Related papers: On Energy Conservation for the Hydrostatic Euler E…
The motion of a compressible inviscid radiative flow can be described by the radiative Euler equations, which consists of the Euler system coupled with a Poisson equation for the radiative heat flux through the energy equation. Although…
Lars Onsager in 1945-1949 made an exact analysis of the high Reynolds-number limit for individual turbulent flow realizations modeled by incompressible Navier-Stokes equations, motivated by experimental observations that dissipation of…
For the compressible Euler equations, even when the initial data are uniformly away from vacuum, solution can approach vacuum in infinite time. Achieving sharp lower bounds of density is crucial in the study of Euler equations. In this…
We prove global existence of appropriate weak solutions for the compressible Navier--Stokes equations for more general stress tensor than those covered by P.-L. Lions and E. Feireisl's theory. More precisely we focus on more general…
The purpose of this note is to give a complete proof of a $C^{0,\alpha}$ regularity result for the pressure for weak solutions of the two-dimensional "incompressible Euler equations" when the fluid velocity enjoys the same type of…
One of the most remarkable features of known nonstationary solutions to the incompressible Euler equations is the phenomenon known as the Taylor hypothesis, which predicts that coarse scale averages of the velocity carry the fine scale…
We construct infinitely many H\"older continuous, global-in-time, and stationary solutions to the stochastic Euler equations and the hypodissipative Navier-Stokes equations, taking values in the space $C(\mathbb{R};C^{\vartheta})$. For the…
In this paper, we investigate nonlinear stability of planar steady Euler flows related to least energy solutions of the Lane-Emden equation in a smooth bounded domain. We prove the orbital stability of these flows in terms of both the $L^s$…
The compressible Euler-Riesz equations are fundamental with wide applications in astrophysics, plasma physics, and mathematical biology. In this paper, we are concerned with the global existence and nonlinear stability of finite-energy…
In this expository work, we present Vishik's theorem on non-unique weak solutions to the two-dimensional Euler equations on the whole space, \[ \partial_t \omega + u \cdot \nabla \omega = f \, , \quad u = \frac{1}{2\pi}…
A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our…
In this paper, we investigate linear stability properties of the 2D isentropic compressible Euler equations linearized around a shear flow given by a monotone profile, close to the Couette flow, with constant density, in the domain…
In the spirit of D. Hoff's weak solution theory for the compressible Navier-Stokes equations (CNS) with bounded density, in this paper we establish the global existence and regularity properties of finite-energy weak solutions to an initial…
We show that for any $\al<\frac 17$ there exist $\al$-H\"older continuous weak solutions of the three-dimensional incompressible Euler equation, which satisfy the local energy inequality and strictly dissipate the total kinetic energy. The…
We investigate the initial-value problem for the relativistic Euler equations governing isothermal perfect fluid flows, and generalize an approach introduced by LeFloch and Shelukhin in the non-relativistic setting. We establish the…
We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t \Omega+U \cdot \nabla \Omega+\Omega\cdot \nabla U=0 \\ &\Omega(x,0)=\Omega_0(x) \end{aligned}\right. \end{equation*} in the…
We examine the blow-up claims of the incompressible Euler equations for several specific flow-fields, (1) the columnar eddies in the vicinity of stagnation; (2) a quasi-three-dimensional structure for illustrating oscillations and…
We prove existence of rotating star solutions which are steady-state solutions of the compressible isentropic Euler-Poisson (EP) equations in 3 spatial dimensions, with prescribed angular momentum and total mass. This problem can be…
We construct solutions to the SQG equation that fail to conserve the Hamiltonian while having the maximal allowable regularity for this property to hold. This result solves the generalized Onsager conjecture on the threshold regularity for…
We show that weak solutions of general conservation laws in bounded domains conserve their generalized entropy, and other respective companion laws, if they possess a certain fractional differentiability of order 1/3 in the interior of the…