Scalars convected by a 2D incompressible flow
Diego Cordoba, Charles Fefferman
我们提供了一个测试的数值模拟，几个二维不可压缩流，出现开发尖锐的前。我们表明，为了有一个前的速度必须有不受控制的速度增长。
We provide a test for numerical simulations, for several two dimensional incompressible flows, that appear to develop sharp fronts. We show that in order to have a front the velocity has to have uncontrolled velocity growth.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Mixing for generic passive scalars by incompressible flows
针对不可压缩流混合的通用性问题，本文证明经典判据在任意小扰动下失效，并建立适用于 $L^\infty$ 数据的 Young 测度理论，刻画一般有界被动标量的混合。
Consider a diffusion-free passive scalar $θ$ being mixed by an incompressible flow $u$ on the torus $\mathbb{T}^d$. Our aim is to study how well this scalar can be mixed under an enstrophy constraint on the advecting velocity field.Our main result shows that the mix-norm ($||θ(t)||_{H^{-1}}$) is bounded below by an exponential function of time. The exponential decay rate we obtain is not universal and depends on the size of the support of the initial data. We also perform numerical simulations and confirm that the numerically observed decay rate scales similarly to the rigorous lower bound, at…
In this paper we study a singularly modified version of the incompressible porous media equation. We investigate the implications for the local well-posedness of the equations by modifying, with a fractional derivative, the constitutive relation between the scalar density and the convecting divergence free velocity vector. Our analysis is motivated by recent work \cite{CCCGW} where it is shown that for the surface quasi-geostrophic equation such a singular modification of the constitutive law for the velocity, quite surprisingly still yields a locally well-posed problem. In contrast, for the s…
On the behaviors of solution near possible blow-up time in the incompressible Euler and related equations
研究了由Euler方程、Navier-斯托克斯方程和地面准地转方程的光滑解组成的标量在可能爆破时刻附近的行为。对标度不变范数的动力学方程进行积分，得到了上述量的可能爆破行为，并由此得到了新的爆破准则和爆破的必要条件.
We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq \infty$. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm $\dot H^{-1}$, the other geometric, related to rearrangements of sets. We study r…
In Ann. Math., 170 (2009), 1417-1436, C. De Lellis and L. Székelyhidi Jr. constructed wild solutions of the incompressible Euler equations using a reformulation of the Euler equations as a differential inclusion together with convex integration. In this article we adapt their construction to the system consisting of adding the transport of a passive scalar to the two-dimensional incompressible Euler equations.
This article addresses a fundamental concern regarding the incompressible approximation of fluid motions, one of the most widely used approximations in fluid mechanics. Common belief is that its accuracy is $O(ε)$ where $ε$ denotes the Mach number. In this article, however, we prove an $O(ε^2)$ accuracy for the incompressible approximation of the isentropic, compressible Euler equations thanks to several decoupling properties. At the initial time, the velocity field and its first time derivative are of $O(1)$ size, but the boundary conditions can be as stringent as the solid-wall type. The fas…
Relaxation enhancement by controlling incompressible fluid flows
提出基于PDE可控性的方法增强被动标量场扩散混合，通过不可压缩Euler方程的状态轨迹构建时间依赖的弛豫增强向量场，并利用近似和精确可控性定理实现弛豫增强。
The Vanishing-Diffusion Limit of an Incompressible Visco-Morphoelastic System: Weak Solutions and a Jaumann Defect
研究三维不可压缩粘性形态弹性系统，通过ε阶扩散正则化构造弱解，证明标量变量强收敛但Jaumann项产生缺陷，并给出缺陷为零的充分条件。
We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible $C^\infty([0,T)\times \mathbb{T}^d)\cap L^1([0,T]; C^{1-}(\mathbb{T}^d))$ velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows non-uniqueness of solutions to the transport equation with an incompressible $L^1([0,T]; C^{1-}(\mathbb{T}^d))$ drift, which is smooth except at one point in time. We also provide three sufficient conditions for anomalous dissipation provided solutions to the inviscid equation become…
We study the mixing properties of a passive scalar advected by an incompressible flow. We consider a class of cellular flows (more general than the class in [Crippa-Schulze M3AS 2017]) and show that, under the constraint that the palenstrophy is bounded uniformly in time, the mixing scale of the passive scalar cannot decay exponentially.
