A Uniqueness Theorem for Incompressible Fluid Flows with Straight Streamlines

Inverse solutions for a second-grade fluid for porous medium channel and Hall current effects

Assuming certain forms of the stream function inverse solutions of an incompressible viscoelastic fluid for a porous medium channel in the presence of Hall currents are obtained. Expressions for streamlines, velocity components and pressure fields are described in each case and are compared with the known viscous and second-grade cases.

Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?

An important open problem in the theory of the Navier-Stokes equations is the uniqueness of the Leray-Hopf weak solutions with $L^2$ initial data. In this paper we give sufficient conditions for non-uniqueness in terms of spectral properties of a natural linear operator associated to scale-invariant solutions recently constructed in \cite{JiaSverak}. If the spectral conditions are satisfied, non-uniqueness and ill-posedness can appear for quite benign compactly supported data, just at the borderline of applicability of the classical perturbation theory. The verification of the spectral conditi…

通过消粘性极限原理选择二维稳态欧拉流

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。

关于周期性涡斑的综述

Survey on periodic vortex patches

该综述聚焦二维欧拉方程和涡斑解，重新审视不可压缩流体动力学中周期解存在性的经典结果，通过结合原著思想与最新进展，对经典构造进行统一简化阐述。

二维不可压缩Euler方程的刚性研究

On the rigidity of the 2D incompressible Euler equations

该研究针对二维有界区域定常Euler流的刚性问题，证明了圆盘、环域中满足切向边界条件的对应驻点条件下流必为环流，还揭示无滑移边界下的绝对刚性，验证了相关猜想并解决公开问题。

Shear flows of an ideal fluid and elliptic equations in unbounded domains

We prove that, in a two-dimensional strip, a steady flow of an ideal incompressible fluid with no stationary point and tangential boundary conditions is a shear flow. The same conclusion holds for a bounded steady flow in a half-plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on one-dimensional symmetry results for solutions of some semilinear elliptic equations. Some related rigidity results of independent interest are also shown in n-dimensional slabs in any dimension n.

On the Serrin problem for ring-shaped domains

In this paper, we deal with the long standing open problem of characterising rotationally symmetric solutions to $Δu = -2$, when Dirichlet boundary conditions are imposed on a ring-shaped planar domain. From a physical perspective, the solution represents the velocity of a homogeneous incompressible fluid, flowing in steady parallel streamlines through a hollow cylindrical pipe and obeying a no-slip condition. In contrast with Serrin's classical result, we show that the simplest possible set of overdetermining conditions, namely the prescription of locally constant Neumann boundary data, is no…

A Liouville theorem for the Euler equations in the plane

This paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two-dimensional plane R^2. We show that any such flow is a shear flow, that is, it is parallel to some constant vector. The proof of this Liouville-type result is firstly based on the study of the geometric properties of the level curves of the stream function and secondly on the derivation of some estimates on the at most logarithmic growth of the argument of the flow. These estimates lead to the conclusion that the streamlines of the flow are all parall…

Low Mach and thin domain limit for the compressible Euler system

We consider the compressible Euler system describing the motion of an ideal fluid confined to a straight layer $Ω_δ=(0,δ)\times\mathbb{R}^2, \ \ δ>0$. In the framework of dissipative measure-valued solutions, we show the convergence to the strong solution of the 2D incompressible Euler system when the Mach number tends to zero and $δ\rightarrow0$.

Weak singularities of 3-D Euler equations and restricted regularity of Navier Stokes equation solutions with time dependent force terms

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…

通道中具有分段常涡量的稳态欧拉方程的孤立解

Solitary solutions to the steady Euler equations with piecewise constant vorticity in a channel

本文在无限通道中构造了具有分段常涡量的两层稳态欧拉方程的局部化孤立解，严格给出精确解曲线及渐近首阶项，并定性描述了含驻点、临界层及边界相交的流线粒子路径。

Time-analyticity of Lagrangian particle trajectories in ideal fluid flow

It is known that the Eulerian and Lagrangian structures of fluid flow can be drastically different; for example, ideal fluid flow can have a trivial (static) Eulerian structure, while displaying chaotic streamlines. Here we show that ideal flow with limited spatial smoothness (an initial vorticity that is just a little better than continuous), nevertheless has time-analytic Lagrangian trajectories before the initial limited smoothness is lost. For proving such results we use a little-known Lagrangian formulation of ideal fluid flow derived by Cauchy in 1815 in a manuscript submitted for a priz…

扰动圆柱中带外力的三维轴对称欧拉流的跨音速激波

Transonic shocks for three-dimensional axisymmetric Euler flows with an external force in a perturbed cylinder

研究在入流超音速流、出口压力、外力和喷嘴壁扰动下，三维轴对称欧拉系统在圆柱中跨音速激波解的存在性与稳定性，通过修正拉格朗日变换和双曲-椭圆模态分解克服奇异性，并利用代数方程唯一确定激波位置。

Existence and stability of viscous vortices

Vorticity plays a prominent role in the dynamics of incompressible viscous flows. In two-dimensional freely decaying turbulence, after a short transient period, evolution is essentially driven by interactions of viscous vortices, the archetype of which is the self-similar Lamb-Oseen vortex. In three dimensions, amplification of vorticity due to stretching can counterbalance viscous dissipation and produce stable tubular vortices. This phenomenon is illustrated in a famous model originally proposed by Burgers, where a straight vortex tube is produced by a linear uniaxial strain field. In real f…

二维欧拉方程稳态解的一个分类

On a classification of steady solutions to two-dimensional Euler equations

本文基于流角集合对二维欧拉方程稳态解进行分类，发现非剪切流需覆盖整个圆或半圆，并证明最小曲率非剪切流的存在性，进而推导出椭圆方程刘维尔定理与剪切流结构稳定性。