Christian Seis
We are concerned with flow enhanced mixing of passive scalars in the presence of diffusion. Under the assumption that the velocity gradient is suitably integrable, we provide upper bounds on the exponential rates of enhanced dissipation. Recent constructions indicate the optimality of our results.
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We consider the passive scalar equations subject to shear flow advection and fractional dissipation. The enhanced dissipation estimates are derived. For classical passive scalar equation ($γ=1$), our result agrees with the sharp one obtained in \cite{Wei18}
Localized Enhanced Dissipation: A Hypocoercivity Approach
本文研究剪切流中被动标量解的局部增强耗散估计，采用详细弱抛物性泛函发展流体方向增强耗散估计。
We consider the solution to the 2D Navier-Stokes equations around the Poiseuille flow $(y^2,0)$ on $\mathbb{T}\times\mathbb{R}$ with small viscosity $ν>0$. Via a hypocoercivity argument, we prove that the $x-$dependent modes of the solution to the linear problem undergo the enhanced dissipation effect with a rate proportional to $ν^{\frac{1}{2}}$. Moreover, we study the nonlinear enhanced dissipation effect and we establish a transition threshold of $ν^{\frac{2}{3}+}$. Namely, when the perturbation of the Poiseuille flow is size at most $ν^{\frac{2}{3}+}$, its size remains so for all times and…
Enhanced dissipation and blow-up suppression in a chemotaxis-fluid system
该研究针对耦合PKS-NS趋化流体系统，证明任意大细胞种群下存在全局正则解，揭示剪切流混合诱导的增强耗散是抑制爆破的核心机制，丰富了趋化-流体耦合系统的全局适定性理论。
A Note on Enhanced Dissipation and Taylor Dispersion of Time-dependent Shear Flows
该研究利用带时间权重的亚强制性泛函，证明临界点缓慢变化的时变剪切流可获得与定常情形一致的尖锐增强耗散及 Taylor 色散估计。
In this article we consider the stability and damping problem for the 2D Boussinesq equations with partial dissipation near a two parameter family of stationary solutions which includes Couette flow and hydrostatic balance. In the first part we show that for the linearized problem in an infinite periodic channel the evolution is asymptotically stable if any diffusion coefficient is non-zero. In particular, this imposes weaker conditions than for example vertical diffusion. Furthermore, we study the interaction of shear flow, hydrostatic balance and partial dissipation. In a second part we adap…
We consider the 2D incompressible Navier-Stokes equations on $\mathbb{T}\times \mathbf{R}$, with initial vorticity that is $δ$ close in $H^{log}_xL^2_{y}$ to $-1$(the vorticity of the Couette flow $(y,0)$). We prove that if $δ\ll ν^{1/2}$, where $ν$ denotes the viscosity, then the solution of the Navier-Stokes equation approaches some shear flow which is also close to Couette flow for time $t\gg ν^{-1/3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\infty$. In particular, we show the nonlinear enhanced dissipation and the inviscid damping results…
Enhanced dissipation and blow-up suppression for the three dimensional Keller-Segel equation with a non-shear incompressible flow
本文研究三维Keller-Segel方程，引入非剪切不可压缩流，通过预解估计证明其增强耗散能抑制爆破，并建立大初始数据下的全局经典解。
In this paper, we prove the linear inviscid damping and voticity depletion phenomena for the linearized Euler equations around the Kolmogorov flow. These results confirm Bouchet and Morita's predictions based on numerical analysis. By using the wave operator method introduced by Li, Wei and Zhang, we solve Beck and Wayne's conjecture on the optimal enhanced dissipation rate for the 2-D linearized Navier-Stokes equations around the bar state called Kolmogorov flow. The same dissipation rate is proved for the Navier-Stokes equations if the initial velocity is included in a basin of attraction of…
Asymptotic stability of the symmetric flow via inviscid damping and enhanced dissipation
该研究针对有限通道无滑移边界下的对称流，建立线性化Navier-Stokes系统的无粘阻尼与增强耗散估计，证明高雷诺数下对称流的渐近稳定性，并给出初始扰动的阈值条件。
Nonlinear enhanced dissipation in viscous Burgers type equations
本文构造无限质量初始数据类，证明粘性 Burgers 方程中非线性输运导致比热方程更快的增强耗散，并刻画带边界层的稳定时变渐近剖面。
Instantaneous Total Enhanced Dissipation For Very Rough Shear Flows
本文研究二维环面上极粗糙水平剪切流平流输运的被动标量增强耗散，推广了负Besov空间中的一般流结果，证明粘度消失时耗散率趋于无穷，并给出非尖锐上下界及无粘方程解的平凡性。
Nonlinear enhanced dissipation in viscous Burgers type equations II
本文利用 Hopf-Cole 变换研究粘性 Burgers 方程中无限质量初始数据解的长时间行为，证明其相较热方程具有一般性的增强耗散，并建立各类的全局吸引子。
