Finite Time Blow-up for the 3D Incompressible Euler Equations

关于不可压缩Navier-Stokes方程的光滑性与有限时间爆破解的注记

Remarks on smoothness and finite-time blowup solutions of the incompressible Navier-Stokes equation

本文从新视角研究了不可压缩Navier-Stokes方程解的光滑性，证明了仅在有限时间内解具有更高光滑性，并针对每种情形证明了解存在爆破时间。

不可压缩欧拉方程轴对称爆破解的不稳定性

Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations

本文证明不可压缩欧拉方程若存在有限时间爆破，其解在轴对称情形下同样会在爆破时刻附近线性不稳定，并建立了仅依赖涡量环向分量的爆破准则。

Blowup of solutions to the thermal boundary layer problem in two-dimensional incompressible heat conducting flow

In this paper, we study the formation of finite time singularities for the solution of the boundary layer equations in the two-dimensional incompressible heat conducting flow. We obtain that the first spacial derivative of the solution blows up in a finite time for the thermal boundary layer problem, for a kind of data which are analytic in the tangential variable but do not satisfy the Oleinik monotonicity condition, by constructing a Lyapunov functional. Moreover, it is observed that the buoyancy coming from the temperature difference in the flow may destabilize the thermal boundary layer.

Well-posedness, Smoothness and Blow-up for Incompressible Navier-Stokes Equations

For any divergence free initial datum $u_0$ with $\|u_0\|_\infty+\|\nabla u_0\|_{L^p}+\|\nabla^2 u_0\|_{L^p}<\infty$ for some $p>d\ (d\ge 2)$, the well-posedness and smoothness are proved for incompressible Navier-Stokes equations on $\mathbb{R}^d$ or $\mathbb{T}^d:=\mathbb{R}^d/\mathbb{Z}^d,$ up to a time explicitly given by the initial datum and three constants coming from the upper bounds of the heat kernel and the Riesz transform. A mild well-posedness is also proved for $L^p$-bounded initial data. The blow-up is proved for both type solutions with finite maximal time.

The Compressible to Incompressible Limit of 1D Euler Equations: the Non Smooth Case

We prove a rigorous convergence result for the compressible to incompressible limit of weak entropy solutions to the isothermal 1D Euler equations.

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.

On 2d incompressible Euler equations with partial damping

We consider various questions about the 2d incompressible Navier-Stokes and Euler equations on a torus when dissipation is removed from or added to some of the Fourier modes.

Uniformly rotating smooth solutions for the incompressible 2D Euler equations

In this paper, we show the existence of a family of compactly supported smooth vorticities, which are solutions of the 2D incompressible Euler equation and rotate uniformly in time and space.

Nonexistence of Local Self-Similar Blow-up for the 3D Incompressible Navier-Stokes Equations

We prove the nonexistence of local self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The local self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region which shrinks to a point dynamically as the time, $t$, approaches the singularity time, $T$. The solution outside the inner core region is assumed to be regular. Under the assumption that the local self-similar velocity profile converges to a limiting profile as $t \to T$ in $L^p$ for some $p \in (3,\infty)$,…

General remarks on the propagation of chaos in wave turbulence and application to the incompressible Euler dynamics

In this paper, we prove propagation of chaos in the context of wave turbulence for a generic quasisolution. We then apply the result to full solutions to the incompressible Euler equation.

Localized non blow-up criterion of the Beale-Kato-Majda type for the 3D Euler equations

We prove a localized non blow-up theorem of the Beale-Kato-Majda type for the solution of the 3D incompressible Euler equations.

Euler Equations of Incompressible Ideal Fluids

This article is a survey concerning the state-of-the-art mathematical theory of the Euler equations of incompressible homogenous ideal fluid. Emphasis is put on the different types of emerging instability, and how they may be related to the description of turbulence.

Singular vorticity solutions of the incompressible Euler equation via inviscid limits

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.

