Related papers: Convergence of the stochastic Navier-Stokes-$\alph…
In this paper, we consider the solvability of the two-dimensional stationary Navier--Stokes equations on the whole plane $\mathbb{R}^2$. In [6], it was proved that the stationary Navier--Stokes equations on $\mathbb{R}^2$ is ill-posed for…
Local behaviors near boundary are analyzed for solutions of the Stokes and Navier-Stoke equations in the half space with localized non-smooth boundary data. We construct solutions of Stokes equations whose velocity field is not bounded near…
We study the two-dimensional stationary Navier-Stokes equations describing the flows around a rotating obstacle. The unique existence of solutions and their asymptotic behavior at spatial infinity are established when the rotation speed of…
We analyze the two-dimensional incompressible Navier-Stokes equations on a smooth, bounded domain with Navier boundary conditions. Starting from an initial vorticity in $L^p$ with $p>2$, we show strong convergence of the vorticity in the…
A new approach to the stochastic theory of turbulence is suggested. The coloured noise that is present in the stochastic Navier-Stokes equation is generated from the delta-correlated noise allowing us to avoid the nonlocal field theory as…
Shockwaves provide a useful and rewarding route to the nonequilibrium properties of simple fluids far from equilibrium. For simplicity, we study a strong shockwave in a dense two-dimensional fluid. Here, our study of nonlinear transport…
Incompressible Navier-Stokes equations on a thin spherical domain $Q_\varepsilon$ along with free boundary conditions under a random forcing are considered. The convergence of the martingale solution of these equations to the martingale…
We consider a stochastic 2D Navier-Stokes equation in a bounded domain. The random force is assumed to be non-degenerate and periodic in time, its law has a support localised with respect to both time and space. Slightly strengthening the…
A proof of existence, uniqueness and smoothness of the Navier-Stokes equations is an actual problem, which solution is important for different branches of science. The subject of this study is obtaining the smooth and unique solutions of…
We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $\Omega\subset\mathbb{R}^{3}$ of class $\mathcal{C}^{1,1}$. We prove existence, uniqueness of weak and strong…
We develop a McKean-Vlasov interpretation of Navier-Stokes equations with external force field in the whole space, by associating with local mild $L^p$-solutions of the 3d-vortex equation a generalized nonlinear diffusion with random…
We consider the Navier-Stokes equations in $\mathbb R^d$ ($d=2,3$) with a stochastic forcing term which is white noise in time and coloured in space; the spatial covariance of the noise is not too regular, so It\^o calculus cannot be…
In the classical work [FK], Fujita and Kato established the local existence of solutions to the 3D Navier-Stokes equations in the critical $\mathbb{H}^{1/2}$-space. In this paper, we are concerned with the global well-posedness of the…
We consider the Navier-Stokes equation on a two dimensional torus with a random force, white noise in time and analytic in space, for arbitrary Reynolds number $R$. We prove probabilistic estimates for the long time behaviour of the…
We consider the barotropic Navier--Stokes system driven by a physically well-motivated transport noise in both continuity as well as momentum equation. We focus on three different situations: (i) the noise is smooth in time and the…
We consider an evolution system modeling a flow of colloidal particles which are suspended in an incompressible fluid and accounts for colloidal crystallization. The system consists of the Navier-Stokes equations for the volume averaged…
A mathematical model describing the flow of two-phase fluids in a bounded container $\Omega$ is considered under the assumption that the phase transition process is influenced by inertial effects. The model couples a variant of the…
In this paper, the system of particles coupled with fluid is considered. The particles are described by a Vlasov equation, and the fluid is governed by a forced Navier-Stokes equations. The interaction with fluid phase governed by…
In this note, we show the existence of regular solutions to the stationary version of the Navier-Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we…
We consider a stochastic model which describes the motion of a 2D incompressible fluid in a unbounded domain with viscosity and memory effects. This model is different from the classical stochastic Navier-Stokes-Voigt equations due to the…