Elaine Cozzi
Assuming that initial velocity has finite energy and initial vorticity is bounded in the plane, we show that for any finite time interval the unique solutions of the Navier-Stokes equations converge uniformly to the unique solution of the Euler equations as viscosity approaches zero. We also establish a rate of convergence.
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Assuming that initial velocity and initial vorticity are bounded in the plane, we show that on a sufficiently short time interval the unique solutions of the Navier-Stokes equations converge uniformly to the unique solution of the Euler equations as viscosity approaches zero. We also establish a rate of convergence.
We say that the vanishing viscosity limit holds in the classical sense if the velocity for a solution to the Navier-Stokes equations converges in the energy norm uniformly in time to the velocity for a solution to the Euler equations. We prove, for a bounded domain in dimension 2 or higher, that the vanishing viscosity limit holds in the classical sense if and only if a vortex sheet forms on the boundary.
Chemin has shown that solutions of the Navier-Stokes equations in the plane for an incompressible fluid whose initial vorticity is bounded and lies in L^2 converge in the zero-viscosity limit in the L^2-norm to a solution of the Euler equations, convergence being uniform over any finite time interval. Yudovich, assuming an initial vorticity lying in L^p for all p >= q for some q, established the uniqueness of solutions to the Euler equations for an incompressible fluid in a bounded domain of n-space assuming a particular bound on the growth of the L^p-norm of the initial vorticity as p grows l…
We study the time evolution of a viscous incompressible fluid with axial symmetry without swirl, when the initial vorticity is very concentrated in $N$ disjoint rings. We show that in a suitable joint limit, in which both the thickness of the rings and the viscosity tend to zero, the vorticity remains concentrated in $N$ disjointed rings, each one of them performing a simple translation along the symmetry axis with constant speed.
We show strong convergence of the vorticities in the vanishing viscosity limit for the incompressible Navier-Stokes equations on the two-dimensional torus, assuming only that the initial vorticity of the limiting Euler equations is in $L^p$ for some $p>1$. This substantially extends a recent result of Constantin, Drivas and Elgindi, who proved strong convergence in the case $p=\infty$. Our proof, which relies on the classical renormalization theory of DiPerna-Lions, is surprisingly simple.
We study vanishing viscosity solutions to the axisymmetric Euler equations with (relative) vorticity in $L^p$ with $p>1$. We show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions. Moreover, we show that the kinetic energy is preserved provided that $p>3/2$ and the vorticity is nonnegative and has finite second moments.
We study the vanishing viscosity limit for the three-dimensional incompressible Navier-Stokes equations in terms of the relative vorticity in the setting of axisymmetric velocity fields without swirl. We show that the weak convergence of relative vorticity to a renormalized solution of the Euler equations, established by Nobili and Seis, can be upgraded to strong convergence.
We consider a sequence of Leray-Hopf weak solutions of the 2D Navier-Stokes equations on a bounded domain, in the vanishing viscosity limit. We provide sufficient conditions on the associated vorticity measures, away from the boundary, which ensure that as the viscosity vanishes the sequence converges to a weak solution of the Euler equations. These assumptions are consistent with vortex sheet solutions of the Euler equations.
Global-in-time inviscid limit of the Navier-Stokes equations for vortex-wave data
本文证明涡-波系统是二维Navier-Stokes方程在任意有限时间区间上的粘性消失极限，并给出两个分量的定量估计，实现全局时间合理性证明。
We consider the zero viscosity limit of the incompressible Navier-Stokes equations with non-slip boundary condition in the half-space for the initial vorticity located away from the boundary. By using the vorticity formulation and Cauchy-Kowaleskaya theorem, Maekawa proved the local in time convergence of the Navier-Stokes equations in the half- plane to the Euler equations outside a boundary layer and to the Prandtl equations in the boundary layer. In this paper, we develop the direct energy method to generalize Maekawa's result to the half-space.
