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The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3-D inhomogeneous incompressible Navier-Stokes system in the whole space~$\R^3$. This class of data is based on functions…
We consider the stochastic Navier--Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear…
We consider the Navier--Stokes--Fourier system in a bounded domain $\Omega \subset R^d$, $d=2,3$, with physically realistic in/out flow boundary conditions. We develop a new concept of weak solutions satisfying a general form of relative…
We consider the incompressible Navier-Stokes equations in the cylinder $\R \times \T$, with no exterior forcing, and we investigate the long-time behavior of solutions arising from merely bounded initial data. Although we do not know if…
This paper addresses the three-dimensional Navier-Stokes equations for an incompressible fluid whose density is permitted to be inhomogeneous. We establish a theorem of global existence and uniqueness of strong solutions for initial data…
We investigate the barotropic compressible Navier-Stokes equations with the Navier-slip boundary conditions in a general two-dimensional bounded simply connected domain. For initial density that is allowed to vanish, we establish the global…
We establish the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for three-dimensional compressible isentropic flow in the whole space. It is shown that there exists a unique regular solution of compressible…
The aim of the note is to discuss different definitions of solutions to the Cauchy problem for the Navier-Stokes equations with the initial data belonging to the Lebesgue space $L_3(\mathbb R^3)$
In this note, we investigate partial regularity of weak solutions of the three dimensional chemotaxis-Navier-Stokes equations, and obtain the $\frac53$-dimensional Hausdorff measure of the possible singular set is vanishing at the first…
We consider the Navier-Stokes equations in a three-dimensional curved thin domain around a given closed surface under Navier's slip boundary conditions. When the thickness of the thin domain is sufficiently small, we establish the global…
In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped nonlinear wave equations and the nonlinear wave equation with nonlinear damping…
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…
A well-known unsolved problem (in the classical theory of fluid mechanics) is to identify a set of initial velocities, which may depend on the viscosity, the body forces and possibly the boundary of the fluid that will allow global in time…
In this paper, we investigate the vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier-Stokes equations with a slip boundary condition. We establish the local well-posedness of the strong solutions for initial boundary…
In the recent paper, the global-in-time inviscid limit of the three-dimensional (3D) isentropic compressible Navier-Stokes equations is considered. First, when viscosity coefficients are given as a constant multiple of density's power…
We prove the global existence and uniqueness of the classical (weak) solution for the 2D or 3D compressible Navier-Stokes equations with a density-dependent viscosity coefficient ($\lambda=\lambda(\rho)$). Initial data and solutions are…
Regularity properties of strong solutions are considered.
The Navier-Stokes-Fourier system is a well established model for describing the motion of viscous compressible heat-conducting fluids. We study the existence of time-periodic weak solutions and improve the known result in the following…
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…
We consider the three-dimensional incompressible Navier--Stokes equations in a curved thin domain with Navier's slip boundary conditions. The curved thin domain is defined as a region between two closed surfaces which are very close to each…