Related papers: Singularity formation of vortex sheets in 2D Euler…
A central but controversial issue in free turbulent shear flows has been the universality (or otherwise) of their growth rates. We resolve this issue here in the special case of a temporal 2D mixing layer in a point vortex gas by extensive…
We construct finite time blow-up solutions to the 2-dimensional harmonic map flow into the sphere $S^2$, \begin{align*} u_t & = \Delta u + |\nabla u|^2 u \quad \text{in } \Omega\times(0,T) \\ u &= \varphi \quad \text{on } \partial…
We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal…
An uncertainty quantification framework is developed for Eulerian-Lagrangian models of particle-laden flows, where the fluid is modeled through a system of partial differential equations in the Eulerian frame and inertial particles are…
Steady fluid flows have very special topology. In this paper we describe necessary and sufficient conditions on the vorticity function of a 2D ideal flow on a surface with or without boundary, for which there exists a steady flow among…
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…
Vortex flows are ubiquitous in both natural processes and engineering applications, including phenomena such as typhoons, water currents, and aerospace fluid dynamics. The vortex particle method, a computational approach grounded in vortex…
Testing the order of accuracy of (very) high order methods for shallow water (and Euler) equations is a delicate operation and the test cases are the crucial starting point of this operation. We provide a short derivation of vortex-like…
We consider two-level finite element discretization methods for the stream function formulation of the Navier-Stokes equations. The two-level method consists of solving a small nonlinear system on the coarse mesh, then solving a linear…
Conventional mathematical models for simulating incompressible fluid flow problems are based on the Navier-Stokes equations expressed in terms of pressure and velocity. In this context, pressure-velocity coupling is a key issue, and…
In this expository work, we present Vishik's theorem on non-unique weak solutions to the two-dimensional Euler equations on the whole space, \[ \partial_t \omega + u \cdot \nabla \omega = f \, , \quad u = \frac{1}{2\pi}…
In type-II superconductors, the dynamics of superconducting vortices determine their transport properties. In the Ginzburg-Landau theory, vortices correspond to topological defects in the complex order parameter. Extracting their precise…
We investigate the nonlinear stability of compressible vortex sheet solutions for three-dimensional (3D) isentropic elastic flows. Building upon previous results on the weakly linear stability of elastic vortex sheets [19], we perform a…
This paper reports several new classes of weakly unstable recurrent solutions of the 2+1-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions have a number of remarkable properties which…
In this work we investigate the statistical mechanics of a family of two dimensional (2D) fluid flows, described by the generalized Euler equations, or $\alpha$-models. These models describe both nonlocal and local dynamics, with one…
Ideal systems of equations such as Euler and MHD may develop singular structures like shocks, vortex/current sheets. Among these, vortical singularities arise due to vortex stretching which can lead to unbounded growth of enstrophy.…
We establish the existence of solutions of the 2D incompressible non-homogeneous Euler equations with $C^{0}_{t}C^{1,\,\sqrt{\frac{4}{3}}-1-\varepsilon}_{x}\cap C^{0}_{t}L^{2}_{x}$ source terms that develop a singularity in finite time. In…
We propose a novel Particle Flow Map (PFM) method to enable accurate long-range advection for incompressible fluid simulation. The foundation of our method is the observation that a particle trajectory generated in a forward simulation…
For any $\alpha \in (0,1/3)$, we construct exact $C^{\alpha}$ self-similar blowup profiles for the vorticity of the 3D incompressible Euler equation without swirl, and build on them to prove asymptotically self-similar blowup from…
We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (AM,2021). These surfaces avoid the Kelvin-Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the…