper we will prove that suitable weak solutions of three dimensional Navier-Stokes equations in bounded domain can be constructed by a particular type of artificial compressibility approximation.

分数多孔介质方程弱解的局部有界性

Local boundedness for weak solutions to fractional porous medium equation

本文研究了分数多孔介质方程在快速扩散情形下弱解的局部有界性，通过最优尾部假设建立了相关结论。

Existence, regularity and uniqueness of weak solutions with bounded magnetic fields to the steady Hall-MHD system

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.

Results for a turbulent system with unbounded viscosities: weak formulations, existence of solutions, boundedness, smoothness'

We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solutions are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solution

Regularity criteria of weak solutions to NSE in some bounded domains involving the pressure

We present a short and elegant proof of an estimate for the pressure in terms of the velocity and external data in bounded domains under the slip and Navier boundary conditions. We also show an application of this result for conditional regularity of weak solutions to the Navier-Stokes equations.

Boundedness of weak solutions to cross-diffusion systems from population dynamics

The global-in-time existence of nonnegative bounded weak solutions to a class of cross-diffusion systems for two population species is proved. The diffusivities are assumed to depend linearly on the population densities in such a way that a certain formal gradient-flow structure holds. The main feature of these systems is that the diffusion matrix may be neither symmetric nor positive definite. The key idea of the proof is to employ the boundedness-by-entropy principle which yields at the same time the existence of global weak solutions and their boundedness. In particular, the uniform bounded…

有界区域内亚椭圆型退化 Kolmogorov 方程弱解的有界性

Boundedness of weak solutions to degenerate Kolmogorov equations of hypoelliptic type in bounded domains

本文针对相容于齐次 Lie 群的亚椭圆型退化 Kolmogorov 方程，结合 De Giorgi 迭代、重整化公式与基本解嵌入估计，证明了有界乘积区域内弱下解的有界性。

Weak transport equation on a bounded domain: stability theory aprés DiPerna-Lions

The article studies the transport equation that governes the motion of a fluid in a bounded domain, under the hypothesis of zero velocity at the boundary and supposing the incompressible nature of the fluid. Together with existence and uniqueness results, we study the DiPerna-Lions stability problem, in the case of a bounded domain.

On the Continuity of Bounded Weak Solutions to Parabolic Equations and Systems with Quadratic Growth in Gradients

We establish the pointwise continuity of bounded weak solutions to of a class of scalar parabolic equations and strongly coupled parabolic systems. Our approach to the regularity theory of parabolic scalar equations is quite elementary and its applications to strongly coupled systems does not require higher $L^p$ integrability of derivatives.

Remarks on Local Boundedness and Local Hölder Continuity of Local Weak Solutions to Anisotropic $p$-Laplacian Type Equations

Locally bounded, local weak solutions to a special class of quasilinear, anisotropic, $p$-Laplacian type elliptic equations, are shown to be locally Hölder continuous. Homogeneous local upper bounds are established for local weak solutions to a general class of quasilinear anisotropic equations.

Strong attractors for weakly damped quintic wave equation in bounded domains

In this paper, we study the longtime dynamics for the weakly damped wave equation with quintic non-linearity in a bounded smooth domain of $\mathbb{R}^3.$ Based on the Strichartz estimates for the case of bounded domains, we establish the existence of a strong global attractor in the phase space $H^2(Ω)\cap H^1_0(Ω)\times H^1_0(Ω)$. Moreover, the finite fractal dimension of the attractor is also shown with the help of the quasi-stable estimation.

The local boundedness of gradients of weak solutions to elliptic and parabolic phi-Laplacian systems

In this thesis, a unified approach to prove the boundedness of gradients of solutions to degenerate and singular elliptic and parabolic phi-Laplacian systems is presented. At first, a Cacciopoli-type energy inequality with an additional function f which can be chosen freely is proven. Then, Di Giorgi's method is applied using level sets which will lead to L-infinity-estimates on the gradient of the weak solution.