Global well-posedness and enhanced dissipation for the 2D stochastic Nernst-Planck-Navier-Stokes equations with transport noise
研究带输运噪声的二维随机 Nernst-Planck-Navier-Stokes 方程，在等扩散率、反价态假设下证明整体适定性，并表明特定噪声可带来任意大的指数收敛速率，从而实现增强耗散。
Mixing, Enhanced Dissipation and Phase Transition in the Kinetic Vicsek Model
本文研究动力学Vicsek模型，分析其长时间行为，发现特定参数下混合与增强耗散可稳定动力学并保障智能体信息传递，使解呈空间均匀系统特征，最终在动力学层面证实了基于智能体的Vicsek模型中的相变。
We examine the phenomenon of enhanced dissipation from the perspective of Hörmander's classical theory of second order hypoelliptic operators [31]. Consider a passive scalar in a shear flow, whose evolution is described by the advection-diffusion equation \[ \partial_t f + b(y) \partial_x f - νΔf = 0 \text{ on } \mathbb{T} \times (0,1) \times \mathbb{R}_+ \] with periodic, Dirichlet, or Neumann conditions in $y$. We demonstrate that decay is enhanced on the timescale $T \sim ν^{-(N+1)/(N+3)}$, where $N-1$ is the maximal order of vanishing of the derivative $b'(y)$ of the shear profile and $N=0…
Enhanced Dissipation and Global Well-Posedness for a Three-Dimensional Flame Propagation Model with Couette Flow
本文研究了在Couette流作用下三维重力诱导火焰前沿模型的全局良好性，通过Couette流诱导的增强耗散证明了Cauchy问题在R^3中的全局良好性，并推导了解及其空间导数在L^p范数下的衰减估计。
In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Poiseuille flow $(1-y^2,0)$ in a finite channel with Navier-slip boundary condition. Based on the resolvent estimates for the linearized operator around the Poiseuille flow, we first establish the enhanced dissipation estimates for the linearized Navier-Stokes equations with a sharp decay rate $e^{-c\sqrtνt}$. As an application, we prove that if the initial perturbation of vortiticy satisfies $$\|ω_0\|_{L^2}\leq c_0ν^{\frac{3}{4}},$$ for some small constant $c_0>0$ independent of the viscosi…
We study the enhanced dissipation for the two-jet Kolmogorov type flow which is a stationary solution to the Navier-Stokes equations on the two-dimensional unit sphere given by the zonal spherical harmonic function of degree two. Based on the pseudospectral bound method developed by Ibrahim, Maekawa, and Masmoudi [15] and a modified version of the Gearhart-Prüss type theorem shown by Wei [48], we derive an estimate for the resolvent of the linearized operator along the imaginary axis and show that a solution to the linearized equation rapidly decays at the rate $O(e^{-\sqrtν\,t})$ when the vis…
In this paper, we study the decay rates for the global small smooth solutions to 3D anisotropic incompressible Navier-Stokes equations. In particular, we prove that the horizontal components of the velocity field decay like the solutions of 2D classical Navier-Stokes equations. While the third component of the velocity field decays as the solutions of 3D Navier-Stokes equations. We remark that such enhanced decay rate for the third component is caused by the interplay between the divergence free condition of the velocity field and the horizontal Laplacian in the anisotropic Navier-Stokes equat…
Enhanced dissipation and blow-up suppression for an aggregation equation with fractional diffusion and shear flow
研究具分数阶扩散和剪切流的聚集方程，通过预解式估计证明剪切流的增强耗散可抑制解的爆破，并在 $α\geq 3/2$ 时建立全局经典解。
This paper is devoted to the stability analysis of the Lamb-Oseen vortex in the regime of high circulation Reynolds numbers. When strongly localized perturbations are applied, it is shown that the vortex relaxes to axisymmetry in a time proportional to $Re^{2/3}$, which is substantially shorter than the diffusion time scale given by the viscosity. This enhanced dissipation effect is due to the differential rotation inside the vortex core. Our result relies on a recent work by Li, Wei, and Zhang, where optimal resolvent estimates for the linearized operator at Oseen's vortex are established. A …
In this article we consider the 2D Navier-Stokes equations with variable viscosity depending on the vertical position. As our main result we establish linear enhanced dissipation near the non-affine stationary states replacing Couette flow. Moreover it turns out that the shear flow overcompensates for weakening viscosity: decreasing viscosity leads to stronger enhanced dissipation and increasing viscosity leads to weaker dissipation than in the constant viscosity case.
