Related papers: Spatial decay/asymptotics in the Navier-Stokes equ…
We show that the solutions to the non-stationary Navier-Stokes equations in $R^d$, $d=2,3$ which are left invariant under the action of discrete subgroups of the orthogonal group $O(d)$ decay much faster as $|x|\to\infty$ or $ t \to\infty$…
We prove that the Navier-Stokes initial value problem is well-posed in the logrithmically refined Besov spaces when the second index is not less than certain critical value, and ill-posed in such spaces when the second index is less than…
In this paper, we prove some results on theexistence and space-time decay rates of global strong solutions of the Cauchy problem for the Navier-Stokes equations in weighed $L^\infty(\mathbb R^d,|x|^\gamma{\rm dx})\cap L^\infty(\mathbb…
We show that the pressure associated with a distributional solution of the Navier-Stokes equations on the whole space satisfies a local expansion defined as a distribution if and only if the solution is mild. This gives a new perspective on…
In this paper, we establish the strong($H^1$) well-posedness of the two dimensional stochastic Navier-Stokes equation with multiplicative noise on moving domains. Due to the nonlocality effect, this equation exhibits a ``piecewise"…
We consider the Navier-Stokes equations in vorticity form in $\mathbb{R}^2$ with a white noise forcing term of multiplicative type, whose spatial covariance is not regular enough to apply the It\^o calculus in $L^q$ spaces, $1<q<\infty$. We…
The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for…
It is known that there has been classified for all $(-1)$-homogeneous axisymmetric no-swirl solutions of the three-dimensional Navier-Stokes equations with a possible singular ray. The main purpose of this paper is to show that the least…
We consider the three-dimensional incompressible Navier-Stokes equation on the whole space. We observe that this system admits a $L^\infty$ family of global spatial plane wave solutions, which are connected with the two-dimensional…
We prove existence of global regular axially-symmetric solutions to the Navier-Stokes equations in a cylindrical domain. We assume the periodic boundary conditions on the top and the bottom of the cylinder, but on the lateral part we assume…
This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted…
We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with $(-1)$-homogeneous initial data has a global scale-invariant solution which is smooth for positive times. Our main technical tools are…
We obtain a global existence result for the three-dimensional Navier-Stokes equations with a large class of initial data allowing growth at spatial infinity. Our work is a continuation of the results by T.-P. Tsai, Z. Bradshaw, I. Kukavica…
These notes are dedicated to the analysis of the one-dimensional free-congested Navier-Stokes equations. After a brief synthesis of the results obtained in [4] related to the existence and the asymptotic stability of partially congested…
We study the long-time behavior of infinite-energy solutions to the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. The initial data we consider are finite-energy perturbations…
New explicit solutions to the incompressible Navier-Stokes equations in $\mathbb{R}^{2}\setminus\left\{ \boldsymbol{0}\right\}$ are determined, which generalize the scale-invariant solutions found by Hamel. These new solutions are invariant…
The decaying speed of a single norm more truly reflects the intrinsic harmonic analysis structure of the solution of the classical incompressible Navier-Stokes equations. No previous work has been able to establish the well-posedness under…
We consider the asymptotic behavior as time goes to infinity of the $L^{2}$-norm of the velocity of the linearized compressible Navier-Stokes equations in ${\bf R}^{n}$ ($n \geq 2$). As an application we shall study the optimality of the…
In this paper, we investigate the convergence of the global large solution to its associated constant equilibrium state with an explicit decay rate for the compressible Navier-Stokes equations in three-dimensional whole space. Suppose the…
A few basic, intuitive, properties of the Navier-Stokes system of equations for incompressible fluid flows are discussed in this paper. We present a rephrased interpretation of the Navier-Stokes equation in a space having an arbitrary…