A sufficient condition for uniqueness of weak solutions of the incompressible Euler system

We give a new sufficient criteria to prove the uniqueness of the incompressible Euler equation in dimension $N\geq2$. In their celebrated works by V. Scheffer [18], A. Shnirelman [19], C. De Lellis and L. Székelyhidi Jr. [7] they have obtained the nonuniqeness of weak solutions of incompressible Euler equation. Here we obtain uniqueness criteria for the same equation under some mild regularity condition on weak solutions. Our proof is simple and can be employed to other equations like inhomogeneous incompressible Euler and Euler-Boussinesq equations. One of the key ingredients in our proof is …

二维非齐次不可压缩欧拉方程的几何爆破准则

Geometric blow-up criteria for the non-homogeneous incompressible Euler equations in 2-D

针对二维变密度不可压缩欧拉方程整体适定性未知的问题，提出仅需控制∇u沿X=∇⊥ρ方向分量的几何爆破准则，覆盖亚临界与临界正则性框架，且能恢复常密度情况的整体适定性。

Energy concentrations and Type I blow-up for the 3D Euler equations

We exclude Type I blow-up, which occurs in the form of atomic concentrations of the $L^2$ norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale.

给定边界条件下不可压静态 Euler 方程正则解的非唯一性及不可压 Navier-Stokes 方程的湍流全局解

Non-uniqueness of Regular Solutions for Incompressible Static Euler Equations with Given Boundary Conditions and Turbulent Global Solutions of Incompressible Navier-Stokes Equations

研究不可压静态 Euler 方程在给定边界条件下的无穷多正则解与随机解，并证明其作为高 Reynolds 数 Navier-Stokes 流长时路径极限可刻画湍流。

A Blow-Up Result for Dyadic Models of the Euler Equations

We partially answer a question raised by Kiselev and Zlatos in \cite{MR2180809}; in the generalized dyadic model of the Euler equation, a blow-up of $H^{1/3+δ}$-norm occurs. We recover a few previous blow-up results for various related dyadic models as corollaries.

Blowup of Solutions of the Hydrostatic Euler Equations

In this paper we prove that for a certain class of initial data, smooth solutions of the hydrostatic Euler equations blow up in finite time.

二维不可压缩欧拉方程在临界Besov空间中的强不适定性

Strong Ill-posedness of the 2d Incompressible Euler Equation in Critical Besov Spaces

本文证明二维不可压缩欧拉方程在临界Besov空间$B^{1}_{\infty, q}$（$1<q<\infty$）中速度场的强不适定性。

Stabilization of the 2D incompressible Euler system in an infinite strip

The paper is devoted to the study of a stabilization problem for the 2D incompressible Euler system in an infinite strip with boundary controls. We show that for any stationary solution (c, 0) of the Euler system there is a control which is supported in a given bounded part of the boundary of the strip and stabilizes the system to (c, 0).

On blow-up shock waves for a nonlinear PDE associated with Euler equations

A second-order PDE is derived from Euler's equaitons under certain assumptions. It is shown that this PDE admits shock and rarefaction waves, and that a single point gradient blow-up admits a unique similarity extension after blow-up that settles uniqueness/entropy issues for such equations.

Traveling vortex pairs for 2D incompressible Euler equations

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.

Quasi-periodic incompressible Euler flows in 3D

We prove the existence of time-quasi-periodic solutions of the incompressible Euler equation on the three-dimensional torus $\T^3$, with a small time-quasi-periodic external force. The solutions are perturbations of constant (Diophantine) vector fields, and they are constructed by means of normal forms and KAM techniques for reversible quasilinear PDEs.

On the Finite-Time Blowup of a 1D Model for the 3D Incompressible Euler Equations

We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a cylindrical domain. We prove the local well-posedness of the model in spaces of zero-mean functions, and study the potential formation of a finite-time singularity under certain convexity conditions for the velocity field. It is hoped that the results obtained on the 1D model will be useful in the analysis of the full 3D problem, whose loss of regularity in finite time has been observed in a recent numerical s…

Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II

In Part II of the paper, we prove linear instability of a certain class of radially symmetric flows of an ideal incompressible fluid in dimension two used in Part I

Remarks on a paper by Gavrilov: Grad-Shafranov equations, steady solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications

We describe a method to construct smooth and compactly supported solutions of 3D incompressible Euler equations and related models. The method is based on localizable Grad-Shafranov equations and is inspired by the recent result \cite{gav}.

On the well-posedness of the incompressible Euler Equation

In this thesis we prove that the homogeneous incompressible Euler equation of hydrodynamics on the Sobolev spaces $H^s(\R^n)$, $n \geq 2$ and $s > n/2+1$, can be expressed as a geodesic equation on an infinite dimensional manifold. As an application of this geometric formulation we prove that the solution map of the incompressible Euler equation, associating intial data in $H^s(\R^n)$ to the corresponding solution at time $t > 0$, is nowhere locally uniformly continuous and nowhere differentiable.

