Related papers: Large time behavior and asymptotic stability of th…
We consider the time-dependent Navier-Stokes equations in a half-space with boundary data on the line $(x,y)=(x_0,y)$ assumed to be time-periodic (or stationary) with a fixed asymptotic velocity ${\bf u}_{\infty}=(1,0)$ at infinity. We show…
In this paper, we present a new framework for the global well-posedness and large-time behavior of a two-phase flow system, which consists of the pressureless Euler equations and incompressible Navier-Stokes equations coupled through the…
This paper is concerned with the 2-dim two-phase interface Euler equation linearized at a pair of monotone shear flows in both fluids. We extend the Howard's Semicircle Theorem and study the eigenvalue distribution of the linearized Euler…
In this article we study the asymptotic behavior of incompressible, ideal, time-dependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small. Our main purpose is to identify the…
We consider the hydrodynamics of an incompressible fluid on a 2D periodic domain. There exists a family of stationary solutions with vorticity given by $\Omega^*=\alpha\cos (\mathbf{p} \cdot \mathbf{x} )+\beta \sin (\mathbf{p} \cdot…
The linear stability of rectilinear compressible vortex sheets is studied for two-dimensional isentropic elastic flows. This problem has a free boundary and the boundary is characteristic. A necessary and sufficient condition is obtained…
Euler's equations govern the behavior of gravity waves on the surface of an incompressible, inviscid, and irrotational fluid of arbitrary depth. We investigate the spectral stability of sufficiently small-amplitude, one-dimensional Stokes…
Accurate simulations of ice sheet dynamics, mantle convection, lava flow, and other highly viscous free-surface flows involve solving the coupled Stokes/free-surface equations. In this paper, we theoretically analyze the stability and…
In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles without the assumption of irrotationality. It is shown that when the variation of Bernoulli's function in the upstream is…
The 2D Euler equations are a simple but rich set of non-linear PDEs that describe the evolution of an ideal inviscid fluid, for which one dimension is negligible. Solving numerically these equations can be extremely demanding. Several…
The two-dimensional (2-D) Euler equations of a perfect fluid possess a beautiful geometric description: they are reduced geodesic equations on the infinite-dimensional Lie group of symplectomorphims with respect to a right-invariant…
Large-scale collective oscillation is discovered in the two-dimensional Euler equations. For initial conditions far from a base stationary flow, the system does not relax to the base stationary flow, but instead shows pairs of coherent…
We study the long-time behavior an extended Navier-Stokes system in $\R^2$ where the incompressibility constraint is relaxed. This is one of several "reduced models" of Grubb and Solonnikov '89 and was revisited recently (Liu, Liu, Pego…
In this paper, we consider steady Euler flows in a planar bounded domain in which the vorticity is sharply concentrated in a finite number of disjoint regions of small diameter. Such flows are closely related to the point vortex model and…
Sinusoidal flows are an important class of explicit stationary solutions of the two-dimensional incompressible Euler equations on a flat torus. For such flows, the steam functions are eigenfunctions of the negative Laplacian. In this paper,…
In convex planar domains, given an initial vorticity with one sign, we study the regularity and geometric properties of the dynamically stable solutions to the Euler equations in the coadjoint orbit of the initial vorticity. These flows…
This paper deals with the dead-water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. More generally, we study the generation of internal waves by a…
In this paper, we provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the…
We introduce a notion of stability for non-autonomous Hamiltonian flows on two-dimensional annular surfaces. This notion of stability is designed to capture the sustained twisting of particle trajectories. The main Theorem is applied to…
In this paper we prove rigidity results for classical solutions to the stationary 2D Euler equations in $\mathbb{R}^2$. Assuming that the velocity field has finite energy and that the stagnation set is connected, we prove that the…