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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…
In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus $\mathbb{T}^d$ with $d \geq 2$. The generalized Navier-Stokes equations is obtained by replacing the standard Laplacian…
In this paper, we establish global strong solutions for arbitrarily large initial data to the 2D and 3D compressible Navier-Stokes-Korteweg system, also referred to as the quantum Navier-Stokes equations, originally derived by Dunn and…
In this paper we investigate well-posedness of the Cauchy problem of the three dimensional generalized Navier-Stokes system. We first establish local well-posedness of the GNS system for any initial data in the Fourier-Herz space…
We study the Navier-Stokes equations with transport noise in critical function spaces. Assuming the initial data belongs to $H^{1/2}$ almost surely, we establish the existence and uniqueness of a local-in-time probabilistically strong…
In this paper, we construct a class of global large solution to the compressible Navier-Stokes equations in the whole space $\R^d$. Precisely speaking, our choice of special initial data whose $\dot{B}^{-1}_{\infty,\infty}$ norm can be…
We construct a family of smooth initial data for the Navier-Stokes equations, bounded in $BMO^{-1}(\mathbb T^3)$, that gives rise to arbitrarily large global solutions. As a consequence, we rule out various hypothetical a priori estimates…
We give a simple proof of the existence of initial data with minimal $L^3$ norm for potential Navier-Stokes singularities, recently established in (arXiv:1012.0145v2) with techniques based on profile decomposition. Our method is based on…
In this paper, we first prove the global well-posedness of 3-D anisotropic Navier-Stokes system provided that the vertical viscous coefficient of the system is sufficiently large compared to some critical norm of the initial data. Then we…
We prove an $\epsilon$-regularity criterion for the 3D Navier-Stokes equations in terms of initial data. It shows that if a scaled local $L^2$ norm of initial data is sufficiently small around the origin, a suitable weak solution is regular…
One proves the existence and uniqueness in $(L^p(\mathbb{R}^3))^3$, $\frac{3}{2}<p<2$, of a global mild solution to random vorticity equations associated to stochastic $3D$ Navier-Stokes equations with linear multiplicative Gaussian noise…
In this paper, we investigate both deterministic and stochastic 2D Navier Stokes equations with anisotropic viscosity. For the deterministic case, we prove the global well-posedness of the system with initial data in the anisotropic Sobolev…
This paper is concerned with the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier-Stokes system describing the one-dimensional motion of a viscous…
In this paper, we derive a new smallness hypothesis of initial data for the three-dimensional incompressible Navier-Stokes equations. That is, we prove that there exist two positive constants $c_0,C_0$ such that if \begin{equation*}…
By using the continuous induction method, we prove that the initial value problem of the three dimensional Navier-Stokes equations is globally well-posed in $L^p(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$ for any $3<p<\infty$. The proof is rather…
We address the global-in-time existence and pathwise uniqueness of solutions for the stochastic incompressible Navier-Stokes equations with a multiplicative noise on the three-dimensional torus. Under natural smallness conditions on the…
We find a global a priori estimate for solutions to the Navier-Stokes equations with periodic boundary conditions guaranteeing in view of the Serrin type condition the existence of global regular solutions. We derive the following estimate…
In this paper, we investigate the global well-posedness of 3-D incompressible inhomogeneous Navier-Stokes equations with ill-prepared large initial data which are slowly varying in one space variable, that is, initial data of the form…
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…
A sufficient condition of regularity for solutions to the Navier-Stokes equations is proved. It generalizes the so-called $L_{3,\infty}$-case.