Related papers: Time-periodic leapfrogging vortex rings in the 3D …
Can every physical system simulate any Turing machine? This is a classical problem which is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore asked in [15] if hydrodynamics is capable…
We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t \Omega+U \cdot \nabla \Omega-\Omega\cdot \nabla U=0 \\ &\Omega(x,0)=\Omega_0(x) \end{aligned}\right. \end{equation*} under…
In this article, time periodic problem of the compressible Euler equations with damping on the whole space is studied. It is well known that in the Euler system, long-time behavior of solutions is a more delicate problem due to lack of the…
The 3D incompressible Euler equation is an important research topic in the mathematical study of fluid dynamics. Not only is the global regularity for smooth initial data an open issue, but the behaviour may also depend on the presence or…
We consider the Lorenz equations, a system of three dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been…
In the field of differential equations, particularly fluid dynamics, many researchers have shown an interest in the behavior of time periodic solutions. In this paper, we study isentropic gas flow in a bounded interval and apply a time…
We consider the evolution of an incompressible two-dimensional perfect fluid as the boundary of its domain is deformed in a prescribed fashion. The flow is taken to be initially steady, and the boundary deformation is assumed to be slow…
The dynamics along the particle trajectories for the 3D axisymmetric Euler equations in an infinite cylinder are considered. It is shown that if the inflow-outflow is highly oscillating in time, the corresponding Euler flow cannot keep the…
We present an Eulerian vortex method based on the theory of flow maps to simulate the complex vortical motions of incompressible fluids. Central to our method is the novel incorporation of the flow-map transport equations for line elements,…
Point vortices on a cylinder (periodic strip) are studied geometrically. The Hamiltonian formalism is developed, a non-existence theorem for relative equilibria is proved, equilibria are classified when all vorticities have the same sign,…
We study the time evolution of an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside $N$ small disjoint rings of thickness $\varepsilon$ and vorticity…
We address the question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows being a promising candidate…
In many occurrences of fluid-structure interaction time-periodic motions are observed. We consider the interaction between a fluid driven by the three dimensional Navier-Stokes equation and a two dimensional linearized elastic Koiter shell…
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…
We start by surveying the planar point vortex motion of the Euler equations in the whole plane, half-plane and quadrant. Then we go on to prove non-collision property of 2-vortex system by using the explicit form of orbits of 2-vortex…
We introduce many families of explicit solutions to the three dimensional incompressible Euler equations for nonviscous fluid flows using the Lagrangian framework. Almost no exact Lagrangian solutions exist in the literature prior to this…
The dynamics of vortex ring pairs in the homogeneous nonlinear Schr\"odinger equation is studied. The generation of numerically-exact solutions of traveling vortex rings is described and their translational velocity compared to revised…
This paper analyzes the space of steady rotating solutions to the two-dimensional incompressible Euler equations nearby vortex patch solutions satisfying a natural nondegeneracy condition. We address the question of desingularization and…
An evolution of a spherical region, subjected to uniform buoyancy force, is investigated. Incompressibility and axial symmetry are assumed, together with a buoyancy discontinuity at the boundary. The boundary turns into a vortex sheet and…
We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time "splash" (or "splat") singularity first introduced in [9], wherein the…