For a homogeneous incompressible 2D fluid confined within a bounded Lipschitz simply connected domain, homogeneous Neumann pressure boundary conditions are equivalent to a constant boundary vorticity. We investigate the rigidity of such conditions.
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.
We construct an initial data for the two-dimensional Euler equation in a bounded smooth symmetric domain such that the gradient of vorticity in $L^{\infty}$ grows as a double exponential in time for all time. Our construction is based on the recent result by Kiselev and Šverák.
By performing estimates on the integral of the absolute value of vorticity along a local vortex line segment, we establish a relatively sharp dynamic growth estimate of maximum vorticity under some assumptions on the local geometric regularity of the vorticity vector. Our analysis applies to both the 3D incompressible Euler equations and the surface quasi-geostrophic model (SQG). As an application of our vorticity growth estimate, we apply our result to the 3D Euler equation with the two anti-parallel vortex tubes initial data considered by Hou-Li. Under some additional assumption on the vorti…
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.
It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations $$\mathrm{St} \partial_t F + v\cdot \nabla_x F = \frac{1}{\mathrm{Kn}} Q(F ,F ) $$ toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded $$\partial_t u + u \cdot \nabla_x u + \nabla_x p = 0,\text{div }u =0.$$ We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macro…
In this paper we consider the motion of a rigid body immersed in a two dimensional unbounded incompressible perfect fluid with vorticity. We prove that when the body shrinks to a massless pointwise particle with fixed circulation, the "fluid+rigid body" system converges to the vortex-wave system introduced by Marchioro and Pulvirenti in [11]. This extends both the paper [2] where the case of a solid tending to a massive pointwise particle was tackled and the paper [3] where the massless case was considered but in a bounded cavity filled with an irrotational fluid.
We prove the global existence of a helical weak solution of the 3D Euler equations, in full space, for an initial velocity with helical symmetry, without swirl and whose initial vorticity is compactly supported in the axial plane and belongs to $L^p$, for some $p>\frac{4}{3}$. This result is an extension of the existence part of the work of B. Ettinger and E. Titi (SIAM J. Math Anal. 41(2009) 269-296), who studied well-posedness of the Euler equations with helical symmetry without swirl, with bounded initial vorticity, in a helical pipe.
This paper is concerned with the global well-posedness of the two-dimensional incompressible vorticity equation in the half plane. Under the assumption that the initial vorticity $ω_0\in W^{k,p}(\R^{2}_+)$ with $k\geq3$ and $1<p<2$, it is shown that the two-dimensional incompressible vorticity equation admits a unique solution $ω\in C([0,T];W^{k,p}(\R^{2}_+))$ for any $T>0$. An elementary and self-contained proof is presented and delicate estimates of the velocity and its derivatives are obtained in this paper. It should be emphasized that the uniform estimate on $\int^t_0\|u(τ)\|_{W^{1,\infty…
Motivated by the works of Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy in [8, Commun. Math. Phys. 348: 129-143, 2016] and Chen and Yu in [5, J. Math. Pures Appl. 131: 1-16, 2019], we address how the $L^p$ control of vorticity could influence the energy conservation for the incompressible homogeneous and nonhomogeneous Euler equations in this paper. For the homogeneous flow in the periodic domain or whole space, we provide a self-contained proof for the criterion $ω=\text{curl}u\in L^{3}(0,T;L^{\frac{3n}{n+2}}(Ω))\,(n=2,3)$, which generalizes the corresponding result in [8] and can be…
In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data We begin by showing that the velocity can be uniquely reconstructed from the vorticity and boundary component circulations, which allows to recast 2D Euler evolution using vorticity and the circulations as dynamic variables. The weak form of this vortex dynamics formulation of the equation…
In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of steady vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal magnitude and opposite signs. The results are obtained by using an improved vorticity method.
In [Math. Meth. Appl. Sci. 19 (1996) 53-62], C. Marchioro examined the problem of vorticity confinement in the exterior of a smooth bounded domain. The main result in Marchioro's paper is that solutions of the incompressible 2D Euler equations with compactly supported nonnegative initial vorticity in the exterior of a connected bounded region have vorticity support with diameter growing at most like $\mathcal{O}(t^{(1/2)+\vare})$, for any $\vare>0$. In addition, if the domain is the exterior of a disk, then the vorticity support is contained in a disk of radius $\mathcal{O}(t^{1/3})$. The purp…
For the $2D$ Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to $L^\infty \cap H^1$ but escapes $H^1$ immediately for $t>0$. Our main observation is that a localized chunk of vorticity bounded in $L^\infty \cap H^1$ with odd-odd symmetry is able to generate a hyperbolic flow with large velocity gradient at least for a short period of time, which stretches the vorticity gradient.
We establish the regularity of weak solutions for the vorticity equation associated to a family of desingularised models for vortex filament dynamics in 3D incompressible viscous flows. These include and generalise the classical model "of an allowance for the thickness of the vortices" due to Louis Rosenhead in 1930. Our approach is based on an interplay between the geometry of vorticity and analytic inequalities in Sobolev spaces.
We prove a continuation critereon for incompressible liquids with free surface boundary. We combine the energy estimates of Christodoulou and Lindblad with an analog of the estimate due to Beale, Kato, and Majda for the gradient of the velocity in terms of the vorticity, and use this to show solution can be continued so long as the second fundamental form and injectivity radius of the free boundary, the vorticity, and one derivative of the velocity on the free boundary remain bounded, assuming that the Taylor sign condition holds.
We consider the two-dimensional, $β$-plane, vorticity equations for an incompressible flow, where the zonally averaged flow varies on scales much larger than the perturbation. We prove global existence and uniqueness of the solution to the equations on periodic settings.
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…
In this article we consider the $α$--Euler equations in the exterior of a small fixed disk of radius $ε$. We assume that the initial potential vorticity is compactly supported and independent of $ε$, and that the circulation of the unfiltered velocity on the boundary of the disk does not depend on $ε$. We prove that the solution of this problem converges, as $ε\to 0$, to the solution of a modified $α$--Euler equation in the full plane where an additional Dirac located at the center of the disk is imposed in the potential vorticity.
We use the vorticity transportation equation as the start point--with the help of stream function for two-dimensional planar incompressible flows--to obtain exact solutions that characterize evolution and dynamics of the flows. These traveling wave solutions, including both continuous and solitary waves, also recover the results existing in the theory of potential flows. We further analyze the flow patterns regarding symmetry solutions generated by Lie transformation groups. Some of the symmetry solutions have not been discussed in the current literature. They could display flow patterns very …
In this paper, we consider turbulence from a geometric perspective based on the vorticity equations for incompressible viscous fluid flows. We derive several quantitative statements about the statistics of turbulent flows. In particular we establish an entropy decay inequality which involves the turbulent kinetic energy and $L^{1}$-enstrophy, and we identify the small time scale of the vorticity broken down and the vorticity creation under the universality assumption of small time scales of turbulence flows.
We investigate a steady planar flow of an ideal fluid in a bounded simple connected domain and focus on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence of solutions for this problem: the vorticity method and the stream function method. A long standing open problem is whether these two entirely different methods result in the same solution. In this paper, we will give a positive answer to this problem by studying the local uniqueness of the solutions. Another result obtained in this paper is that if the domain is convex, then the vor…
We consider the incompressible 2D Euler equation in an infinite cylinder $\mathbb{R}\times \mathbb{T}$ in the case when the initial vorticity is non-negative, bounded, and compactly supported. We study $d(t)$, the diameter of the support of vorticity, and prove that it allows the following bound: $d(t)\leq Ct^{1/3}\log^2 t$ when $t\rightarrow\infty$.
This article is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small data solutions.
In this paper, we are interested in the global persistence regularity for the 2D incompressible Euler equations in some function spaces allowing unbounded vorticities. More precisely, we prove the global propagation of the vorticity in some weighted Morrey-Campanato spaces and in this framework the velocity field is not necessarily Lipschitz but belongs to the log-Lipschitz class $L^αL,$ for some $α\in (0,1).$
In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is locally integrable with values in the same Lebesgue space.
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.
