Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains

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…

Energy-balance for the incompressible Euler equations with stochastic forcing

We establish energy-balance for weak solutions of the stochastically forced incompressible Euler equations, enjoying Hölder regularity $C^α$, $α>1/3$. It is well known as the Onsager's conjecture for the deterministic incompressible Euler equations, which describes the energy conservation of weak solutions having Hölder regularity $C^α$, $α>1/3$. Additionally, we obtain energy-balance for the inhomogeneous incompressible Euler system driven by a cylindrical Wiener process.

具有紧凑支撑横截面涡量的三维欧拉方程中的簇状涡旋螺旋

Clustered vortex helices with compactly supported cross-sectional vorticity in the 3D Euler equations

本文基于三维不可压缩欧拉方程，通过粘合技术构造了首个光滑多涡旋解，展示了 collapsing 的螺旋纤维簇，横截面涡量始终在 $\mathbb{R}^2$ 中保持紧凑支撑。

Uniqueness and energy balance for isentropic Euler equation with stochastic forcing

In this article, we prove uniqueness and energy balance for isentropic Euler system driven by a cylindrical Wiener process. Pathwise uniqueness result is obtained for weak solutions having Hölder regularity $C^α,α>1/2$ in space and satisfying one-sided Lipschitz bound on velocity. We prove Onsager's conjecture for isentropic Euler system with stochastic forcing, that is, energy balance equation for solutions enjoying Hölder regularity $C^α,α>1/3$. Both the results have been obtained in a more general setting by considering regularity in Besov space.

Solution properties of a 3D stochastic Euler fluid equation

We prove local well-posedness in regular spaces and a Beale-Kato-Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton's 2nd Law in every Lagrangian domain.

On the well-posedness of stochastic Boussinesq equations with cylindrical multiplicative noise

The Boussinesq equations are fundamental in meteorology. Among other aspects, they aim to model the process of front formation. We use the approach presented in [Hol15] to introduce stochasticity into the incompressible Boussinesq equations. This is, we introduce cylindrical transport noise in a way that the geometric properties in the Euler-Poincaré formulation are preserved. One of our main results establishes the local well-posedness of regular solutions for these new stochastic Boussinesq equations. We also construct a blow-up criterion and derive some general estimates, which are crucial …

A priori estimates and analytical construction of radially symmetric solutions in the gas dynamics

In this article we derive C^1-a priori estimates on the Riemann invariants of the Euler compressible equations in the case of cylindrical or spherical symmetry. These estimates allow then to construct shock waves with a time of existence proportional to the distance to the origin at the initial time.

Reduction of Stratified Axi-Symmetric Euler-Poisson Equations Under Symmetry

The paper considers Euler-Poisson equations which govern the steady state of a self gravitating, rotating, axi-symmetric fluid under the additional assumption that it is incompressible and stratified. In this setting we show that the original system of six nonlinear partial differential equations can be reduced to two equations, one for the mass density and the other for gravitational field. This reduction is carried out in cylindrical coordinates. As a result we are able to derive also expressions for the pressure as a function of the density. The resulting equations are then solved analytica…

三维扰动下柱形跨声速激波解的存在性与稳定性

Existence and stability of cylindrical transonic shock solutions under three dimensional perturbations

本文研究三维入流与出口压力扰动下柱形跨声速激波的存在稳定性，利用 deformation-curl 分解和 Rankine-Hugoniot 条件重构，唯一确定自由激波锋面并获得最优正则性。

不可压缩 Euler 方程的置换对称解

Permutation symmetric solutions of the incompressible Euler equation

本文研究不可压缩 Euler 方程的置换对称解，将动力学约化为单涡量分量方程，并利用 Elgindi 爆破解构造具有多重离散对称性的有限时间爆破 $C^{1,\alpha}$ 解。

Global solvability of the rotating Navier-Stokes equations with fractional Laplacian in a periodic domain

We consider existence of global solutions to equations for three-dimensional rotating fluids in a periodic frame provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. In the spatially decaying case, Koh, Lee and Takada (2014) showed existence for the large times of solutions of the rotating Euler equations provided by the large Coriolis force. In this case the resonant equation does not appear anymore. In the periodic case, however, the resonant equation appears, and thus the mai…

关于不存在连续欧几里得对称性的三维完美流体平衡纤维化的存在性

On the existence of fibered three-dimensional perfect fluid equilibria without continuous Euclidean symmetry

本文构造了无连续欧几里得对称性的三维理想不可压缩欧拉方程稳态解族，具有离散m重对称性，由非退化伯努利压力水平集纤维化，缩小了Grad猜想的适用范围。

带有奇异局部Hölder连续漂移项的随机阻尼波动方程与Euler-Bernoulli方程

Stochastic damped wave and Euler-Bernoulli equations with singular locally Hölder continuous drift

该研究针对含奇异局部Hölder连续漂移项的随机微分方程，证明其满足弱唯一性与路径wise唯一性，覆盖1维随机阻尼波动方程及3维内的随机阻尼Euler-Bernoulli梁方程。

具有跳跃扩散噪声的Allen-Cahn方程用于生物损伤与修复过程

An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes

本文提出了一种包含跳跃扩散噪声的随机Allen-Cahn方程，用于建模生物损伤与修复过程，并分析了其数学性质与数值模拟方法。