不可压缩 $3$ 维 Euler 方程中的涡旋螺旋簇

Cluster of vortex helices in the incompressible $3$d Euler equations

该研究针对三维无粘不可压缩 Euler 方程，证明了多个螺旋丝状涡旋结构可相互坍缩，形成涡旋螺旋簇。

三维无旋不可压缩欧拉方程的渐近自相似爆破：I. $C^{\infty}$ 一维极限型

Asymptotically Self-Similar Blowup for 3D Incompressible Euler with $C^{1, 1/3-}$ Velocity I: $C^{\infty}$ 1D Limiting Profiles

本文研究了三维轴对称不可压缩欧拉方程的一维模型，通过构造自相似爆破型，探讨了不同α值下的爆破行为及极限特性。

A Lagrangian Interior Regularity Result for the Incompressible Free Boundary Euler Equation with Surface Tension

We consider the three-dimensional incompressible free-boundary Euler equations in a bounded domain and with surface tension. Using Lagrangian coordinates, we establish a priori estimates for solutions with minimal regularity assumptions on the initial data.

Finite time blow up for a model of incompressible Navier-Stokes Equations

T. Tao constructed an averaged Navier-Stokes equations which obey an energy identity. Nevertheless, he proved that smooth solutions can blow up in finite time. This demonstrates that any proposed positive solution to the famous regularity problem for the three dimensional incompressible Navier-Stokes equations which does not use the finer structure of the nonlinearity cannot possibly be successful. In this paper, we propose a very simple model of the Navier Stokes equation which also obey the energy identity and establish a similar result of finite-time blow up . The drawback of our result is …

Necessary conditions for blow-up solutions to the restricted Euler--Poisson equations

In this work, we study the behavior of blow-up solutions to the multidimensional restricted Euler--Poisson equations which are the localized version of the full Euler--Poisson system. We provide necessary conditions for the existence of finite-time blow-up solutions in terms of the initial data, and describe the asymptotic behavior of the solutions near blow up times. We also identify a rich set of the initial data which yields global bounded solutions.

A fully non-linear version of the Euler incompressible equations: the semi-geostrophic system

This work gathers new results concerning the semi-geostrophic equations: existence and stability of measure valued solutions, existence and uniqueness of solutions under certain continuity conditions for the density, convergence to the incompressible Euler equations. Meanwhile, a general technique to prove uniqueness of sufficiently smooth solutions to non-linearly coupled system is introduced, using optimal transportation.

Blow-up of Smooth Solutions to the Euler-Poisson Equations

In this paper, the finite time blow-up of smooth solutions to the Cauchy problem for full Euler-Poisson equations and isentropic Euler-Poisson equations with repulsive forces or attractive forces in high dimensions $(n\geq3)$ is proved for a large class of initial data. It is not required that the initial data has compact support or contains vacuum in any finite regions.

具有均匀 $C^{1,\sqrt{\frac{4}{3}}-1-\varepsilon}$ 力的二维不可压缩非齐次欧拉方程的涡度爆破

Vorticity blow-up for the 2D incompressible non-homogeneous Euler equations with uniform $C^{1,\sqrt{\frac{4}{3}}-1-\varepsilon}$ force

通过将Boussinesq爆破方法适应到非齐次欧拉方程，证明了在有限时间内产生奇异性的解的存在性，并揭示了强迫系统中两种不同类型奇异性的潜在存在。

Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I

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.

The incompressible Euler limit of the Boltzmann equation for a gas of fermions

We are interested in the hydrodynamic limit of the Boltzmann equation for a gas of fermions in the incompressible Euler regime. We use the relative entropy method as improved by Saint-Raymond. Our result is analogous to what is obtained in the classical case, but the treatment is slightly complicated by the cubic nonlinearity of the collision operator.

不可压缩多孔介质方程的一维模型中的有限时间爆破

Finite time blow-up in a 1D model of the incompressible porous media equation

本文推导了一个一维不可压缩多孔介质（IPM）模型，证明其局部适定性，并发现一类初始数据存在有限时间爆破。

The finite time blow-up for the Euler-Poisson equations in $\Bbb R^n$

We prove the finite time blow-up for $C^1$ solutions to the Euler-Poisson equations in $\Bbb R^n$, $n\geq 1$, with/without background density for initial data satisfying suitable conditions. We also find a sufficient condition for the initial data such that $C^3$ solution breaks down in finite time for the compressible Euler equations for polytropic gas flows.