Enhanced Dissipation via the Malliavin Calculus
该研究利用Malliavin Calculus技术研究增强耗散现象，通过简洁初等的论证，重现了剪切流的经典增强耗散时间尺度结果并得到x方向精确亚椭圆正则化性质。
A pathwise approach to the enhanced dissipation of passive scalars advected by shear flows
本研究提出一种基于随机微分方程粒子轨迹分析的概率框架，研究剪切流中被动标量的增强耗散，可复现已有结果并拓展至奇异/无穷阶临界点情形，还能给出局部流线衰减速率估计。
Enhanced dissipation and Taylor dispersion in higher-dimensional parallel shear flows
研究任意空间维度下被动标量在平行剪切流中的演化，通过预解式估计与亚强制方法，在增强耗散和Taylor色散区域获得了大时间最优衰减估计并比较了两种方法的效率。
Enhanced dissipation and Taylor dispersion by a parallel shear flow in an infinite cylinder with unbounded cross section
本文研究无界截面无限圆柱体中平行剪切流对被动标量的输运，发现高频下出现增强耗散，低频下仅Taylor弥散衰减，并需无穷远非退化条件。
We consider the 3D incompressible Navier-Stokes equations under the following $2+\frac{1}{2}$-dimensional situation: vertical vortex blob (quasi-streamwise vortices) being stretched by two-dimensional shear flow. We prove enhanced dissipation induced by such quasi-streamwise vortices.
Enhanced dissipation and stability threshold for the Navier--Stokes equations near Poiseuille flow
本文证明在加权Sobolev空间中，二维不可压缩Navier-Stokes方程在Poiseuille流附近的初始扰动大小为O(ν^{2/3})时，流动全局非线性稳定，非零模态以增强耗散指数衰减，零模态受控，且该稳定性尺度无对数损失。
We study the 2D Navier-Stokes equations linearized around the Couette flow $(y,0)^t$ in the periodic channel $\mathbb T \times [-1,1]$ with no-slip boundary conditions in the vanishing viscosity $ν\to 0$ limit. We split the vorticity evolution into the free evolution (without a boundary) and a boundary corrector that is exponentially localized to at most an $O(ν^{1/3})$ boundary layer. If the initial vorticity perturbation is supported away from the boundary, we show inviscid damping of both the velocity and the vorticity associated to the boundary layer. For example, our $L^2_t L^1_y$ estimat…
Sharp linear stability and the absence of enhanced dissipation for Kolmogorov flow in the 2D Navier--Stokes equations with horizontal dissipation
研究水平耗散二维Navier--Stokes方程中Kolmogorov流的线性稳定性，证明无增强耗散，并给出各模态的精确衰减率与不稳定条件。
We consider the Navier-Stokes equations on the two-dimensional unit sphere and study the linear stability of the two-jet Kolmogorov type flow which is a stationary solution given by the zonal spherical harmonic function of degree two. We prove the linear stability of the two-jet Kolmogorov type flow for an arbitrary viscosity coefficient by showing the exponential decay of a solution to the linearized equation towards an equilibrium which grows as the viscosity coefficient tends to zero. The main result of this paper is the nonexistence of nonzero eigenvalues of the perturbation operator appea…
A Stochastic RAGE Theorem and Enhanced Dissipation for Transport Noise
本文证明了适用于噪声传输方程两点运动的随机 RAGE 定理，识别了扩散方程增强耗散的充要条件，并证明了随机剪切流的精确增强耗散率。
Enhanced dissipation by advection and applications to PDEs
本综述概述了平流与扩散相互作用导致的增强耗散现象，从物理原理过渡到数学分析，并展示了其在 Cahn-Hilliard 等非线性偏微分方程中的应用。
Finite time mixing and enhanced dissipation for 2D Navier--Stokes equations by Ornstein--Uhlenbeck flow
该研究以路径依赖方式分析受输运型Ornstein–Uhlenbeck流扰动的二维Navier–Stokes方程涡量形式，证明合适的流可使其具备有限时间混合与增强耗散性质，区别于以往将扰动视为Stratonovich输运噪声的研究框架。
Enhanced Dissipation, Taylor Dispersion, and Inviscid Damping of Couette flow in the Boussinesq system on the Plane
本文在平面全无界区域上证明了具有大 Richardson 数的稳定分层 Couette 流在二维全耗散 Boussinesq 系统中的定量渐近稳定性，首次给出显式增强耗散、Taylor 色散和无粘阻尼估计。
