Related papers: 3D incompressible Euler: Geometric formalism and a…
We establish a result concerning the so-called Lagrangian controllability of the Euler equation for incompressible perfect fluids in dimension 3. More precisely we consider a connected bounded domain of R^3 and two smooth contractible sets…
We discuss general incompressible inviscid models, including the Euler equations, the surface quasi-geostrophic equation, incompressible porous medium equation, and Boussinesq equations. All these models have classical unique solutions, at…
The motion of an incompressible fluid in Lagrangian coordinates involves infinitely many symmetries generated by the left Lie algebra of group of volume preserving diffeomorphisms of the three dimensional domain occupied by the fluid.…
We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. The new approach evolves the flow map using the gradient-augmented level set method (GALSM).…
This paper is devoted to the geometric analysis of the incompressible averaged Euler equations on compact Riemannian manifolds with boundary. The equation also coincides with the model for a second-grade non-Newtonian fluid. We study the…
Building on the recent work of C. De Lellis and L. Sz\'{e}kelyhidi, we construct global weak solutions to the three-dimensional incompressible Euler equations which are zero outside of a finite time interval and have velocity in the…
We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We construct unique local-in-time solutions in the Lagrangian setting for $u_0 \in H^{2.5+\delta }$ such that the Rayleigh-Taylor…
In this paper we are interested in geometric aspects of blowup in the axisymmetric 3D Euler equations with swirl on a cylinder. Writing the equations in Lagrangian form for the flow derivative along either the axis or the boundary and…
We define various notions of Lagrangian solution in physical space for 3-d incompressible geostrophic system with free upper boudary under different conditions for initial data,then prove their existence via the minimization with respect to…
We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial…
A novel semi-Lagrangian method is introduced to solve numerically the Euler equation for ideal incompressible flow in arbitrary space dimension. It exploits the time-analyticity of fluid particle trajectories and requires, in principle,…
We present here a constructive method of Lagrangian approximate control- lability for the Euler equation. We emphasize on different options that could be used for numerical recipes: either, in the case of a bi-dimensionnal fluid, the use of…
In this paper we derive a representation of the deterministic 3-dimensional Navier-Stokes equations based on stochastic Lagrangian paths. The particle trajectories obey SDEs driven by a uniform Wiener process; the inviscid Weber formula for…
We present a semi-Lagrangian characteristic mapping method for the incompressible Euler equations on a rotating sphere. The numerical method uses a spatio-temporal discretization of the inverse flow map generated by the Eulerian velocity as…
We derive a priori estimates for the incompressible free-boundary Euler equations with surface tension in three spatial dimensions. Working in Lagrangian coordinates, we provide a priori estimates for the local existence when the initial…
In this note, we establish Yudovich's existence and uniqueness result for bounded (as well as mildly unbounded) vorticity weak solution of the two-dimensional incompressible Euler equations. As a biproduct of our proof, we establish some…
In this paper, we study the inhomogeneous incompressible Euler equation (IIE in short) from a Lagrangian perspective. We establish a geodesic description of this equation and discuss the associated geometric structures. We also find the…
We propose an efficient semi-Lagrangian Characteristic Mapping (CM) method for solving the three-dimensional (3D) incompressible Euler equations. This method evolves advected quantities by discretizing the flow map associated with the…
Two prized papers, one by Augustin Cauchy in 1815, presented to the French Academy and the other by Hermann Hankel in 1861, presented to G\"ottingen University, contain major discoveries on vorticity dynamics whose impact is now quickly…
In this paper, we study the control system associated with the incompressible 3D Euler system. We show that the velocity field and pressure of the fluid are exactly controllable in projections by the same finite-dimensional control.…