Related papers: Loss of smoothness and energy conserving rough wea…
This paper focuses on the study of the density-dependent incompressible Euler equations in space dimension $d=2$, for low regularity (\textsl{i.e.} non-Lipschitz) initial data satisfying assumptions in spirit of the celebrated Yudovich…
The Cauchy problem for the two-dimensional incompressible Euler equation is globally well-posed for smooth initial data. In this paper, we show that for a dense $G_\delta$ set of initial data, the solutions lose regularity in infinite time,…
We study the mathematical properties of time-dependent flows of incompressible fluids that respond as an Euler fluid until the modulus of the symmetric part of the velocity gradient exceeds a certain, a-priori given but arbitrarily large,…
In [25], Moffatt introduced the concept of helicity in an inviscid fluid and examined the helicity preservation of smooth solution to barotropic compressible flow. In this paper, it is shown that the weak solutions of the above system in…
In their seminal paper "Oscillations and concentrations in weak solutions of the incompressible fluid equations", R. DiPerna and A. Majda introduced the notion of measure-valued solution for the incompressible Euler equations in order to…
A statistical method for calculating equilibrium solutions of the shallow water equations, a model of essentially 2-d fluid flow with a free surface, is described. The model contains a competing acoustic turbulent {\it direct} energy…
We prove an extension theorem for local solutions of the 3d incompressible Euler equations. More precisely, we show that if a smooth vector field satisfies the Euler equations in a spacetime region $\Omega\times(0,T)$, one can choose an…
In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of H\"older continuous functions, relaxing some of the assumptions on the time variable (both…
We show the short-time existence and nonlinear stability of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on the weakly linear stability result of Morando--Trebeschi (2008) [20]. The…
In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $u = \nabla^\perp \Psi$, $\Psi(r,\theta) = r^{\lambda} \psi(\theta)$, for $\lambda >0$, we show…
In the dynamics of viscous fluid, the case of vanishing kinematic viscosity is actually equivalent to the Reynolds number tending to infinity. Hence, in the limit of vanishing viscosity the fluid flow is essentially turbulent. On the other…
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…
In this work we study the asymptotic behavior of solutions of the incompressible two-dimensional Euler equations in the exterior of a single smooth obstacle when the obstacle becomes very thin tending to a curve. We extend results by…
It has been known for some time that a 3D incompressible Euler flow that has initially a barely smooth velocity field nonetheless has Lagrangian fluid particle trajectories that are analytic in time for at least a finite time (Ph. Serfati…
Onsager conjectured that solutions of the incompressible Euler equations possessing a certain degree of roughness do not conserve the kinetic energy. Since, within the physical frame of Onsager's conjecture, the kinetic energy is the only…
For any positive integer $k$, we prove the existence of nontrivial $C^k$-smooth uniformly rotating solutions to the 2D incompressible Euler equations with compact spatial support. These solutions, which can be chosen to be small…
Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of solutions for the Navier-Stokes equations in two dimensions have been known for a long time. Leray showed…
The fundamental "two-fluid" model for describing plasma dynamics is given by the Euler-Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. We prove global stability of…
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 study the local balance of momentum for weak solutions of incompressible Euler equations obtained from the zero-viscosity limit in the presence of solid boundaries, taking as an example flow around a finite, smooth body. We show that…