Keith Leitmeyer
An enstrophy cascade is exhibited for the Navier-Stokes equations in physical scales independently of boundary conditions under physically reasonable assumptions on the flow.
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Vanishing of long time average p-enstrophy dissipation rate in the inviscid limit of the 2D damped Navier-Stokes equations
本文将二维阻尼 Navier-Stokes 方程无粘极限下长时间平均 enstrophy 耗散率消失的结果推广至 p-enstrophy，并借助全局吸引子不变测度与涡量强收敛结论简化证明。
On the effect of the Coriolis force on the enstrophy cascade
研究了 $β$-平面近似下受科里奥利力作用的 2D Navier-Stokes 方程中的直接涡度拟能级联，证明小尺度下三阶结构函数渐近行为遵循无科里奥利力的 2D 湍流三阶普适定律，且涡度拟能通量不受其影响，首次提供了数学严格证明。
By direct numerical simulation to the two-dimensional Navier-Stokes equations with small-scale forcing and large-scale damping, Xiao-Wan-Chen-Eyink (2009) found an evidence that inverse energy cascade may proceed with the vortex thinning mechanism. On the other hand, Alexakis-Doering (2006) calculated upper bound of the bulk averaged enstrophy dissipation rate of the steady-state two dimensional turbulence. {In this paper, we show that vortex thinning induces enhanced dissipation with strictly slower vanishing order of the enstrophy dissipation than $Re^{-1}$.}
This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes system --- as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no estimates available with finite a priori bounds on the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. In order to quantify the maximum possible growth of enstrophy, w…
Bounding curves in the energy,enstrophy-plane are derived for the 3D Navier-Stokes equations under an assumption on coherence of the vorticity direction. The analysis in the critical case where the direction is Hölder continuous with exponent $r=1/2$ results in a curve with extraordinarily large maximal enstrophy (exponential in Grashof), in marked contrast to the subcritical case, $r>1/2$ (algebraic in Grashof).
We consider enstrophy dissipation in two-dimensional (2D) Navier-Stokes flows and focus on how this quantity behaves in thelimit of vanishing viscosity. After recalling a number of a priori estimates providing lower and upper bounds on this quantity, we state an optimization problem aimed at probing the sharpness of these estimates as functions of viscosity. More precisely, solutions of this problem are the initial conditions with fixed palinstrophy and possessing the property that the resulting 2D Navier-Stokes flows locally maximize the enstrophy dissipation over a given time window. This pr…
Existence of 2D enstrophy cascade in a suitable mathematical setting, and under suitable conditions compatible with 2D turbulence phenomenology, is known both in the Fourier and in the physical scales. The goal of this paper is to show that the same geometric condition preventing the formation of singularities - 1/2-Hölder coherence of the vorticity direction - coupled with a suitable condition on a modified Kraichnan scale, and under a certain modulation assumption on evolution of the vorticity, leads to existence of 3D enstrophy cascade in physical scales of the flow.
Current theoretical results for the three-dimensional Navier--Stokes equations only guarantee that solutions remain regular for all time when the initial enstrophy ($\|Du_0\|^2:=\int|{\rm curl} u_0|^2$) is sufficiently small, $\|Du_0\|^2\leχ_0$. In fact, this smallness condition is such that the enstrophy is always non-increasing. In this paper we provide a numerical procedure that will verify regularity of solutions for any bounded set of initial conditions, $\|Du_0\|^2\leχ_1$. Under the assumption that the equations are in fact regular we show that this procedure can be guaranteed to termina…
Computational simulations of turbulent flows indicate that the regions of low dissipation/enstrophy production feature high degree of local alignment between the velocity and the vorticty, i.e., the flow is locally near-Beltrami. Hence one could envision a geometric scenario in which the persistence of the local near-Beltramy property might be consistent with a (possible) finite-time singularity formation. The goal of this note is to show that this scenario is in fact prohibited if the sine of the angle between the velocity and the vorticty is small enough with respect to the local enstrophy.
We consider the inviscid limit of the stochastic damped 2D Navier- Stokes equations. We prove that, when the viscosity vanishes, the stationary solution of the stochastic damped Navier-Stokes equations converges to a stationary solution of the stochastic damped Euler equation and that the rate of dissipation of enstrophy converges to zero. In particular, this limit obeys an enstrophy balance. The rates are computed with respect to a limit measure of the unique invariant measure of the stochastic damped Navier-Stokes equations.
We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in ${\mathbb R}^2$. We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enstrophy balance.