Global well-posedness for the incompressible Euler equations in an endpoint Sobolev space
Univ Paris Est Creteil, Univ Gustave Eiffel, CNRS, LAMA UMR8050（巴黎东克雷泰伊大学、古斯塔夫·埃菲尔大学、法国国家科学研究中心、LAMA UMR8050）；Korea Institute for Advanced Study（韩国高等科学研究院）
In this work we study the asymptotic behavior of viscous incompressible 2D flow in the exterior of a small material obstacle. We fix the initial vorticity $ω_0$ and the circulation $γ$ of the initial flow around the obstacle. We prove that, if $γ$ is sufficiently small, the limit flow satisfies the full-plane Navier-Stokes system, with initial vorticity $ω_0 + γδ$, where $δ$ is the standard Dirac measure. The result should be contrasted with the corresponding inviscid result obtained by the authors in [Comm P.D.E. 28 (2003) 349-379], where the effect of the small obstacle appears in the coeffi…
We consider the motion of several rigid bodies immersed in a two-dimensional incompress-ible perfect fluid, the whole system being bounded by an external impermeable fixed boundary. The fluid motion is described by the incompressible Euler equations and the motion of the rigid bodies is given by Newton's laws with forces due to the fluid pressure. We prove that, for smooth solutions, Newton's equations can be recast as a second-order ODE for the degrees of freedom of the rigid bodies with coefficients depending on the fluid vorticity and on the circulations around the bodies, but not anymore o…
In this paper we study the solvability of different boundary value problems for the two dimensional steady incompressible Euler equation. Two main methods are currently available to study those problems, namely the Grad-Shafranov method and the vorticity transport method. We describe for which boundary value problems these methods can be applied. The obtained solutions have non-vanishing vorticity.
We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated to vector fields with gradient given by a singular integral in order to define Lagrangian solutions, for which the vorticity is transported by the flow. We prove strong stability of these solutions via strong convergence of the flow, under only the assumption of L^1 weak convergence of the initial vorticity. The existence of Lagrangian solutions to the Euler …
In [Commun Math Phys 348(1), 129-143, 2016], Cheskidov et al. proved that physically realizable weak solutions of the incompressible 2D Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions are those that can be obtained as limits of vanishing viscosity. The key hypothesis was boundedness of the initial vorticity in $L^p$, $p>1$. In this work we extend their result, by adding forcing to the flow.
We consider the 2D incompressible Euler equation on a corner domain $Ω$ with angle $νπ$ with $\frac{1}{2}<ν<1$. We prove that if the initial vorticity $ω_0 \in L^{1}(Ω)\cap L^{\infty}(Ω)$ and if $ω_0$ is non-negative and supported on one side of the angle bisector of the domain, then the weak solutions are unique. This is the first result which proves uniqueness when the velocity is far from Lipschitz and the initial vorticity is nontrivial around the boundary.
We consider the motion of several rigid bodies immersed in a two-dimensional incompressible perfect fluid. The motion of the rigid bodies is given by the Newton laws with forces due to the fluid pressure and the fluid motion is described by the incompressible Euler equations. Our analysis covers the case where the circulations of the fluid velocity around the bodies are nonzero and where the fluid vorticity is bounded. The whole system occupies a bounded simply connected domain with an external fixed boundary which is impermeable except on an open non-empty part where one allows some fluid to …
We construct a family of steady solutions to the two-dimensional incompressible Euler equation in a general bounded domain, such that the vorticity is supported in two well-separated regions of small diameter and converges to a pair of point vortices with opposite signs. Compared with previous results, we do not need to assume the existence of an isolated local minimum point of the Kirchhoff-Routh function. Moreover, due to their variational nature, the solutions obtained are Lyapunov stable in $L^p$ norm of the vorticity. The proofs are achieved by maximizing the kinetic energy over an approp…
It is well-known that the dynamics of vortices in an ideal incompressible two-dimensional fluid contained in a bounded not necessarily simply connected smooth domain is described by the Kirchhoff--Routh point vortex system. In this paper, we revisit the classical problem of how well solutions to the Euler equations approximate these vortex dynamics and extend previous rigorous results to the case where the vorticity field is unbounded. More precisely, we establish estimates for the $2$-Wasserstein distance between the vorticity and the empirical measure associated with the point vortex dynamic…
In this paper we consider the incompressible Euler equation in a simply-connected bounded planar domain. We study the confinement of the vorticity around a stationary point vortex. We show that the power law confinement around the center of the unit disk obtained in [2] remains true in the case of a stationary point vortex in a simply-connected bounded domain. The domain and the stationary point vortex must satisfy a condition expressed in terms of the conformal mapping from the domain to the unit disk. Explicit examples are discussed at the end.
Toward P.-L. Lions' open question in \cite{Lions96} concerning the propagation of regularity for density patch, we establish the global existence of solutions to the 2-D inhomogeneous incompressible Navier-Stokes system with initial density given by $(1-η){\bf 1}_{\Om_0}+{\bf 1}_{\Om_0^c}$ for some small enough constant $η$ and some $W^{k+2,p}$ domain $\Om_0,$ and with initial vorticity belonging to $L^1\cap L^p$ and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain $\Om_0$ is preserved by time evolution.
