Related papers: Energy conservation in the 3D Euler equations on $…
We study weak solutions of the incompressible Euler equations on $\mathbb{T}^2\times \mathbb{R}_+$; we use test functions that are divergence free and have zero normal component, thereby obtaining a definition that does not involve the…
Energy conservations are studied for inhomogeneous incompressible and compressible Euler equations with general pressure law in a torus or a bounded domain. We provide sufficient conditions for a weak solution to conserve the energy. By…
We consider the 3D Euler equations for incompressible homogeneous fluids and we study the problem of energy conservation for weak solutions in the space-periodic case. First, we prove the energy conservation for a full scale of Besov…
In this paper we consider the incompressible 3D Euler and Navier-Stokes equations in a smooth bounded domain. First, we study the 3D Euler equations endowed with slip boundary conditions and we prove the same criteria for energy…
In this paper, we study the problem of energy conservation for the solutions to the incompressible viscoelastic flows. First, we consider Leray-Hopf weak solutions in the bounded Lipschitz domain $\Omega$ in $\mathbb{R}^d\,\, (d\geq 2)$. We…
Motivated by the works of Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy in [8, Commun. Math. Phys. 348: 129-143, 2016] and Chen and Yu in [5, J. Math. Pures Appl. 131: 1-16, 2019], we address how the $L^p$ control of vorticity…
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…
The 3D incompressible Euler equations in a bounded domain are most often supplemented with impermeable boundary conditions, which constrain the fluid to neither enter nor leave the domain. We establish well-posedness with inflow, outflow of…
The goal of this note is to show that, also in a bounded domain $\Omega \subset \mathbb{R}^n$, with $\partial \Omega\in C^2$, any weak solution, $(u(x,t),p(x,t))$, of the Euler equations of ideal incompressible fluid in $\Omega\times (0,T)…
This paper investigates the stochastic 3D Euler equations on a periodic domain $\mathbb{T}^3$, driven by a $GG^*$-Wiener process $B$ of trace class: \begin{align*} \mathrm{d} u+\mathrm{div}(u\otimes u)\,\mathrm{d} t+\nabla…
We consider the incompressible Euler equations in a bounded domain in three space dimensions. Recently, the first two authors proved Onsager's conjecture for bounded domains, i.e., that the energy of a solution to these equations is…
The first half of Onsager's conjecture states that the Euler equations of an ideal incompressible fluid conserve energy if $u (\cdot ,t) \in C^{0, \theta} (\mathbb{T}^3)$ with $\theta > \frac{1}{3}$. In this paper, we prove an analogue of…
In this paper, we study the energy equality for weak solutions to the non-resistive MHD equations with physical boundaries. Although the equations of magnetic field $b$ are of hyperbolic type, and the boundary effects are considered, we…
In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions with velocity satisfying additional assumptions in fractional Sobolev spaces with respect to the space variables,…
Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in 3D conserve energy only if they have a certain minimal smoothness, (of order of 1/3 fractional derivatives) and that they dissipate energy if they…
It is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space $B^{1/3}_{3,\infty}$. When the singular set of the solution is (or belongs to) a smooth manifold, we…
We investigate sufficient H\"older continuity conditions on Leray-Hopf (weak) solutions to the in unsteady Navier-Stokes equations in three dimensions guaranteeing energy conservation. Our focus is on the half-space case with homogeneous…
Onsager's conjecture states that the conservation of energy may fail for $3D$ incompressible Euler flows with H\"{o}lder regularity below $1/3$. This conjecture was recently solved by the author, yet the endpoint case remains an interesting…
This paper addresses the problem of energy conservation for the two- and three-dimensional density-dependent Euler equations. Two types of sufficient conditions on the regularity of solutions are provided to ensure the conservation of total…
We give a localized regularity condition for energy conservation of weak solutions of the Euler equations on a domain $\Omega\subset \mathbb{R}^d$, $d\ge 2$, with boundary. In the bulk of fluid, we assume Besov regularity of the velocity…