The paper is concerned with the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space $L^q$ with $2<q\le\infty$, we show the degenerate viscous lake equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In the special case when the vorticity is in $L^\infty$, an explicit convergence rate is obtained.
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in $\R^n$ with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure $p$ can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolut…
Singular vorticty solutions of the incompressible 3D-Euler equation are constructed which satisfy the BKM criterion (cf. [2]). The construction is done by inviscid limits of vorticity solutions of transformed incompressible Navier Stokes type equations with a damping potential term, where the latter equations admit a global regular solution for positive viscosity. The inviscid limit vorticity solution of the incompressible Euler vorticity equation becomes singular at a point of the boundary of a finite domain.
In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in \cite{BouchutCrippa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in \cite{Seis16a} and some tools from harmonic analysis introduced in \cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in \cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtai…
We deal with the incompressible Navier-Stokes equations, in two and three dimensions, when some vortex patches are prescribed as initial data i.e. when there is an internal boundary across which the vorticity is discontinuous. We show -thanks to an asymptotic expansion- that there is a sharp but smooth variation of the fluid vorticity into a internal layer moving with the flow of the Euler equations; as long as this later exists and as t * nu is small, where nu is the viscosity coefficient.
The Navier-Stokes equations in $\mathbb R^2_+$ with point vortex initial data: Zero-viscosity limit
本文研究半空间中纳维-斯托克斯方程在点涡初始数据下的渐进行为，通过精确匹配点涡与边界层剖面，分析涡量的三个组成部分，建立精细估计并证明无粘极限，显示解在边界外趋近于拉姆-奥塞恩涡，在近边界区域趋近于普朗特边界层系统。
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…
We consider the inviscid limit for the two-dimensional Navier--Stokes equations in the class of integrable and bounded vorticity fields. It is expected that the difference between the Navier--Stokes and Euler velocity fields vanishes in $L^2$ with an order proportional to the square root of the viscosity constant $ν$. Here, we provide an order $\left(ν/|\logν|\right)^{\frac12\exp(-Ct)}$ bound, which slightly improves upon earlier results by Chemin.
In this paper we prove the uniform-in-time $L^p$ convergence in the inviscid limit of a family $ω^ν$ of solutions of the $2D$ Navier-Stokes equations towards a renormalized/Lagrangian solution $ω$ of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $ω^ν$ to $ω$ in $L^p$. Finally, we show that solutions of the Euler equations with $L^p$ vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Euler…
In a recent paper [5], the global well-posedness of the two-dimensional Euler equation with vorticity in \mbox{$L^1\cap LBMO$} was proved, where $ LBMO$ is a Banach space which is strictly imbricated between \mbox{$L^\infty$} and $BMO$. In the present paper we prove a global result of inviscid limit of the Navier-stokes system with data in this space and other spaces with the same BMO flavor. Some results of local uniform estimates on solutions of the Navier-Stokes equations, independent of the viscosity, are also obtained.
We consider solutions to the Navier-Stokes equations with Navier boundary conditions in a bounded domain in the plane with a C^2-boundary. Navier boundary conditions can be expressed in the form w = (2 K - A) v . T and v . n = 0 on the boundary, where v is the velocity, w the vorticity, n a unit normal vector, T a unit tangent vector, and A is a bounded measurable function on the boundary. Such solutions have been considered for simply connected domains by Clopeau, Mikelić, and Robert under the assumption that the initial vorticity is bounded, and by Lopes Filho, Nussenzveig Lopes, and Planas …
The focus of this paper is on the analysis of the boundary layer and the associated vanishing viscosity limit for two classes of flows with symmetry, namely, Plane-Parallel Channel Flows and Parallel Pipe Flows. We construct explicit boundary layer correctors, which approximate the difference between the Navier-Stokes and the Euler solutions. Using properties of these correctors, we establish convergence of the Navier-Stokes solution to the Euler solution as viscosity vanishes with optimal rates of convergence. In addition, we investigate vorticity production on the boundary in the limit of va…
Classical vorticity solution branches of the three dimensional incompressible Euler equation are constructed where a velocity component can blow up at some point after finite time for regular data in H2. Furthermore, vorticity can blow up after finite time for data in H2, and there are classical solution branches with regular data which develop weak singularities at some point of space time after finite time. The construction of these time-local solution branches is by viscosity limits of viscosity extensions of time-reversed Euler-type equations. The short time (weak) singularities are then i…
In this paper we prove that solutions of the 2D Euler equations in vorticity formulation obtained via vanishing viscosity approximation are renormalized.