Instantaneous blowup of incompressible flow with passive tracer
通过逆级联机制构造一族解，使得速度场和被动标量场同时发生L∞范数爆破，并解决了被动分量带来的相容性约束。
This work deals with the entropic regularization of the Brenier problem for perfect incompressible fluids introduced by Arnaudon, Cruzeiro, Léonard and Zambrini. We show that as in the original setting, there exists a scalar pressure field which is interpreted as the Lagrange multiplier associated to the incompressibility constraint. The proof goes through a reformulation of the problem in PDE terms.
In this article, we study the homogenization limit of a family of solutions to the incompressible 2D Euler equations in the exterior of a family of $n_k$ disjoint disks with centers $\{z^k_i\}$ and radii $\varepsilon_k$. We assume that the initial velocities $u_0^k$ are smooth, divergence-free, tangent to the boundary and that they vanish at infinity. We allow, but we do not require, $n_k \to \infty$, and we assume $\varepsilon_k \to 0$ as $k\to \infty$. Let $γ^k_i$ be the circulation of $u_0^k$ around the circle $\{|x-z^k_i|=\varepsilon_k\}$. We prove that the homogenization limit retains inf…
Statistical conservation laws for scalar model problems: Hierarchical evolution equations
本文针对带随机初始数据的标量守恒律，提出多点PDF和单点高阶导数PDF两种分层演化方程框架，以刻画统计相关性并指导封闭策略。
Support growth of vorticity for bi-rotational Euler flows in high dimensions
该研究在高维空间中探讨了双旋转对称性下欧拉方程的涡量支持增长，证明了特定初始条件下的涡量支持直径无限增长。
Anomalous dissipation and Euler flows
该研究严格证明了具有$C^{(\frac{1}{3})^-}$正则性的不可压缩欧拉方程弱解平流的标量存在反常耗散，首次推导了无外力确定性流体力学方程下的标量湍流第零定律，还得到了漂移典型性及符合理查森假设的标量方差下界等湍流性质。
Introduction to the theory of mixing for incompressible flows
本文从偏微分方程角度介绍了不可压缩流混合理论，讨论了混合尺度的定义及下界估计，分析了其对流体几何和正则性的影响。
Exact Lagrangian Realization and Robust Strain Sensing in Incompressible Flow
该研究证明三维欧拉与纳维-斯托克斯方程单壳解的单粒子变形梯度可实现整个\boldsymbol{\rm SL}(3,\boldsymbol{\rm R})群，分类了确定无迹应变的有限材料方向系统，明确了鲁棒应变传感的精确标量通道数。
We study the mixing properties of a scalar $ρ$ advected by a certain incompressible velocity field $u$ on the two dimensional unit ball, which is a stationary radial solution of the Euler equation. The scalar $ρ$ solves the continuity equation with velocity field $u$ and we can measure the degree of mixedness of~$ρ$ with two different scales commonly used in this setting, namely the geometric and the functional mixing scale. We develop a physical space approach well adapted for the quantitative analysis of the decay in time of the geometric mixing scale, which turns out to be polynomial for a …
Our recent work identifies material surfaces in incompressible flows that extremize the transport of an arbitrary, weakly diffusive scalar field relative to neighboring surfaces. Such barriers and enhancers of transport can be located directly from the deterministic component of the velocity field without diffusive or stochastic simulations. Here we extend these results to compressible flows and to diffusive concentration fields affected by sources or sinks, as well as by spontaneous decay. We construct diffusive transport extremizers with and without constraining them on a specific initial co…
We consider a class of invariant measures for a passive scalar $f$ driven by an incompressible velocity field $\boldsymbol{u}$, on a $d$-dimensional periodic domain, satisfying $$ \partial_t f + \boldsymbol{u} \cdot \nabla f = 0, \qquad f(0)=f_0. $$ The measures are obtained as limits of stochastic viscous perturbations. We prove that the span of the $H^1$ eigenfunctions of the operator $\boldsymbol{u}\cdot\nabla$ contains the support of these measures. We also analyze several explicit examples: when $\boldsymbol{u}$ is a shear flow or a relaxation enhancing flow (a generalization of weakly mi…
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。
We address the challenge of optimal incompressible stirring to mix an initially inhomogeneous distribution of passive tracers. As a quantitative measure of mixing we adopt the $H^{-1}$ norm of the scalar fluctuation field, equivalent to the (square-root of the) variance of a low-pass filtered image of the tracer concentration field. First we establish that this is a useful gauge even in the absence of molecular diffusion: its vanishing as $t --> \infty$ is evidence of the stirring flow's mixing properties in the sense of ergodic theory. Then we derive absolute limits on the total amount of mix…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。