The purpose of this paper is to derive the anisotropic averaged Euler equations and to study their geometric and analytic properties. These new equations involve the evolution of a mean velocity field and an advected symmetric tensor that captures the fluctuation effects. Besides the derivation of these equations, the new results in the paper are smoothness properties of the equations in material representation, which gives well-posedness of the equations, and the derivation of a corrector to the macroscopic velocity field. The numerical implementation and physical implications of this set of equations will be explored in other publications.
We define a Gaussian invariant measure for the two-dimensional averaged-Euler equation and show the existence of its solution with initial conditions on the support of the measure. An invariant surface measure on the level sets of the energy is also constructed, as well as the corresponding flow. Poincaré recurrence theorem is used to show that the flow returns infinitely many times in a neighbourhood of the initial state.
Using the Mountain Pass Theorem, we establish the existence of periodic solution for Euler-Lagrange equation. Lagrangian consists of kinetic part (an anisotropic G-function), potential part $K-W$ and a forcing term. We consider two situations: $G$ satisfying $Δ_2\cap\nabla_2$ in infinity and globally. We give conditions on the growth of the potential near zero for both situations.
In this paper, we consider the derivation of the Kadomtsev-Petviashvili (KP) equation for cold ion-acoustic wave in the long wavelength limit of the two-dimensional quantum Euler-Poisson system, under different scalings for varying directions in the Gardner-Morikawa transform. It is shown that the types of the KP equation depend on the scaled quantum parameter $H>0$. The QKP-I is derived for $H>2$, QKP-II for $0<H<2$ and the dispersive-less KP (dKP) equation for the critical case $H=2$. The rigorous proof for these limits is given in the well-prepared initial data case, and the norm that is ch…
We study the multiphasic formulation of the incompressible Euler equation introduced by Brenier: infinitely many phases evolve according to the compressible Euler equation and are coupled through a global in-compressibility constraint. We are able to prove that the entropy, when averaged over all phases, is a convex function of time, a result that was conjectured by Brenier. The novelty in our approach consists in introducing a time-discretization that allows us to import a flow interchange inequality previously used by Matthes, McCann and Savar{é} to study first order in time PDE, namely the …
The Euler-$α$ equations model the averaged motion of an ideal incompressible fluid when filtering over spatial scales smaller than $α$. We show that there exists $β>1$ such that weak solutions to the two and three dimensional Euler-$α$ equations in the class $C^0_t H^β_x$ are not unique and may not conserve the Hamiltonian of the system, thus demonstrating flexibility in this regularity class. The construction utilizes a Nash-style intermittent convex integration scheme. We also formulate an appropriate version of the Onsager conjecture for Euler-$α$, postulating that the threshold between rig…
Piecewise smooth stationary Euler flows with support in a neighborhood of a helix
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（西班牙高等科学研究委员会数学科学研究所）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
By introducing a new averaged quantity with a fast decay weight to perform Sideris's argument (Commun Math Phys, 1985) developed for the Euler Equations, we extend the formation of singularities of classical solution to the 3D Euler Equations established in Sideris (1985) and Makino et al. (Jpn J Appl Math, 1986) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler-Poisson Equations, but remove the …
We construct a class of global, dynamical solutions to the 3d Euler equations near the stationary state given by uniform "rigid body" rotation. These solutions are axisymmetric, of Sobolev regularity, have non-vanishing swirl and scatter linearly, thanks to the dispersive effect induced by the rotation. To establish this, we introduce a framework that builds on the symmetries of the problem and precisely captures the anisotropic, dispersive mechanism due to rotation. This enables a fine analysis of the geometry of nonlinear interactions and allows us to propagate sharp decay bounds, which is c…
We present a new hydrodynamic model consisting of the pressureless Euler equations and the isentropic compressible Navier-Stokes equations where the coupling of two systems is through the drag force. This coupled system can be derived, in the hydrodynamic limit, from the particle-fluid equations that are frequently used to study the medical sprays, aerosols and sedimentation problems. For the proposed system, we first construct the local-in-time classical solutions in an appropriate $L^2$ Sobolev space. We also establish the \emph{a priori} large-time behavior estimate by constructing a Lyapun…
We study uniaxial solutions of the Euler-Lagrange equations for a Landau-de Gennes free energy for nematic liquid crystals, with a fourth order bulk potential, with and without elastic anisotropy. In the elastic isotropic case, we show that (i) all uniaxial solutions of the Euler-Lagrange equations, with a director field of certain symmetry, necessarily have the radial-hedgehog structure modulo an orthogonal transformation, (ii) the "escape into third dimension" director cannot correspond to a purely uniaxial solution of the Landau-de Gennes Euler-Lagrange equations and we do not use artificia…
We are concerned with supersonic vortex sheets for the Euler equations of compressible inviscid fluids in two space dimensions. For the problem with constant coefficients, in [10] the authors have derived a pseudo-differential equation which describes the time evolution of the discontinuity front of the vortex sheet. In agreement with the classical stability analysis, the problem is weakly stable if $|[v\cdotτ]|>2\sqrt{2}\,c$, and the well-posedness was obtained in standard weighted Sobolev spaces. The aim of the present paper is to improve the result of [10], by showing the existence of the s…
In this paper, we study a second-order, nonlinear evolution equation with damping arising in elastodynamics. The nonlinear term is monotone and possesses a convex potential but exhibits anisotropic and nonpolynomial growth. The appropriate setting for such equations is that of monotone operators in Orlicz spaces. Global existence of solutions in the sense of distributions is shown via convergence of the backward Euler scheme combined with an internal approximation. Moreover, we show uniqueness in a class of sufficiently smooth solutions and provide an a priori error estimate for the temporal s…
While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in $\mathbb{R}^3$ with a fixed speed of rotation. We show that for any $M>0$, axisymmetric initial data of sufficiently small size $\varepsilon$ lead to solutions that exist for a long time at least $\varepsilon^{-M}$ and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this w…
Global Existence Results for the Anisotropic Boussinesq System in Dimension Two
研究了具有水平粘性的二维Boussinesq方程组的整体解的存在性问题。我们首先研究的情况下，Navier-斯托克斯方程没有垂直粘性耦合的输运方程。其次，我们考虑了经典的二维不可压Euler方程和只在水平方向上有扩散的输运扩散方程之间的耦合。对于这两个系统和任意大的数据，我们构造整体弱解`a la Leray。接下来，我们陈述更规则数据的全局适定性结果。我们的结果强烈依赖于这样一个事实，即扩散发生在垂直于浮力的方向。
The first half of Onsager's conjecture states that the Euler equations of an ideal incompressible fluid conserve energy if $u (\cdot ,t) \in C^{0, θ} (\mathbb{T}^3)$ with $θ> \frac{1}{3}$. In this paper, we prove an analogue of Onsager's conjecture for several subgrid scale $α$-models of turbulence. In particular we find the required Hölder regularity of the solutions that ensures the conservation of energy-like quantities (either the $H^1 (\mathbb{T}^3)$ or $L^2 (\mathbb{T}^3)$ norms) for these models. We establish such results for the Leray-$α$ model, the Euler-$α$ equations (also known as t…
Nested invariant tori foliating a vector field and its curl: toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries
