A simple one-dimensional model for the three-dimensional vorticity equation

A simple qualitative one-dimensional model for the 3-D vorticity equation of incompressible fluid flow is developed. This simple model is solved exactly; despite its simplicity, this equation retains several of the most important structural features in the vorticity equations and its solutions exhibit some of the phenomena observed in numerical computations for breakdown for the 3-D Euler equations.     

Remarks on the Regularity to 3-D Ideal Magnetohydrodynamic Equations

In this paper we are interested in the sufficient conditions which guarantee the regularityof solutions of 3-D ideal magnetohydrodynamic equations in the arbitrary time interval [0,T].Fivesufficient conditions are given.Our results are motivated by two main ideas:one is to control theaccumulation of vorticity alone;the other is to generalize the corresponding geometric conditions of3-D Euler equations to 3-D ideal magnetohydrodynamic equations.     

有关二维Euler方程的一些估计

首先得到Ｌｏｒｅｎｔｚ空间中的一些结果,然后在此基础上得到了有关二维Ｅｕｌｅｒ方程解的一些估计。这些估计与该方程当初始旋度ω０∈Ｌ＾－１∩Ｌ＾ｐ（ｐ〉１）时解的唯一性有关。     

EXISTENCE OF WEAK SOLUTIONS OF 2－D EULER EQUATIONS WITH INITIAL VORTICITY ω_0∈E（log~＋L）~α（α＞0）

ＥＸＩＳＴＥＮＣＥＯＦＷＥＡＫＳＯＬＵＴＩＯＮＳＯＦ２－ＤＥＵＬＥＲＥＱＵＡＴＩＯＮＳＷＩＴＨＩＮＩＴＩＡＬＶＯＲＴＩＣＩＴＹω＿０∈Ｅ（ｌｏｇ￣＋Ｌ）￣α（α＞０）ＪＩＵＱＵＡＮＳＥＮ（ＩｎｓｔｉｔｕｔｅＯｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ｔ...     

On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain

We show that a smooth solution of the 3-D Euler equations in a bounded domain breaks down, if and only if a certain norm of vorticity blows up at the same time. Here this norm is weaker than bmo-norm.     

The weak vorticity formulation of the 2-d euler equations and concentration-cancellation

The weak limit of a sequence of approximate solutions of the 2-D Euler equations will be a solution if the approximate vorticities concentrate only along a curve x(t) that is Holder continuous with exponent ½.

A new proof is given of the theorem of DiPerna and Majda that weak limits of steady approximate solutions are solutions provided that the singularities of the inhomogeneous forcing term are sufficiently mild. An example shows that the weaker condition imposed here on the forcing term is sharp.

A simplified formula for the kernel in Delort's weak vorticity formulation of the two-dimensional Euler equations makes the properties of that kernel readily apparent, thereby simplying Delort's proof of the existence of one-signed vortex sheets.     

CONVERGENCE OF VORTEX METHODS FOR 3-D EULER EQUATIONS

1. IntroductionThe convergence problem of vortex methods for the Euler equations has been studied by many authors. Hald and Delprete proved the convergence for two--dimensionalinitial value problems [3]. Three-dimensional initial value problems were studied byBeale and Majda [2] and Beale [1]. Ying [4] and Ying and Zhang [sl, [61 provedthe convergence of vortex methods for two--dimensional initial-boundary value problems of the Euler equations. Ying [7] proved the convergence of vortex met…     

Convergence of the point vortex method for the 3-D euler equations

We prove consistency, stability, and convergence of a point vortex approximation to the 3-D incompressible Euler equations with smooth solutions. The 3-D algorithm we consider here is similar to the corresponding 3-D vortex blob algorithm introduced by Beale and Majda; see [3]. We first show that the discretization error is second-order accurate. Then we show that the method is stable in l^p norm for the particle trajectories and in w^?1.p norm for discrete vorticity. Consequently, the method converges up to any time for which the Euler equations have a smooth solution. One immediate application of our convergence result is that the vortex filament method without smoothing also converges.     

Solutions to the 2D Euler equations with velocity unbounded at infinity

We prove existence of solutions to the two-dimensional Euler equations with vorticity bounded and with velocity locally bounded but growing at infinity at a rate slower than a power of the logarithmic function. We place no integrability conditions on the initial vorticity. This result improves upon a result of Serfati which gives existence of a solution to the two-dimensional Euler equations with bounded velocity and vorticity.     

The point-vortex method for periodic weak solutions of the 2-D Euler equations

Almost-sure convergence of a subsequence of the vorticity to a weak solution is proven for the point-vortex method for 2-D, inviscid, incompressible fluid flow. Here “almost-sure” is with respect to sequences of random components included in the initial position and strength of each vortex. The initial vorticity is assumed to be periodic and, depending on the initialization scheme, to lie in L log L or L^p with p > 2. The randomization of the initial data is not needed when the initial vorticity is nonnegative; such initial data also need not be periodic, and is only required to be a bounded measure lying in H⁻¹. All these results are also valid for the “vortex-blob” method with the smoothing parameter vanishing at an arbitrary rate. The sense in which solutions of point-vortex dynamics are weak solutions of the Euler equations is also discussed.     

Self-similar point vortices and confinement of vorticity

This papers deals with the large time behavior of solutions of the incompressible Euler equations in dimension 2. We consider a self-similar configuration of point vortices which grows like the square root of the time. We study the confinement properties of a blob of vorticity initially located around the first point vortex and moving in the velocity field produced by itself and by the other point vortices. We find a su?cient condition on the point vortices such that the vorticity stays confined around the first point vortex at a rate better than the square root of the time. The relevance to the large time behavior of the Euler equations is discussed.     

On the continuation principles for the Euler equations and the quasi-geostrophic equation

We obtain new continuation principle of the local classical solutions of the 3D Euler equations, where the regularity condition of the direction field of the vorticiy and the integrability condition of the magnitude of the vorticity are incorporated simultaneously. The regularity of the vorticity direction field is most appropriately measured by the Triebel-Lizorkin type of norm. Similar result is also obtained for the inviscid 2D quasi-geostrophic equation.     

Shock Formation for 2D Isentropic Euler Equations with Self-similar Variables

The author studies the 2D isentropic Euler equations with the ideal gas law. He exhibits a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry. These solutions to the 2D isentropic Euler equations are associated with non-zero vorticity at the shock and have uniform-in-time 1 3-H¨older bound. Moreover, these point shocks are of self-similar type and share the same profile, which is a solution to the 2D self-similar Burgers equation. The proof of the solutions, following the 3D construction of Buckmaster, Shkoller and Vicol (in 2023), is based on the stable 2D self-similar Burgers profile and the modulation method.     

A nonlinear Hamiltonian structure for the Euler equations

The Euler equations for inviscid incompressible fluid flow have a Hamiltonian structure in Eulerian coordinates, the Hamiltonian operator, though, depending on the vorticity. Conservation laws arise from two sources. One parameter symmetry groups, which are completely classified, yield the invariance of energy and linear and angular momenta. Degeneracies of the Hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point-wise conservation of vorticity. It is conjectured that no further conservation laws exist, indicating that the Euler equations are not completely integrable, in particular, do not have soliton-like solutions.     

Construction of solutions to the two-dimensional stationary Euler equations by the pseudo-advection method

The finite difference approximation of a nonstationary pseudo-advected vorticity equation is proved to yield generalized solutions to the two-dimensional stationary Euler equations with nonvanishing vorticity. This result is obtained by the simultaneous limiting of lattice scale and time.Received: 15 May 2002     

Existence of global weak solutions to one-component vlasov-poisson and fokker-planck-poisson systems in one space dimension with measures as initial data

We consider Cauchy problems for the 1-D one component Vlasov-Poisson and Fokker-Planck-Poisson equations with the initial electron density being in the natural space of arbitrary non-negative finite measures. In particular, the initial density can be a Dirac measure concentrated on a curve, which we refer to as “electron sheet” initial data. These problems resemble both structurally and functional analytically Cauchy problems for the 2-D Euler and Navier-Stokes equations (in vorticity formulation) with vortex sheet initial data. Here, we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved. We give a natural definition of the product, establish its weak stability, and existence of weak solutions follows. Our concept of weak solutions through the newly defined product is justified since solutions to the Fokker-Planck-Poisson equation, the analogue of Navier-Stokes equation, are shown to converge to weak solutions of the Vlasov-Poisson equation as the Fokker-Planck term vanishes. The main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations. This is needed because the problem studied here is beyond the range of applicability of the “velocity averaging” compactness methods of DiPerna-Lions. © 1994 John Wiley & Sons, Inc.     

THE VANISHING PRESSURE LIMIT OF SOLUTIONS TO THE SIMPLIFIED EULER EQUATIONS FOR ISENTROPIC FLUIDS

In this paper,the Riemann problem of the 1-D reduced model for the 2-D Euler equations is considered and the Riemann solutions are obtained.It is proved that,as the pressure vanishes,they converge to two kinds of Riemann solutions to the 1D reduced model for the 2-D transport equations:one contains δ-shocks,the other contains vacuum.     

On Some Model Equations of Euler and Navier-Stokes Equations

The author proposes a two-dimensional generalization of Constantin-Lax-Majda model. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line (vorticity formulation), the author presents some further model equations. He possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. Some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations are made.     

光滑区域上二维无黏性无热传导Boussinesq方程组与三维轴对称不可压Euler方程组的指数增长全局光滑解

研究二维无黏性无热传导Boussinesq方程组和三维轴对称不可压Euler方程组光滑解的增长情况,找各种区域使其上的方程组有快增长的解。对Boussinesq方程组,通过选取初始温度和速度的一个分量,可以把方程去耦为两部分。从关于涡量的部分求出涡量、速度场和使结论成立的区域,从关于温度的部分,可见温度的高阶导的增长仅依赖于速度场的一个分量。通过适当选取该分量,得到温度高阶导有指数增长的全局光滑解。对轴对称Euler方程组做类似的处理,适当选取速度场的径向分量,可把方程组去耦,最终得到一类光滑区域,在其上方程组有指数增长全局光滑解。该研究把Chae、Constantin、Wu对一个二维锥形区域上无黏性无热传导Boussinesq方程的结果,推广到一类光滑区域上, 并把他们的方法应用到三维轴对称不可压Euler方程组, 得到了类似的结果。     

Algebraic spiral solutions of 2d incompressible Euler

We consider self-similar solutions of the 2d incompressible Euler equations. We construct a class of solutions with vorticity forming algebraic spirals near the origin, in analogy to vortex sheets rolling up into algebraic spirals.