On the regularity of the solution map of the incompressible Euler equation

In this paper we consider the incompressible Euler equation on the Sobolev space $H^s(\R^n)$, $s > n/2+1$, and show that for any $T > 0$ its solution map $u_0 \mapsto u(T)$, mapping the initial value to the value at time $T$, is nowhere locally uniformly continuous and nowhere differentiable.

Velocity blow-up in $C^1\cap H^2$ for the 2D Euler equations

We give a vorticity-dynamical proof of $C^1\cap H^2$-illposedness of the 2D Euler equations. Our construction shows that the unique Yudovich solution escapes both $C^1$ and $H^2$ instantaneously.

Blow up for the 2D Euler Equation on Some Bounded Domains

We find a smooth solution of the 2D Euler equation on a bounded domain which exists and is unique in a natural class locally in time, but blows up in finite time in the sense of its vorticity losing continuity. The domain's boundary is smooth except at two points, which are interior cusps.

Incompressible Euler Equations: the blow-up problem and related results

The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent approaches to the problem. We first review Kato's classical local well-posedness result in the Sobolev space and derive the celebrated Beale-Kato-Majda criterion for finite time blow-up. Then, we discuss recent refinements of the criterion as well as geometric type of theorems on the sufficiency condition for the regularity of solutions. After that we review resu…

A simple proof of ill-posedness for incompressible Euler equations in critical Sobolev spaces

We provide a simple proof that the Cauchy problem for the incompressible Euler equations in $\mathbb{R}^{d}$ with any $d\ge3$ is ill-posed in critical Sobolev spaces, extending an earlier work of Bourgain and Li in the case $d = 3$. The ill-posedness is shown for certain critical Lorentz spaces as well.

Nonlinear inviscid damping for 2-D inhomogeneous incompressible Euler equations

We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressible Euler equations are globally well-posed and the velocity converges strongly to a shear flow close to the Couette flow, and the vorticity will be driven to small scales by a linear evolution and weakly converges as $t\to \infty$. To our knowledge, t…

Fast rotation limit for the 2-D non-homogeneous incompressible Euler equations

In the present paper, we study the fast rotation limit for the density-dependent incompressible Euler equations in two space dimensions with the presence of the Coriolis force. In the case when the initial densities are small perturbation of a constant profile, we show the convergence of solutions towards the solutions of a quasi-homogeneous incompressible Euler system. The proof relies on a combination of uniform estimates in high regularity norms with a compensated compactness argument for passing to the limit. This technique allows us to treat the case of ill-prepared initial data.

A sharp local blow-up condition for Euler-Poisson equations with attractive forcing

We improve the recent result of Chae & Tadmor in [Comm. Math. Sci. 6(3) (2008) 785-789], proving a one-sided threshold condition which leads to finite-time breakdown of the Euler-Poisson equations in arbitrary dimension n.

Co-rotating and traveling vortex sheets for the 2D incompressible Euler equation

We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff-Rott operator, and accomplished by using implicit function theorem at point vortex solutions with suitably chosen function spaces.

Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations

We consider solutions to the Cauchy problem for the incompressible Euler equations on the 3-dimensional torus which are continuous or Hölder continuous for any exponent $θ<\frac{1}{16}$. Using the techniques introduced in \cite{DS12} and \cite{DS12H}, we prove the existence of infinitely many (Hölder) continuous initial vector fields starting from which there exist infinitely many (Hölder) continuous solutions with preassigned total kinetic energy.

Existence of Weak Solutions for the Incompressible Euler Equations

Using a recent result of C. De Lellis and L. Székelyhidi Jr. we show that, in the case of periodic boundary conditions and for dimension greater or equal 2, there exist infinitely many global weak solutions to the incompressible Euler equations with initial data $v_0$, where $v_0$ may be any solenoidal $L^2$-vectorfield. In addition, the energy of these solutions is bounded in time.

Small scale creation for solutions of the incompressible two dimensional Euler equation

We construct an initial data for two-dimensional Euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times. This estimate is known to be sharp - the double exponential growth is the fastest possible growth rate.

Withdraw a paper entitled "On the growth rate of solutions for 2D incompressible Euler equations"

This paper has been withdrawn by the author due to a crucial definition error of Triebel space.

关于随机不可压缩密度依赖欧拉方程的局部适定性

On local well-posedness of the stochastic incompressible density-dependent Euler equations

本文研究三维空间中随机非齐次不可压缩欧拉方程，通过将问题转化为随机问题并利用输运方程估计，证明了加性和乘性噪声下局部解的存在性与路径唯一性。