Related papers: Invariant measures for the non-periodic two-dimens…
Non-stationary Euler flows of gases are studied. The system of differential equations describing such flows can be represented by means of 2-forms on zero-jet space and we get some exact solutions by means of such a representation.…
We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the flows of completely integrable Hamiltonian systems are investigated.…
We study the Neumann and Dirichlet problems for the total variation flow in metric measure spaces. We prove existence and uniqueness of weak solutions and study their asymptotic behaviour. Furthermore, in the Neumann problem we provide a…
We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered…
We consider the inhomogeneous (or density dependent) incompressible Euler equations in a three-dimensional periodic domain. We construct density $\varrho$ and velocity $u$ such that, for any $\alpha<1/7$, both of them are $\alpha $-H\"older…
An analytical linear solution of the fully compressible Euler equations is found, in the particular case of a stationary two dimensional flow that passes over an orographic feature with small height-width ratio. A method based on the…
We study the long-time behaviour of a stochastic Allen-Cahn-Navier-Stokes system modelling the dynamics of binary mixtures of immiscible fluids. The model features two stochastic forcings, one on the velocity in the Navier-Stokes equation…
We prove stability for arbitrarily long times of the zero solution for the so-called $\beta$-plane equation, which describes the motion of a two-dimensional inviscid, ideal fluid under the influence of the Coriolis effect. The Coriolis…
In this paper, we proved the well-posedness theory of compressible subsonic jet flows for two-dimensional steady Euler system with {\it general} incoming horizontal velocity as long as the flux is larger than a critical value. One of the…
The periodic KdV equation u_t=u_{xxx}+\beta uu_x arises from a Hamiltonian system with infinite-dimensional phase space L^2(T). Bourgain has shown that there exists a Gibbs measure \nu on balls \{\phi :\Vert\Phi\Vert^2_{L^2}\leq N\} in the…
We consider the Euler equations on the two-dimensional torus and construct invariant measures for the dynamics of these equations, concentrated on sufficiently regular Sobolev spaces so that strong solutions are also known to exist. The…
Incompressible flows of an ideal two-dimensional fluid on a closed orientable surface of positive genus are considered. Linear stability of harmonic, i.e. irrotational and incompressible, solutions to the Euler equations is shown using the…
Local structures, beyond the well-known `frozen-in' to the barotropic flows of the generalized vorticities, of the two-fluid model of plasma flows are presented. More general non-barotropic situations are also considered. A modified Euler…
This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter…
In this work we consider a finite dimensional approximation for the 2D Euler equations on the sphere, proposed by V. Zeitlin, and show their convergence towards a solution to Euler equations with marginals distributed as the enstrophy…
In fluid mechanics, a lot of authors have been reporting analytical solutions of Euler and Navier-Stokes equations. But there is an essential deficiency of non-stationary solutions indeed. In our presentation, we explore the case of…
In this paper, we establish two stability theorems for steady or traveling solutions of the two-dimensional incompressible Euler equation in a finite periodic channel, extending Arnold's classical work from the 1960s. Compared to Arnold's…
We construct global-in-time, unique solutions of the two-dimensional Euler equations in a Yudovich type space and a $\rm bmo$-type space. First, we show the regularity of solutions for the two-dimensional Euler equations in the Spanne space…
For the one dimensional Burgers equation with a random and periodic forcing, it is well-known that there exists a family of invariant measures, each corresponding to a different average velocity. In this paper, we consider the coupled…
We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and…