Finite-time blowup for a Navier-Stokes model equation for the self-amplification of strain
本文研究了Navier-Stokes应变方程的模型方程，证明了其有限时间爆破机制源于应变自放大，表明涡度拟能增长等式与应变约束空间不足以保证全局正则性，并给出了完整Navier-Stokes方程的条件爆破结果。
Filtered Vortex Stretching and Subgrid Defects for the Three-Dimensional Navier-Stokes Equations
针对三维Navier-Stokes方程，通过滤波涡旋拉伸的有限尺度估计，利用方向缺陷和扩散吸收，建立了滤波涡度平衡中各项的严格界。
Local analysis of the two dimensional Navier-Stokes equations is used to obtain estimates on the energy and enstrophy fluxes involving Taylor and Kraichnan length scales and the size of the domain. In the framework of zero driving force and non-increasing global energy, these bounds produce sufficient conditions for existence of the direct enstrophy and inverse energy cascades. Several manifestations of locality of the fluxes under these conditions are obtained. All the scales involved are actual physical scales in R^2 and no homogeneity assumptions are made.
Energy dissipation near the outflow boundary in the vanishing viscosity limit
该研究针对有界区域非特征边界下的不可压缩Navier-Stokes与Euler方程，推导了零黏度极限下出流边界附近的能量耗散率、边界层拟涡能的量级关系及拟涡能产生率的变化规律。
Enstrophy dynamics for flow past a solid body with no-slip boundary condition
本文研究边界涡量分布对绕流体涡量动力学的影响，推导了包含涡函数边界值的新能量恒等式，并证明了斯托克斯系统的涡量耗散性，获得纳维-斯托克斯系统的新涡量动力学方程。
Quantitative enstrophy bounds for measure vorticities
本文通过改进的纳什不等式，建立了二维纳维-斯托克斯方程中测度初始涡度的涡量定量界，并探讨了其在耗散速率估计中的应用。
We introduce a rough perturbation of the Navier-Stokes system and justify its physical relevance from balance of momentum and conservation of circulation in the inviscid limit. We present a framework for a well-posedness analysis of the system. In particular, we define an intrinsic notion of solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. In two space dimensions, we prove that well-posedness and enstrophy balance holds. Moreover, we derive rough path continuity of the equation, which yields a Wong-Zakai result for Brownian driving…
We provide sufficient conditions for mathematically rigorous proofs of the third order universal laws capturing the energy flux to large scales and enstrophy flux to small scales for statistically stationary, forced-dissipated 2d Navier-Stokes equations in the large-box limit. These laws should be regarded as 2d turbulence analogues of the $4/5$ law in 3d turbulence, predicting a constant flux of energy and enstrophy (respectively) through the two inertial ranges in the dual cascade of 2d turbulence. Conditions implying only one of the two cascades are also obtained, as well as compactness cri…
Anomalous smoothing effect on the incompressible Navier-Stokes-Fourier limit from Boltzmann with periodic velocity
本文借助带周期速度与特制碰撞算子的 Boltzmann 方程流体动力学极限，并引入微结构非平凡项，证明三维不可压缩 Navier-Stokes-Fourier 系统对任意初值存在拟涡能一致有界的全局弱解。
In this paper, we deal with the controllability properties of a system of $m$ coupled Stokes systems or $m$ coupled Navier-Stokes systems. We show the null-controllability of such systems in the case where the coupling is in a cascade form and when the control acts only on one of the systems. Moreover, we impose that this control has a vanishing component so that we control a $m\times N$ state (corresponding to the velocities of the fluids) by $N-1$ distributed scalar controls. The proof of the controllability of the coupled Stokes system is based on a Carleman estimate for the adjoint system.…
We consider the behaviour of weak solutions of the unforced three-dimensional Navier-Stokes equations, under the assumption that the initial condition has finite energy ($\|u\|^2=\int|u|^2$) but infinite enstrophy ($\|Du\|^2=\int|{\rm curl} u|^2$). We show that this has to be reflected in the solution for small times, so that in particular $\|Du(t)\|\to+\infty$ as $t\to0$. We also give some limitations on this `backwards blowup', and give an elementary proof that the upper box-counting dimension of the set of singular times can be no larger than one half. Although similar in flavour, this fina…
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…
Cascade mechanisms for Navier-Stokes blow-up
本文通过Obukhov二进模型阐明三维Navier-Stokes方程的逆能量级联导致瞬时I型爆破的机制，并证明在高间歇性下有限能量解中同样发生，同时给出混合模型的有限时间爆破定理。