We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a time-scale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation time-scales. The proofs are based on a …
Enhanced dissipation and temporal decay in the Euler-Poisson-Navier-Stokes equations
本文研究三维耦合流体模型的全局适定性与长时间行为，利用 Poisson 方程耗散实现密度以 $(1+t)^{-11/4}$ 快速衰减，快于速度差和速度，并在耦合流消失时证明指数衰减。
A Harris theorem for enhanced dissipation, and an example of Pierrehumbert
本文利用满足Harris条件的随机动力系统框架，证明扩散强度为$κ$时增强耗散以至少$1-o(κ^N)$的概率在$|\ln κ|$时间尺度上发生，并由此得到Pierrehumbert随机剪切流的增强耗散。
We consider absorbing chemical reactions in a fluid flow modeled by the coupled advection-reaction-diffusion equations. In these systems, the interplay between chemical diffusion and fluid transportation causes the enhanced dissipation phenomenon. We show that the enhanced dissipation time scale, together with the reaction coupling strength, determines the characteristic time scale of the reaction.
Exponentially mixing flows with slow enhanced dissipation
该研究针对小扩散系数下的被动标量耗散问题，构造了无增强耗散的C0指数混合不可压缩流，还改进了带显式常数的混合流耗散时间界，支持时间非齐次混合速率。
Mixing in incompressible flows: transport, dissipation, and their interplay
本文从偏微分方程角度探讨流体动力学中的混合问题，综述了最优混合、输运方程正则性丧失、增强耗散及反常耗散。
Let $H\in C^1\cap W^{2,p}$ be an autonomous, non-constant Hamiltonian on a compact $2$-dimensional manifold, generating an incompressible velocity field $b=\nabla^\perp H$. We give sharp upper bounds on the enhanced dissipation rate of $b$ in terms of the properties of the period $T(h)$ of the close orbits $\{H=h\}$. Specifically, if $0<ν\ll 1$ is the diffusion coefficient, the enhanced dissipation rate can be at most $O(ν^{1/3})$ in general, the bound improves when $H$ has isolated, non-degenerate elliptic point. Our result provides the better bound $O(ν^{1/2})$ for the standard cellular flow…
Accelerated and fast magnetic reconnection through enhanced resistive dissipation for MHD equations
本文研究三维周期 MHD 方程中的快速磁重联，证明平流诱导的增强耗散可使重联快于电阻时间尺度，并提出具有独立价值的高 Sobolev 范数增强扩散估计。
Exponential mixing and enhanced dissipation on the unit sphere with Rossby-Haurwitz flows
研究单位球面上光滑不可压速度场族，证明其被动标量指数混合，并给出伴随平流-扩散方程的最优增强耗散率。
Uniform linear inviscid damping and enhanced dissipation near monotonic shear flows in high Reynolds number regime (I): the whole space case
本文研究了高雷诺数下单调剪切流附近二维Navier-Stokes方程的线性化动力学，通过将非局部项视为紧致微扰精确控制Orr-Sommerfeld方程解，从而获得了关于粘性均匀的线性无粘阻尼和增强耗散衰减估计。
Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on $\mathbb{T}^2$
本文研究二维环面上时间周期 Lipschitz 速度场驱动的对流-扩散方程，证明其在 $|\log ν|$ 时间尺度上具有最优增强耗散，并进一步得到无扩散情形的指数混合结果。
Stabilization by transport noise and enhanced dissipation in the Kraichnan model
本文证明在 Kraichnan 模型中，输运噪声可使不稳定线性算子全局渐近稳定，并实现任意大的指数收敛速率（增强耗散），且强制四个模态即可达到稳定化。
We consider solutions to the 2d Navier-Stokes equations on $\mathbb{T}\times\mathbb{R}$ close to the Poiseuille flow, with small viscosity $ν>0$. Our first result concerns a semigroup estimate for the linearized problem. Here we show that the $x$-dependent modes of linear solutions decay on a time-scale proportional to $ν^{-1/2}|\logν|$. This effect is often referred to as \emph{enhanced dissipation} or \emph{metastability} since it gives a much faster decay than the regular dissipative time-scale $ν^{-1}$ (this is also the time-scale on which the $x$-independent mode naturally decays). We ach…
