A note on statistical solutions of the three-dimensional Navier–Stokes equations: The time-dependent case

Time-dependent statistical solutions of the three-dimensional Navier–Stokes equations for incompressible fluids are considered. They are a mathematical formalization of the notion of ensemble averages in turbulence theory and form the backbone for a mathematical foundation of the theory of turbulence. The two main notions of statistical solutions, previously introduced, are revisited and a new formulation of one of them is given. An existence proof for this new formulation is given, along with a number of useful properties.     

Self-similar solutions of stationary Navier–Stokes equations

In this paper, we mainly study the existence of self-similar solutions of stationary Navier–Stokes equations for dimension $n=3,4$. For $n=3$, if the external force is axisymmetric, scaling invariant, $C^{1,\alpha }$ continuous away from the origin and small enough on the sphere $S^2$, we shall prove that there exists a family of axisymmetric self-similar solutions which can be arbitrarily large in the class $C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$. Moreover, for axisymmetric external forces without swirl, corresponding to this family, the momentum flux of the flow along the symmetry axis can take any real number. However, there are no regular ($U\in C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$) axisymmetric self-similar solutions provided that the external force is a large multiple of some scaling invariant axisymmetric F which cannot be driven by a potential. In the case of dimension 4, there always exists at least one self-similar solution to the stationary Navier–Stokes equations with any scaling invariant external force in $L^{4/3,\infty }({\mathbb{R}}^4)$.     

On symmetric Leray solutions of the stationary Navier–Stokes equations

A classical result of Amick (Acta Math 161:71–130, 1988) on the nontriviality of the symmetric Leray solutions of the steady-state Navier–Stokes equations in the plane is extended to Lipschitz domains. This results is compared with the famous Stokes paradox of linearized hydrodynamics and applied to a mixed problem of some interest in the applications.     

On very weak solutions of the stationary Navier–Stokes equations

We study the stationary Navier–Stokes equations in a bounded domain Ω of R ³ with smooth connected boundary. The notion of very weak solutions has been introduced by Marušić-Paloka (Appl. Math. Optim. 41:365–375, 2000), Galdi et al. (Math. Ann. 331:41–74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117–152, 2009) to obtain solvability results for the Navier–Stokes equations with very irregular data. In this article, we prove a complete solvability result which unifies those in Marušić-Paloka (Appl. Math. Optim. 41:365–375, 2000), Galdi et al. (Math. Ann. 331:41–74, 2005) and Kim (Arch. Ration. Mech. Anal. 193:117–152, 2009) by adapting the arguments in Choe and Kim (Preprint) and Kim and Kozono (Preprint).     

Stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations

This paper is devoted to the investigation of stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations. For a Leray weak solution of the Navier–Stokes equations in a critical Besov space, it is shown that the Leray weak solution is uniformly stable with respect to a small perturbation of initial velocity and external forcing. If the perturbation is not small, the perturbed weak solution converges asymptotically to the original weak solution as the time tends to the infinity. Additionally, an energy equality and weak–strong uniqueness for the three-dimensional Navier–Stokes equations are derived. The findings are mainly based on the estimations of the nonlinear term of the Navier–Stokes equations in a Besov space framework, the use of special test functions and the energy estimate method.     

Existence of stationary solutions of the Navier–Stokes equations in the presence of a wall

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here, we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.     

On uniqueness of stationary solutions to the Navier–Stokes equations in exterior domains

The aim of this paper is to prove a uniqueness criterion for solutions to the stationary Navier–Stokes equation in 3-dimensional exterior domains within the class $u\in L_{3,\infty }$ with $?u\in L_{3/2,\infty }$, where $L_{3,\infty }$ and $L_{3/2,\infty }$ are the Lorentz spaces. Our criterion asserts that if $u$ and $v$ are the solutions, $u$ is small in $L_{3,\infty }$ and $u,v\in L_p$ for some $p>3$, then $u=v$. The proof is based on analysis of the dual equation with the aid of the bootstrap argument.     