Strong Solutions and Applications to the Inviscid Limit for 2D Navier-Stokes with Navier Slip
该研究针对带Navier滑移条件的二维不可压缩Navier-Stokes方程，证明其速度弱解为强解，推导了消失粘性极限下涡量的一致强收敛性，且整体体积粘性涡量耗散消失。
Orbit-Level Stretching in Cubic Fourier-Galerkin Navier-Stokes: Sharp Incidence, Spectral Decay, and a Continuation Criterion
研究立方傅里叶-伽辽kin截断下的纳维-斯托克斯方程的轨道级涡量伸展，通过轨道-三元组 incidence 估计和面归一化分解，建立尖锐的界，并验证轨道级持续性准则。
This manuscript derives an evolution equation for the symmetric part of the gradient of the velocity (the strain tensor) in the incompressible Navier-Stokes equation on $\mathbb{R}^3$, and proves the existence of $L^2$ mild solutions to this equation. We use this equation to obtain a simplified identity for the growth of enstrophy for mild solutions that depends only on the strain tensor, not on the nonlocal interaction of the strain tensor with the vorticity. The resulting identity allows us to prove a new family of scale-critical, necessary and sufficient conditions for the blow-up of a solu…
Full-Tail Dynamical Rigidity Forced by Atomic Navier-Stokes Energy Concentration
针对趋近有限终端时间的三维纳维-斯托克斯流，证明点原子强制全尾动力学刚性，得到受约束奥森传播子饱和等结果，揭示延迟二阶算子预算在端点附近失效。
In this paper, we deal with the existence of insensitizing controls for the Navier-Stokes equations in a bounded domain with Dirichlet boundary conditions. We prove that there exist controls insensitizing the $L^2$ -norm of the observation of the solution in an open subset $\mathcal{O}$ of the domain, under suitable assumptions on the data. This problem is equivalent to an exact controllability result for a cascade system. First we prove a global Carleman inequality for the linearized Navier-Stokes system with right-hand side, which leads to the null controllability at any time $T>0$. Then, we…
Fractional Stochastic Navier-Stokes Equations: Local Well-Posedness and Enstrophy Balance
研究三维环面上具有Caputo分数阶时间记忆和分数阶Laplacian空间耗散的随机不可压缩流体，证明局部适定性和涡度平衡，并给出临界指数和全局存在性条件。
Invisible Defect Cascades for Navier-Stokes Regularity
针对三维不可压缩Navier-Stokes方程局部正则性问题，提出条件性尺度临界缺陷级联约化定理，通过二进尺度重标度、粗粒化、压力分裂等技术，将奇异点归约为不可见缺陷级联，并给出正则性判据。
This article is devoted to incompressible Euler equations (or to Navier-Stokes equations in the vanishing viscosity limit). It describes the propagation of quasi-singularities. The underlying phenomena are consistent with the notion of a cascade of energy.
We define the Ladyzhenskaya-Lions exponent $α_{\rm {\tiny \sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-Δ)^α$ in ${\Bbb R}^n$, for all $n\geq 2$. We review the proof of strong global solvability when $α\geq α_{\rm {\tiny \sc l}} (n)$, given smooth initial data. If the corresponding Euler equations for $n>2$ were to allow uncontrolled growth of the enstrophy ${1\over 2} \|\nabla u \|^2_{L^2}$, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for $α<α_{\rm {\tiny \sc l}} (n)$. The ener…
Spatial analyticity and exponential decay of Fourier modes for the stochastic Navier-Stokes equation
该研究构造了仅需初始涡量有界的二维、三维随机Navier-Stokes方程局部时间空间实解析解，还证明了解全局存在时有限维Galerkin近似的指数衰减性，且该衰减在时间、初始涡量、噪声系数上均一致。
In this investigation we perform a systematic computational search for potential singularities in 3D Navier-Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity $\int_0^T \| \mathbf{u}(t) \|_{L^q(Ω)}^p \, dt$, where $2/p+3/q \le 1$, $q > 3$, is bounded, then the solution $\mathbf{u}(t)$ of the Navier-Stokes system is smooth on the interval $[0,T]$. In other words, if a singularity should occur at some time $t \in [0,T]$, then this quantity must be unbounded. We have probed this condition by studying a family of variational PDE optimization problems …
On the interaction of strain and vorticity for solutions of the Navier--Stokes equation
本文证明了一个新的恒等式，揭示了应变与涡量在纳维-斯托克斯方程非线性方面的相互作用，为理解非线性耗散提供了新视角，并推导了新的正则性准则。
We show - in the framework of physical scales and $(K_1,K_2)$-averages - that Kolmogorov's dissipation law combined with the smallness condition on a Taylor length scale are sufficient to guarantee energy cascades in the forced Navier-Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws - in terms of Grashof number - for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.
