Related papers: Gaining two derivatives on a singular force in the…
The weak solution to the Navier-Stokes equations in a bounded domain $D \subset \mathbb{R}^3$ with a smooth boundary is proved to be unique provided that it satisfies an additional requirement. This solution exists for all $t \geq 0$. In a…
We establish the existence and uniqueness of solutions to stochastic 2D Navier-Stokes equations in a time-dependent domain driven by Brownian motion. A martingale solution is constructed through domain transformation and appropriate…
Exponential stabilization to time-dependent trajectories for the incompressible Navier-Stokes equations is achieved with explicit feedback controls. The fluid is contained in two-dimensional spatial domains and the control force is, at each…
We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with vacuum as far-field density. It is proved that if the initial density decays not too slow at infinity, the 2D…
By applying Wiegner' method in \cite{Wiegner}, we first prove the large time decay estimate for the global solutions of a 2.5 dimensional Navier-Stokes system, which is a sort of singular perturbed 2-D Navier-Stokes system in three space…
This paper deals with the uniqueness of mild solutions to the forced or unforced Navier-Stokes equations in the whole space. It is known that the uniqueness of mild solutions to the unforced Navier-Stokes equations holds in…
The local well-posedness theory for the incompressible Navier-Stokes equations in $\BMO^{-1}$ has attracted considerable attention over the past two decades. In a recent breakthrough, Coiculescu and Palasek (Invent. Math., 2025) settled the…
This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier-Stokes equations, complexified in time. Specifically, for periodic boundary conditions on $[0,L]^2$, and a force…
In this paper, we give a sufficient condition to guarantee the existence of a smooth solution of the Navier-Stokes Equation with the nice decreasing properties at infinity. In this way, we prove the existence of smooth physically reasonable…
In to previous papers by the authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen…
The present paper deals with existence and uniqueness of global mild solutions for the 2D Navier-Stokes equations with impulses. Using the framework of nonautonomous dynamical systems, we extend previous results considering the 2D…
We study 2D Navier-Stokes equations with a constraint on $L^2$ energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier-Stokes equation on $\R^2$ and $\T$, by a fixed point argument. We…
In this paper, we use forward-backward stochastic differential systems to study the solution of two and d dimensional ($d\geq 3$) Navier-Stokes-$\alpha$ equation. For the two dimensional Navier-Stokes-$\alpha$ equation with space periodic…
In recent literature several derivations of incompressible Navier-Stokes type equations that model the dynamics of an evolving fluidic surface have been presented. These derivations differ in the physical principles used in the modeling…
This paper discussed the existence and uniqueness of the smoothing solution of the Navier-Stokes equations. At first, we construct the theory of the linear equations which is about the unknown four variables functions with constant…
We construct a local in time spatially real-analytic solution to the 2D and 3D stochastic Navier--Stokes equation driven by a spatially real-analytic multiplicative and transport noise but emanating from an initial condition that is only…
In this paper, we improve some known uniqueness results of weak solutions for the 3D Navier-Stokes equations. The proof uses the Fourier localization technique and the losing derivative estimates.
Considering the stochastic Navier-Stokes system in $\mathbb{R}^d$ forced by a multiplicative white noise, we establish the local existence and uniqueness of the strong solution when the initial data take values in the critical space…
Consider the equations of Navier-Stokes in $\R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided the initial data is small with respect to the norm…
In this paper we establish a sharp non-uniqueness result for stochastic $d$-dimensional ($d\geq2$) incompressible Navier-Stokes equations. First, for every divergence free initial condition in $L^2$ we show existence of infinite many global…