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A stochastic description of solutions of the Navier-Stokes equation is investigated. These solutions are represented by laws of finite dimensional semi-martingales and characterized by a weak Euler- Lagrange condition. A least action…
The continuity of the kinetic energy is an important property of incompressible viscous fluid flows. We show that for any prescribed finite energy divergence-free initial data there exist infinitely many global in time weak solutions with…
The paper is concerned with the IBVP of the Navier-Stokes equations. The result of the paper is in the wake of analogous results obtained by the authors in previous articles [4, 5]. The goal is to estimaste the possible gap between the…
We are concerned with the question of well-posedness of stochastic three dimensional incompressible Euler equations. In particular, we introduce a novel class of dissipative solutions and show that (i) existence; (ii) weak--strong…
We prove some estimates for suitable weak solutions to the non-stationary three-dimensional Navier-Stokes equations under assumptions that certain invariant functionals of the velocity are bounded.
Stochastic Navier--Stokes equations in a thin three-dimensional domain are considered, driven by additive noise. The convergence of martingale solution of the stochastic Navier--Stokes equations in a thin three-dimensional domain to the…
We prove existence of weak solutions (in the probabilistic sense) for a general class of stochastic semilinear wave equations on bounded domains of $R^d$ driven by a possibly discontinuous square integrable martingale.
The paper is concerned with the IBVP of the Navier-Stokes equations. The goal is the attempt to construct a weak solution enjoying an energy equality. This result is a natural continuation and improvement of the one obtained by the same…
The purpose of this paper is to prove the existence of global in time local energy weak solutions to the Navier-Stokes equations in the half-space $\mathbb R^3_+$. Such solutions are sometimes called Lemari\'e-Rieusset solutions in the…
We consider the incompressible inhomogeneous Navier-Stokes equations with constant viscosity coefficient and density which is bounded and bounded away from zero. We show that the energy balance relation for this system holds for weak…
We prove the existence and uniqueness of weak solutions of the inhomogeneous incompressible Navier--Stokes equations without vacuum using the relative energy method. We present a novel and direct proof of the existence of weak solutions…
Under the assumption of an initial datum divergence free and in L2, we prove the existence of a weak solution to the Navier-Stokes initial boundary value problem enjoying the energy equality on (0,t), almost everywhere in t>0, in…
We prove the equivalence of being a Leray-Hopf weak solution to the Navier-Stokes equations in $\mathbb{R}^m, m \ge 3$, to satisfying a well known integral equation. We use this equation to derive some properties of these weak solutions.
We prove the existence of strong solutions to Navier-Stokes equations in three dimensional thin domains. Our proof is based on the energy and the Poincar\'e inequalities as well as contraction principle argument and is free of the mean…
We prove uniqueness of weak solutions of the three-dimensional compressible Navier-Stokes equations with potential force. We make use of the Lagrangean framework in comparing the instantaneous states of corresponding fluid particles in two…
We introduce a concept of dissipative measure valued martingale solutions for stochastic compressible Navier-Stokes equations. These solutions are weak from a probabilistic perspective, since they include both the driving Wiener process and…
We consider vanishing viscosity approximations to solutions of the stochastic incompressible Euler equations in two space dimensions with additive noise. We identify sufficient and necessary conditions under which martingale solutions of…
Weak solutions of incompressible Navier-Stokes Equations re-obtained variationally
Large weak solutions to Navier--Stokes--Maxwell systems are not known to exist in their corresponding energy space in full generality. Here, we mainly focus on the three-dimensional setting of a classical incompressible…
It is well known that a Leray-Hopf weak solution enjoys an energy inequality. Here, we investigate the energy equality related to a suitable weak solution to the Navier-Stokes initial boundary value problem. The term suitable is meant in…