Solutions Of Incompressible Euler Equations Research Articles

Statistical solutions of the two-dimensional incompressible Euler equations in spaces of unbounded vorticity

We study statistical solutions of the incompressible Euler equations in two dimensions with vorticity in Lp, 1≤p≤∞, and in the class of vortex-sheets with a distinguished sign. Our notion of statistical solution is based on the framework due to Bronzi, Mondaini and Rosa in [4]. Existence in this setting is shown by approximation with discrete measures, concentrated on deterministic solutions of the Euler equations. Additionally, we provide arguments to show that the statistical solutions of the Euler equations may be obtained in the inviscid limit of statistical solutions of the incompressible Navier-Stokes equations. Uniqueness of trajectory statistical solutions is shown in the Yudovich class.

Sharp energy regularity and typicality results for Hölder solutions of incompressible Euler equations

This paper is devoted to show a couple of typicality results for weak solutions $v\in C^\theta$ of the Euler equations, in the case $\theta 0}W^{\frac{2\theta}{1-\theta} + \varepsilon,p}(I)$ for any open $I \subset [0,T]$, are a residual set in $X_\theta$. This, in particular, partially solves [9, Conjecture 1]. We also show that smooth solutions form a nowhere dense set in the space of all the $C^\theta$ weak solutions. The technique is the same and what really distinguishes the two cases is that in the latter there is no need to introduce a different complete metric space with respect to the natural one.

Asymptotic limits of dissipative turbulent solutions to a compressible two-fluid model

In this paper we study the asymptotic limits of a compressible two-fluid model in an unbounded domain Ω with general initial data. By applying refined related entropy method and carrying out detailed analysis on the oscillations of velocity, we derive rigorously that the dissipative turbulent solutions (velocity) of the compressible two-fluid model converge to the strong solution of the quasi-geostropic equation when there is a Coriolis force and Ω=R2×T1, and while the two-fluid model has no Coriolis force term and Ω=R3, its solutions converge to the strong solution of incompressible Euler equations.

Iterative Bregman Projections for Regularized Transportation Problems

This paper details a general numerical framework to approximate solutions to linear programs related to optimal transport. The general idea is to introduce an entropic regularization of the initial linear program. This regularized problem corresponds to a Kullback--Leibler Bregman divergence projection of a vector (representing some initial joint distribution) on the polytope of constraints. We show that for many problems related to optimal transport, the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form. This allows us to make use of iterative Bregman projections (when there are only equality constraints) or, more generally, Bregman--Dykstra iterations (when inequality constraints are involved). We illustrate the usefulness of this approach for several variational problems related to optimal transport: barycenters for the optimal transport metric, tomographic reconstruction, multimarginal optimal transport, and in particular its application to Brenier's relaxed solutions of incompressible Euler equations, partial unbalanced optimal transport, and optimal transport with capacity constraints.

From quantum Euler–Maxwell equations to incompressible Euler equations

The combined quasi-neutral and non-relativistic limit of compressible quantum Euler–Maxwell equations for plasmas is studied in this paper. For well-prepared initial data, it is shown that the smooth solution of compressible quantum Euler–Maxwell equations converges to the smooth solution of incompressible Euler equations by using the modulated energy method. Furthermore, the associated convergence rates are also obtained.

Regularization of Point Vortices Pairs for the Euler Equation in Dimension Two

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem [ -\ep^2 \Delta u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p, \quad & x\in\Omega, u=0, \quad & x\in\partial\Omega, ] where $p>1$, $\Omega\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $\Omega$ is simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1,...,x_m)$ with the same strength $\kappa>0$, there is a stationary classical solution approximating stationary $m$ points vortex solution of incompressible Euler equations with vorticity $m\kappa$. Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by D. Smets and J. Van Schaftingen (2010).

Convergence of compressible Euler–Poisson system to incompressible Euler equations

In this paper, we study the quasi-neutral limit of compressible Euler–Poisson equations in plasma physics in the torus T d . For well prepared initial data the convergence of solutions of compressible Euler–Poisson equations to the solutions of incompressible Euler equations is justified rigorously by an elaborate energy methods based on studies on an λ-weighted Lyapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to λ is to use the curl–div decomposition of the gradient.

A new Savage–Hutter type model for submarine avalanches and generated tsunami

In this paper, we present a new two-layer model of Savage–Hutter type to study submarine avalanches. A layer composed of fluidized granular material is assumed to flow within an upper layer composed of an inviscid fluid (e.g. water). The model is derived in a system of local coordinates following a non-erodible bottom and takes into account its curvature. We prove that the model verifies an entropy inequality, preserves water at rest for a sediment layer and their solutions can be seen as particular solutions of incompressible Euler equations under hydrostatic assumptions. Buoyancy effects and the centripetal acceleration of the grain movement due to the curvature of the bottom are considered in the definition of the Coulomb term. We propose a two-step Roe type solver to discretize the presented model. It exactly preserves water at rest and no movement of the sediment layer, when its angle is smaller than the angle of repose, and up to second order all stationary solutions. Finally, some numerical tests are performed by simulating submarine and sub-aerial avalanches as well as the generated tsunami.

Convergence of Compressible Euler–Maxwell Equations to Incompressible Euler Equations

In this paper we study the combined quasineutral and non-relativistic limit of compressible Euler–Maxwell equations. For well prepared initial data the convergences of solutions of compressible Euler–Maxwell equations to the solutions of incompressible Euler equations are justified rigorously by an analysis of asymptotic expansions and a careful use of ε-weighted Liapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to ε is to use the curl–div decomposition of the gradient.

Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system

The combined quasineutral and inviscid limit for the Vlasov-Poisson-Fokker-Planck (VPFP) system is rigorously derived in this paper. It is shown that the solution of VPFP system converges to the solution of incompressible Euler equations with damping. The proof of convergence result is based on compactness arguments and the so-called relative-entropy method.

Incompressible, inviscid limit of the compressible Navier–Stokes system

We prove some asymptotic results concerning global (weak) solutions of compressible isentropic Navier–Stokes equations. More precisely, we establish the convergence towards solutions of incompressible Euler equations, as the density becomes constant, the Mach number goes to 0 and the Reynolds number goes to infinity.

Weak solution of incompressible Euler equations with decreasing energy

Weak solution of incompressible Euler equations are L 2-vector fields, satisfying integral relations, which express the mass and momentum balance. They are believed to describe the turbulent fluid motion at high Reynolds numbers. We justify this conjecture by constructing a weak solution with decreasing kinetic energy. The construction is based on Generalized Flows, introduced by Y. Brenier.

Energy dissipation without viscosity in ideal hydrodynamics I. Fourier analysis and local energy transfer

We outline a proof and give a discussion at a physical level of an assertion of Onsager's: namely, that a solution of incompressible Euler equations with Hölder continuous velocity of order h > 1 3 conserves kinetic energy, but not necessarily if h ≤ 1 3 . We prove the result under a “ ∗-Hölder condition” which is somewhat stronger than usual Hölder continuity. Our argument establishes also the fundamental result that the instantaneous (sub-scale) energy transfer is dominated by local triadic interactions for a ∗-Hölder solution with exponent h in the range 0 < h < 1. However, we must use a “band-averaged” energy flux for the proof: as we explain, the ordinary definition of the flux fails to adequately measure transport in wavenumber space (scale), since it is insensitive to the distance through which energy is displaced by individual interactions. We discuss some connections of the results with phenomenological theories of fully-developed turbulence, the 1941 Kolmogorov theory and the “multifractal model” of Parisi and Frisch.