Trajectory statistical solutions for the 3D Navier–Stokes equations: The trajectory attractor approach

In this article, we construct the trajectory statistical solution for the 3D incompressible Navier–Stokes equations via the natural translation semigroup and trajectory attractor. In our construction, the trajectory statistical solution is an invariant space–time probability measure which is carried by the trajectory attractor of the natural translation semigroup defined on the trajectory space, and the trajectory statistical solution possesses the invariant property under the acting of the translation semigroup.     

A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier–Stokes equations

The partial regularity of the suitable weak solutions to the Navier–Stokes equations in Rⁿ${\mathbb{R}}^n$ with n=2,3,4$n=2,3,4$ and the stationary Navier–Stokes equations in Rⁿ${\mathbb{R}}^n$ for n=2,3,4,5,6$n=2,3,4,5,6$ are investigated in this paper. Using some elementary observation of these equations together with De Giorgi iteration method, we present a unified proof on the results of Caffarelli, Kohn and Nirenberg [1], Struwe [17], Dong and Du [5], and Dong and Strain [7]. Particularly, we obtain the partial regularity of the suitable weak solutions to the 4d non-stationary Navier–Stokes equations, which improves the previous result of [5], where Dong and Du studied the partial regularity of smooth solutions of the 4d Navier–Stokes equations at the first blow-up time.     

Remark on the stability of the large stationary solutions to the Navier–Stokes equations under the general flux condition

Consider stationary weak solutions of the Navier–Stokes equations in a bounded domain in R³${\mathbb{R}}^3$ under the nonhomogeneous boundary condition. We give a new approach for the stability of the stationary flow in the L²$L^2$-framework. Furthermore, we give some examples of stable solutions which may be large in L³(Ω)$L^3\left(\Omega \right)$ or W^1,3/2(Ω)$W^{1,3/2}\left(\Omega \right)$.     

Downscaling data assimilation algorithm with applications to statistical solutions of the Navier–Stokes equations

Based on a previously introduced downscaling data assimilation algorithm, which employs a nudging term to synchronize the coarse mesh spatial scales, we construct a determining map for recovering the full trajectories from their corresponding coarse mesh spatial trajectories, and investigate its properties. This map is then used to develop a downscaling data assimilation scheme for statistical solutions of the two-dimensional Navier–Stokes equations, where the coarse mesh spatial statistics of the system is obtained from discrete spatial measurements. As a corollary, we deduce that statistical solutions for the Navier–Stokes equations are determined by their coarse mesh spatial distributions. Notably, we present our results in the context of the Navier–Stokes equations; however, the tools are general enough to be implemented for other dissipative evolution equations.     

On the existence of singular solutions of the stationary Navier–Stokes problem

We consider the steady Navier–Stokes equations in the punctured regions (?) Ω?=?Ω ₀ \ {o} (with {o} ∈ Ω ₀) and (??) $ \varOmega ={{\mathbb{R}}^2}\backslash \left( {{{\overline{\varOmega}}_0}\cup \left\{ o \right\}} \right) $ (with $ \left\{ o \right\}\notin {{\overline{\varOmega}}_0} $ ), where Ω ₀ is a simple connected Lipschitz bounded domain of $ {{\mathbb{R}}^2} $ . We regard o as a sink or a source in the fluid. Accordingly, we assign the flux $ \mathcal{F} $ through a small circumference surrounding o and a boundary datum a on Γ?=??Ω ₀ such that the total flux $ \mathcal{F}+\int\nolimits_{\varGamma } {\boldsymbol{a}\cdot \boldsymbol{n}} $ is zero in case (?). We prove that if $ \left| \mathcal{F} \right|<2\pi \nu $ and $ \left| \mathcal{F} \right|+\left| {\int\nolimits_{\varGamma } {\boldsymbol{a}\cdot \boldsymbol{n}} } \right|<2\pi \nu $ in (?) and (??), respectively, where ν is the kinematical viscosity, then the problem has a C ^∞ solution in Ω, which behaves at o like the gradient of the fundamental solution of the Laplace equation.