Sharp norm inflation for 3D Navier-Stokes equations in supercritical spaces
本文证明了三维不可压缩Navier-Stokes方程在标度临界线附近的超临界Besov空间中存在范数膨胀，并揭示了正负正则性指标下不同的能量级联增长机制，同时首次构造出涡量有限时间任意大增长的例子。
Critical Ledgers and Scale-Defect Cascades for Navier-Stokes
针对三维不可压缩Navier-Stokes方程在局部正则性阈值附近的适定弱解，提出有限尺度供给-税收缩减定理，通过账本坐标将持续临界坏性转化为明确的会计替代方案。
Statistical Estimates for 2D stochastic Navier-Stokes Equations
本文研究二维随机湍流的统计性质，推导了能量和涡度耗散率均值的严格界，证明无粘极限下能量耗散趋于零而涡度耗散有界，符合二维湍流的双级联理论。
On Turbulent Behavior of the Generalized Surface Quasigeostrophic Equations
本文通过级联论证和数值模拟，为广义表面准地转方程推导出更优的能量谱标度，并证明了涡度级联条件及耗散率上界，揭示了参数依赖关系。
Rigorous estimates for the total - (kinetic) energy plus pressure - flux in R^3 are obtained from the three dimensional Navier-Stokes equations. The bounds are used to establish a condition - involving Taylor length scale and the size of the domain - sufficient for existence of the inertial range and the energy cascade in decaying turbulence (zero driving force, non-increasing global energy). Several manifestations of the locality of the flux under this condition are obtained. All the scales involved are actual physical scales in R^3 and no regularity or homogeneity/scaling assumptions are mad…
A Principle of Maximum Entropy for the Navier-Stokes Equations
本文针对粘性不可压缩流动提出最大熵原理，通过最大化能量-拟涡能面上的相对熵泛函，建立与纳维-斯托克斯方程定态统计解及柯尔莫哥洛夫湍流理论的联系。
Renormalized Lambert-W Cascade and Finite-Time Amplification and Blowup for reconstructed b Dynamics on T^3 for the 3D Navier Stokes Equations
本文整合重整化Lambert-W级联及有限时间放大机制，重构T^3上Navier-Stokes方程b_i动力学，支持其有限时间失去光滑性的结论。
In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier-Stokes equations, in the periodic, smooth finite energy, smooth $H^1$, Schwartz, or mild $H^1$ categories, and with or without a forcing term. In particular, we show that if one has global well-posedness in $H^1$ for the periodic Navier-Stokes problem with a forcing term, then one can obtain global regularity both for periodic and for Schwartz initial data (thus yielding a positive answer to both official formulations of the problem for the C…
Navier-Stokes Equations with Fractional Dissipation and Associated Doubly Stochastic Yule Cascades
本文研究了与分数耗散纳维-斯托克斯方程关联的自相似双重随机尤尔级数，探讨了其在不同参数下的爆炸行为，并利用随机解过程建立了偏微分方程的非唯一性和有限时间爆破结果。
Global smooth solutions by high mode Lie-Transport noise for Logarithmically Hyperdissipative Navier-Stokes equations
研究三维环面上带Lie-transport噪声的对数超黏性Navier-Stokes模型，证明当噪声强度足够大且频率足够高时，系统以任意接近1的概率存在唯一全局光滑解。
The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier-Stokes equations. In their work Li and Sinai have proposed a renormalization approach to the problem of existence of finite-time blow up solutions of this equation. In this setting, existence of finite time blow ups is a consequence of existence of a fixed point for a certain renormalization operator on an appropriate functional space. They have provided a proof of existence of complex-valued finite time blow up solutions of the quasi-geostrophic equation. In this paper we revisit …
The existence of weak solutions to the stationary Navier-Stokes equations in the whole plane $\mathbb{R}^2$ is proven. This particular geometry was the only case left open since the work of Leray in 1933. The reason is that due to the absence of boundaries the local behavior of the solutions cannot be controlled by the enstrophy in two dimensions. We overcome this difficulty by constructing approximate weak solutions having a prescribed mean velocity on some given bounded set. As a corollary, we obtain infinitely many weak solutions in $\mathbb{R}^2$ parameterized by this mean velocity, which …
The paper explores the symbiotic relation between the Navier-Stokes equations and the associated stochastic cascades. Specifically, we examine how some well-known existence and uniqueness results for the Navier-Stokes equations can inform about the probabilistic features of the associated stochastic cascades, and how some probabilistic features of the stochastic cascades can, in turn, inform about the existence and uniqueness (or the lack thereof) of solutions. Our method of incorporating the stochastic explosion gives a simpler and more natural method to construct the solution compared to the…
The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the…
Orbit-Level Transfer Matrix for the 3D Fourier-Galerkin Navier-Stokes System on the Periodic Torus: Explicit Orbit-Triad Incidence Bounds and Deterministic Row-Sum Estimates
本文研究了在周期托勒密上利用全八面体对称群Oh对三维不可压缩纳维-斯托克斯方程进行立方傅里叶-伽辽金截断后的非线性相互作用，通过轨道级转移矩阵M_N(u)和壳切片中的轨道三元组发生次数估计，得出N^{4+ε}的界及确定性行和界。
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