We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel $\mathbb{T} \times \mathbb{R}$. Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order $ν^{-1/3}$, $ν$ being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conju…
Suppression of blow-up in 3-D Keller-Segel model via Couette flow in whole space
本文通过Green函数方法研究全空间中Couette流对三维Keller-Segel模型的增强耗散效应，证明大Couette流可抑制爆破并获得解的全局存在性。
Optimal mixing and enhanced dissipation for a smooth velocity field in any spatial dimension
本文在任意维度环面上构造光滑随机速度场，证明其作为通用指数混合器，并在小分子扩散下以最优速率增强耗散，方法基于随机动力系统和两点马尔可夫过程的均匀遍历性。
Weak collision effect on nonlinear Landau damping for the Vlasov-Poisson-Fokker-Planck system
本文研究环面上Vlasov-Poisson-Fokker-Planck系统中弱碰撞对非线性Landau阻尼的影响，针对不同Gevrey指标建立了小初值的全局稳定性与增强耗散，并揭示了相位混合、增强耗散和等离子体回声的作用。
We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic $\mathbb{T}^2$ setting and the case of a bounded channel $\mathbb{T}\times [0,1]$ with no-flux boundary conditions. In the infinite Péclet number limit (diffusivity $ν\to 0$), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Ge…
In this paper, we study the nonlinear stability of a steady circular flow created between two rotating concentric cylinders. The dynamics of the viscous fluid are described by 2D Navier-Stokes equations. We adopt scaling variables. For the rescaled equations, we prove that the steady flow (Taylor-Couette flow) is asymptotically stable up to a large perturbation of initial data. Back to the original 2D Navier-Stokes equations, this implies an improved transition threshold for the Taylor-Couette flow. The improvement is due to enhanced dissipation and new observations and constructions of weight…
In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel equation with a fractional dissipation and advection by a weakly mixing (see Definition \ref{def:2.4}). Here the attractive kernel has strong singularity, namely, the derivative appears in the nonlinear term by singular integral. Without advection, the solution of equation blows up in finite time. Under a suitable mixing condition on the advection, we show the global existence of classical solution with large initial data in the case of the derivative of dissipative term is higher than that of nonli…
We study the mixing effect for a generalized Keller-Segel system with degenerate dissipation and advection by a weakly mixing. Here the attractive operator has weak singularity, namely, the negative derivative appears in the nonlinear term by singular integral. Without advection, the solution of equation blows up in finite time. We show that the global well-posedness of solution with large advection. Since dissipation term degenerate into the damping, the enhanced dissipation effect of mixing no longer occurs, we prove that the mixing effect can weak the influence of nonlinear term. In this ca…
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。
Enhanced Dissipation via time-modulated velocity fields
本研究针对非自治流动中的增强耗散问题，通过带时间相关权重的欠强制性框架推导平流-扩散方程的能量衰减估计，发现一类被递增函数界定的调制函数可实现超增强耗散，开关型速度场的耗散速率则与自治流相当。
The Existence of Anomalous Dissipation over Bounded Interior Domains
本文通过构造球体内一个线性Stokes流的实例，证明了有界内部区域上反常耗散的存在性，为理解三维湍流中反常耗散现象提供了新的数学证据。
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