Related papers: Smooth or singular solutions to the Navier--Stokes…

We develop mathematical methods which allow us to study asymptotic properties of solutions to the three dimensional Navier-Stokes system for incompressible fluid in the whole three dimensional space. We deal either with the Cauchy problem…

This article examines the smoothness of the solution to the Navier-Stokes equation from a novel perspective. Here, the existence of the smoother solution relative to x and to the time t was shown only for a finite time. Moreover, for each…

This article studies the uniqueness of the weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case. Here, the investigation is provided using two different approaches. The first (the main) result is obtained…

In this article the question on uniqueness of weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case is studied. Here the investigation is carried out with use of another approach. The uniqueness of velocity…

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…

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…

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…

We analyze the forced incompressible stationary Navier-Stokes flow in $\mathbb{R}^n_+$, $n>2$. Existence of a unique solution satisfying a global integrabilty property measured in a scale of tent spaces is established for small data in…

For the N>=2 dimensional incompressible Naver-Stokes Equation, We have got its solution as a power series of time t, in which the coefficients are all known functions determined only by the initial velocity v0. We also prove that the…

We show that, for a given H\"older continuous curve in $\{(\gamma(t),t)\,:\, t>0\} \subset R^3\times R^+$, there exists a solution to the Navier-Stokes system for an incompressible fluid in $R^3$ which is smooth outside this curve and…

Consider the unforced incompressible homogeneous Navier-Stokes equations on the $d$-torus $\mathbb{T}^d$ where $d\geq 4$ is the space dimension. It is shown that there exist nontrivial steady-state weak solutions $u\in L^{2}(\mathbb{T}^d)$.…

We show the unique existence of solutions to stationary Navier-Stokes equations with small singular external forces belonging to a critical space. To the best of our knowledge, this is the largest critical space that is available up to now…

In this paper, we investigate the existence of a unique global smooth solution to the three-dimensional incompressible Navier-Stokes equations and provide a concise proof. We establish a new global well-posedness result that allows the…

We consider the Cauchy problem for the incompressible Navier-Stokes equations in dimension three and construct initial data in the critical space $BMO^{-1}$ from which there exist two distinct global solutions, both smooth for all $t>0$.…

Regularity properties of strong solutions are considered.

In this paper, we prove existence of smooth solutions of the Navier-Stokes equations that gives a positive answer to the problem proposed by Fefferman [3].

We discuss the regularity of solutions of 2D incompressible Navier-Stokes equations forced by singular forces. The problem is motivated by the study of complex fluids modeled by the Navier-Stokes equations coupled to a nonlinear…

In this paper we describe a method to derive solutions of the incompressible Navier- Stokes system of equations for non-stationary initial value problems in $\mathbb{R}^n$. We show that for a given smooth solenoidal initial velocity vector…

The paper is aimed at analysing a singular perturbation of the Navier-Stokes equations on a compact closed manifold. The case of compact smooth manifolds with boundary under the Dirichlet conditions is also included. Global existence and…

First, the solution uniqueness and existence of a stationary anisotropic (linear) Stokes system with constant viscosity coefficients in a compressible framework on $n$-dimensional flat torus are analysed in a range of periodic Sobolev…