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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 present a study by computer simulations of a class of complex-valued solutions of the three-dimensional Navier-Stokes equations in the whole space, which, according to Li and Sinai, present a blow-up (singularity) at a finite time. The…
We consider the motion of an axisymmetric column of Navier-Stokes fluid with a free surface. Due to surface tension, the thickness of the fluid neck goes to zero in finite time. After the singularity, the fluid consists of two halves, which…
We investigate the hydrostatic approximation of a hyperbolic version of Navier-Stokes equations, which is obtained by using Cattaneo type law instead of Fourier law, evolving in a thin strip $\R\times (0,\varepsilon)$. The formal limit of…
We note that the equations of relativistic hydrodynamics reduce to the incompressible Navier-Stokes equations in a particular scaling limit. In this limit boundary metric fluctuations of the underlying relativistic system turn into a…
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…
We prove geometrically improved version of Prodi-Serrin type blow-up criterion. Let $v$ and $\omega$ be the velocity and the vorticity of solutions to the 3D Navier-Stokes equations and denote $\{f\}_+=\max\{f, 0\}$ , $Q_T=\Bbb R^3\times…
We consider complex-valued solutions of the three-dimensional Navier-Stokes system without external forcing on $R^3$. We show that there exists an open set in the space of 10-parameter families of initial conditions such that for each…
This paper deals with the finite-time blow-up phenomena of classical solutions for Vlasov/Navier-Stokes equations under suitable assumptions on the initial configurations. We show that a solution to the coupled kinetic-fluid system may be…
We study an initial boundary value problem for the 3-dimensional compressible Navier-Stokes equations with hyperbolic heat conduction, where the classical Fourier law is replaced by the Cattaneo-Christov constitutive relation. We focus on…
In this work we investigate the question of preventing the three-dimensional, incompressible Navier-Stokes equations from developing singularities, by controlling one component of the velocity field only, in space-time scale invariant…
We study the Navier-Stokes system describing the motion of a compressible viscous fluid driven by a nonlinear multiplicative stochastic force. We establish local in time existence (up to a positive stopping time) of a unique solution, which…
In this article, we study the solutions of the damped Navier--Stokes equation with Navier boundary condition in a bounded domain $\Omega$ in $\mathbb{R}^3$ with smooth boundary. The existence of the solutions is global with the damped term…
We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\|r^{1-\frac3q}u\|_{L_t^\infty…
In this paper, we establish the existence and uniqueness of local strong solutions to the kinetic Cucker--Smale model coupled with the isentropic compressible Navier--Stokes equation in the whole space. Moreover, the blowup mechanism for…
We discuss the applicability of a unified hyperbolic model for continuum fluid and solid mechanics to modeling non-Newtonian flows and in particular to modeling the stress-driven solid-fluid transformations in flows of viscoplastic fluids,…
A forced solution $v$ of the Navier-Stokes equation in any open domain with no slip boundary condition is constructed. The scaling factor of the forcing term is the critical order $-2$. The velocity, which is smooth until its final blow up…
Systems of hydrodynamic type equations derived from the Navier-Stokes equations and the boundary layer equations are considered. A transformation of the Crocco type reducing the equation order for the longitudinal velocity component is…
We consider the hyperbolic version of three-dimensional anisotropic Naver-Stokes equations in a thin strip and its hydrostatic limit that is a hyperbolic Prandtl type equations. We prove the global-in-time existence and uniqueness for the…
We study the strong solution to the 3-D compressible Navier--Stokes equations. We propose a new blow up criterion for barotropic gases in terms of the integral norm of density $\rho$ and the divergence of the velocity $\bu$ without any…