On a class of forced active scalar equations with small diffusive parameters

Well-posedness for the diffusive 3D Burgers equations with initial data in $H^{1/2}$

In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encou…

Near identity transformations for the Navier-Stokes equations

The Navier-Stokes equations and their various approximations can be described in terms of near identity maps, that are diffusive particle path transformations of physical space. The active velocity is obtained from the diffusive path transformation and a virtual velocity using the Weber formula. The active vorticity is obtained from the diffusive path transformation and a virtual vorticity using a Cauchy formula. The virtual velocity and the virtual vorticity obey diffusive equations, which reduce to passive advection formally, if the viscosity is zero. Apart from being proportional to the vis…

Solutions to a class of forced drift-diffusion equations with applications to the magneto-geostrophic equations

We prove the global existence of classical solutions to a class of forced drift-diffusion equations with $L^2$ initial data and divergence free drift velocity $\{u^ν\}_{ν_\ge0}\subset L^\infty_t BMO^{-1}_x$, and we obtain strong convergence of solutions as the viscosity $ν$ vanishes. We then apply our results to a family of active scalar equations which includes the three dimensional magneto-geostrophic $\{$MG$^ν\}_{ν\ge0}$ equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth's fluid core. We prove the existence of a compact global attractor $\{\…

On the supercritically diffusive magneto-geostrophic equations

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 …

反应-扩散系统中通过扩散正规型进行波前选择

Front selection in reaction-diffusion systems via diffusive normal forms

本文证明了多分量反应-扩散系统中入侵过程的传播速度由边际谱稳定性条件决定，通过引入坐标变换和自相似变量刻画波前前沿的扩散动力学并控制匹配误差，从而完整证明了边际稳定性猜想。

主动布朗粒子模型的稳定性分析

Stability analysis for active Brownian particle models

本文通过线性稳定性分析，揭示了主动布朗粒子模型在非扩散和小角扩散情况下的稳定性特征，并发现其相分离机制与Landau阻尼类似但衰减更弱。

Compressible fluids and active potentials

We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of th…

Non-uniqueness in law for two-dimensional Navier-Stokes equations with diffusion weaker than a full Laplacian

We study the two-dimensional Navier-Stokes equations forced by random noise with a diffusive term generalized via a fractional Laplacian that has a positive exponent strictly less than one. Because intermittent jets are inherently three-dimensional, we instead adapt the theory of intermittent form of the two-dimensional stationary flows to the stochastic approach presented by Hofmanov$\acute{\mathrm{a}}$, Zhu $\&$ Zhu (2019, arXiv:1912.11841 [math.PR]) and prove its non-uniqueness in law.

活性悬浮体中的取向混合

Orientation mixing in active suspensions

研究活性悬浮体动力学模型中取向混合现象，证明无扩散情形下类似朗达阻尼的混合及最优衰减率，扩散情形下混合估计持续至ν^{-1/2}时间尺度，为集体动力学模型提供新工具。

On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations

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.

具有长程噪声的临界随机热方程

A critical stochastic heat equation with long-range noise

研究空间维数至少为3且受时间白噪声、空间协方差按$|x|^{-2}$衰减的临界半线性随机热方程，证明在带对数衰减的扩散标度下解的逐点统计可由FBSDE刻画。

粘弹性流动的几何爆破准则：Oldroyd-B 和 FENE-P 模型

Pressure Quotients and Endpoint Velocity-Clock Criteria for Non-Diffusive Viscoelastic Flows

针对无应力扩散的二维 Oldroyd-B 和 FENE-P 流动，利用对数构型变量和有限可扩展性障碍变量，建立了包含速度梯度 Besov 模、对数构型范数和有限可扩展性障碍范数的封闭延续准则。

一种简化模型用于耦合达里厄斯-兰道不稳定性与扩散-热不稳定性

A simplified model for coupling Darrieus-Landau and diffusive-thermal instabilities

本文提出简化模型耦合达里厄斯-兰道不稳定性与扩散-热不稳定性，通过识别立方耦合项，揭示两种不稳定性在不同渐进行 regime 中的相互作用，推导出包含非局部稳定项的广义演化方程，解释火焰前沿不稳定性及细观结构。