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We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type…
We obtain existence and conormal Sobolev regularity of strong solutions to the 3D compressible isentropic Navier-Stokes system on the half-space with a Navier boundary condition, over a time that is uniform with respect to the viscosity…
We present several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. We first focus on…
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…
In this paper, we prove the existence of global weak solutions for 3D compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation. The main contribution of this paper…
This present paper is dedicated to the study of the Cauchy problem of the two-dimensional Euler-Boussinesq-B$\rm\acute{e}$nard equations which couple the incompressible Euler equations for the velocity and a transport equation with critical…
We consider the 2-D incompressible Euler equations in a bounded domain and show that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity…
By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of…
We prove weak existence of Euler equation (or Navier-Stokes equation) perturbed by a multiplicative noise on bounded domains of $\mathbb R^2$ with Dirichlet boundary conditions and with periodic boundary conditions. Solutions are $H^1$…
This paper is concerned with a two dimensional Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We prove that the associated Cauchy problem is well-posed for initial data of low regularity, with…
We consider the Riemann problem composed of two shocks for the 1D Euler system. We show that the Riemann solution with two shocks is stable and unique in the class of weak inviscid limits of solutions to the Navier-Stokes equations with…
We consider the compressible Navier-Stokes equation with density dependent viscosity coefficients, focusing on the case where those coefficients vanish on vacuum. We prove the stability of weak solutions both in the torus and in the whole…
We prove the conservation of energy for weak and statistical solutions of the two-dimensional Euler equations, generated as strong (in an appropriate topology) limits of the underlying Navier-Stokes equations and a Monte Carlo-Spectral…
In this paper, the Cauchy problem for the three-dimensional (3-D) full compressible Navier-Stokes equations (CNS) with zero thermal conductivity is considered. First, when shear and bulk viscosity coefficients both depend on the absolute…
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)…
We establish local balance equations for smooth functions of the vorticity in the DiPerna-Majda weak solutions of 2D incompressible Euler, analogous to the balance proved by Duchon and Robert for kinetic energy in 3D. The anomalous term or…
In this paper, we consider the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Euler system for two-phase flow motion, which consists of the isentropic compressible Navier-Stokes equations and the isothermal compressible…
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…
This paper is concerned with the existence of compactly supported admissible solutions to the Cauchy problem for the isentropic compressible Euler equations. In more than one space dimension, convex integration techniques developed by De…
We consider the three-dimensional incompressible Euler equations on a bounded domain $\Omega$ with $C^4$ boundary. We prove that if the velocity field $u \in C^{0,\alpha} (\Omega)$ with $\alpha > 0$ (where we are omitting the time…