We develop the concept of an infinite-energy statistical solution to the Navier-Stokes and Euler equations in the whole plane. We use a velocity formulation with enough generality to encompass initial velocities having bounded vorticity, which includes the important special case of vortex patch initial data. Our approach is to use well-studied properties of statistical solutions in a ball of radius R to construct, in the limit as R goes to infinity, an infinite-energy solution to the Navier-Stokes equations. We then construct an infinite-energy statistical solution to the Euler equations by ma…
On the dynamics of the boundary vorticity for incompressible viscous flows
本研究推导出不可压缩粘性流的边界涡量动力学方程，发现壁面处粘度加倍，可在无需主流细节的情况下有界估计边界涡量，并通过随机直接数值模拟的两种更新方法验证了方程的有效性。
In 1895, Korteweg and de Vries (KdV), derived their celebrated equation describing the motion of waves of long wavelength in shallow water. In doing so they made a number of quite reasonable assumptions, incompressibility of the water and irrotational fluid. The resulting equation, the celebrated KdV equation, has been shown to be a very reasonable description of real water waves. However there are other phenomena which have an impact on the shape of the wave, that of vorticity and viscosity. This paper examines how a constant vorticity affects the shape of waves in electrohydrodynamics. For c…
In this paper, we investigate the local-in-time well-posedness for the two-dimensional Prandtl equations in weighted Sobolev spaces under the Oleinik's monotonicity condition.Due to the loss of tangential derivative caused by vertical velocity appearing in convective term, we add with artificial horizontal viscosity term to construct an approximate system that can obtain local-in-time well-posedness results easily. For this approximate system, we construct a new weighted norm for the vorticity to derive a positive life time(independent of artificial viscosity coefficient) and obtain uniform bo…
We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes [LMN], on the vanishing viscosity limit of circularly symmetric viscous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in $L^2$-norm as long as the prescribed angular velocity $α(t)$ of the boundary has bounded total variation. Here we establish convergence in stronger $L^2$ and $L^p$-Sobolev spaces, allow for more singular angular velocities $α$, and address the issue of analyzing the behavior of the boundary layer. This includes an analysis of concentration of vorticity in the vanishing…
We study the 2D Navier--Stokes solution starting from an initial vorticity mildly concentrated near $N$ distinct points in the plane. We prove quantitative estimates on the propagation of concentration near a system of interacting point vortices introduced by Helmholtz and Kirchhoff. Our work extends the previous results in the literature in three ways: The initial vorticity is concentrated in a weak (Wasserstein) sense, it is merely $L^p$ integrable for some $p>2$, and the estimates we derive are uniform with respect to the viscosity.
In this work we extend the results in [6,32] on the 2D IPM system with constant viscosity (Atwood number $A_μ=0$) to the case of viscosity jump ($|A_μ|<1$). We prove a h-principle whereby (infinitely many) weak solutions in $C_tL_{w^*}^{\infty}$ are recovered via convex integration whenever a subsolution is provided. As a first example, non-trivial weak solutions with compact support in time are obtained. Secondly, we construct mixing solutions to the unstable Muskat problem with initial flat interface. As a byproduct, we check that the connection, established by Székelyhidi for $A_μ=0$, betwe…
The two-dimensional Navier-Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the moments of an expansion of the vorticity with respect to Hermite functions and of the centers of vorticity concentrations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit we formally recover the Helmholtz-Kirchhoff model for the evolution of point-vortices. The present expansion systematically incorporates the effects of both viscosity and finite vortex core size. We also…
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…
We establish the inviscid limit of the incompressible Navier-Stokes equations on the whole plane $\mathbb{R}^2$ for initial data having vorticity as a superposition of point vortices and a regular component. In particular, this rigorously justifies the vortex-wave system from the physical Navier-Stokes flows in the vanishing viscosity limit, a model that was introduced by Marchioro and Pulvirenti in the early 90s to describe the dynamics of point vortices in a regular ambient vorticity background. The proof rests on the previous analysis of Gallay in his derivation of the vortex-point system.
Uniform vorticity depletion and inviscid damping for periodic shear flows in the high Reynolds number regime
研究高雷诺数下非方形环面上二维Navier-Stokes方程线性化剪切流的动力学，获得关于粘性均匀的无粘阻尼与涡度耗散估计，核心在于分析三种谱参数区域下的Orr-Sommerfeld方程解的行为。
Global Yudovich-type solutions to a reduced model for micropolar fluids with zero viscosity
本文研究了二维微极流体简化模型在低正则性下的适定性，证明了全局唯一性Yudovich型解，扩展了Yudovich框架到扰动不可压欧拉方程的系统。
The quasi-geostrophic equation or the Euler equation with dissipation studied in the present paper is a simplified form of the atmospheric circulation model introduced by Charney and DeVore [J. Atmos. Sci. 36(1979), 1205-1216] on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking. The fluid motion defined by the equation is driven by a zonal thermal forcing and an Ekman friction forcing measured by $κ>0$. It is proved that the steady-state solution is unique for $κ>1$ while multiple steady-state solutions exist for $κ<κ_{crit}$ with respect …
On the Role of the Viscosity Parameters in the Large Time Asymptotics of 2D Micropolar Flows
本文研究了二维微极流中四个粘性参数的作用，发现大时间行为仅依赖于运动粘性系数μ，而非其他参数。通过新的涡量类似恒等式，揭示了微旋转效应增强耗散的作用。
Vanishing viscosity non-unique solutions to the forced 2D Euler Equations
本文研究受迫二维 Navier–Stokes 方程在黏性趋于零时能否选择唯一 Euler 解，通过初始数据的 $O(\varepsilon)$ 扰动揭示阈值 $\varepsilon\sim\nu^{\kappa_{\rm c}}$ 及非唯一非径向极限。
Consider the dynamics of a layer of viscous incompressible fluid under the influence of gravity. The upper boundary is a free boundary with the effect of surface tension taken into account, and the lower boundary is a fixed boundary on which the Navier-slip condition is imposed. It is proved that there is a uniform time interval on which the estimates independent of both viscosity and surface tension coefficients of the solution can be established. This then allows one to justify the vanishing viscosity and surface tension limits by the strong compactness argument. In the presence of surface t…
In this paper, we investigate the global existence and uniqueness of strong solutions to 2D Boussinesq system with anisotropic thermal diffusion or anisotropic viscosity and with striated initial data. Using the key idea of Chemin to solve 2-D vortex patch of ideal fluid, namely the striated regularity can help to bound the gradient of the velocity, we can prove the global well-posedness of the Boussinesq system with anisotropic thermal diffusion with initial vorticity being discontinuous across some smooth interface. In the case of an anisotropic horizontal viscosity we can study the propagat…
The Muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media. In this paper, we consider both the 2D and 3D setting of two fluids of different constant densities and different constant viscosities. In this situation, the related contour equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available for the amplitude and the slopes of the interface in $L^\infty$. We prove …
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…
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。
Asymptotic Limits for Strain-Gradient Viscoelasticity with Nonconvex Energy
本文研究多空间维度中具有高阶梯度与非凸能量的粘弹性系统，建立了粘性趋于零和色散系数趋于零时的渐近极限，并在二维情形下给出了收敛速率。
Strong Uniqueness by Kraichnan Transport Noise for the 2D Boussinesq Equations with Zero Viscosity
本文研究了无粘二维Boussinesq方程在Kraichnan型传输噪声下的强唯一性，利用噪声引起的反常正则性补偿非线性项和耦合项的奇异性，证明了概率强解的存在性与唯一性。
We construct global weak solutions of the Euler equations in an infinite cylinder $Π=\{x\in \mathbb{R}^{3}\ |\ x_h=(x_1,x_2),\ r=|x_h|<1\}$ for axisymmetric initial data without swirl when initial vorticity $ω_{0}=ω^θ_{0}e_θ$ satisfies $ω^θ_{0}/r\in L^{q}$ for $q\in [3/2,3)$. The solutions constructed are Hölder continuous for spatial variables in $\overlineΠ$ if in addition that $ω^θ_{0}/r\in L^{s}$ for $s\in (3,\infty)$ and unique if $s=\infty$. The proof is by a vanishing viscosity method. We show that the Navier-Stokes equations subject to the Neumann boundary condition is globally well-po…
On a generalized incompressible model in two dimensions
研究Ohkitani提出的广义不可压缩模型，通过旋转速度梯度将二维Burgers方程与不可压缩Navier-Stokes方程关联，证明了在H^3初始数据下无粘模型局部解的存在唯一性及有限时间爆破条件，以及粘性模型全局解的存在唯一性、长时间行为和时间衰减率。
A selection principle for 2D steady Euler flows via the vanishing viscosity limit
本文研究了二维稳态欧拉流的消粘性极限，发现有限域内仅存在常涡量流，且在无限长带状域中，周期性粘性速度的极限仅包括常流、Couette流和Poiseuille流，证明依赖于对粘性和无粘流流线的细致分析和一个强大的刚性定理。
No anomalous dissipation in two-dimensional incompressible fluids
本文证明在初始涡度含正奇异部分时，二维不可压缩 Navier-Stokes 方程的消失粘性解无非异常耗散，揭示了二维湍流与三维的鲜明对比。
We construct and study global solutions for the 3-dimensional incompressible MHD systems with arbitrary small viscosity. In particular, we provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution initially behaves like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number), thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects, eventually, the total energy of the system becomes extremely small compared to the viscosity so that the d…
First, we consider Kolmogorov flow (a shear flow with a sinusoidal velocity profile) for 2D Navier-Stokes equation on a torus. Such flows, also called bar states, have been numerically observed as one type of metastable states in the study of 2D turbulence. For both rectangular and square tori, we prove that the non-shear part of perturbations near Kolmogorov flow decays in a time scale much shorter than the viscous time scale. The results are obtained for both the linearized NS equations with any initial vorticity in L^2, and the nonlinear NS equation with initial L^2 norm of vorticity of the…
Motivated by applications to vortex rings, we study the Cauchy problem for the three-dimensional axisymmetric Navier-Stokes equations without swirl, using scale invariant function spaces. If the axisymmetric vorticity is integrable with respect to the two-dimensional measure dr dz, where (r,θ,z) denote the cylindrical coordinates in R^3, we show the existence of a unique global solution, which converges to zero in L^1 norm as time goes to infinity. The proof of local well-posedness follows exactly the same lines as in the two-dimensional case, and our approach emphasizes the similarity between…
Vanishing viscosity limit for axisymmetric vortex rings
本文研究了 $R^3$ 中低粘性不可压缩 Navier-Stokes 方程的 Cauchy 问题，证明了在小粘性下圆形涡丝解可由热方程解与 Kelvin 公式速度近似，从而严格论证了弱粘性流体中涡丝的双法向运动，并利用 Arnold 几何稳定性方法验证了近似解的稳定性。
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