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.     

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 the uniqueness of weak solutions for the 3D Navier–Stokes equations

In this paper, we improve some known uniqueness results of weak solutions for the 3D Navier–Stokes equations. The proof uses the Fourier localization technique and the losing derivative estimates.     

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.     

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.     

On local regularity for suitable weak solutions of the Navier–Stokes equations near the boundary

A class of sufficient conditions for local boundary regularity of suitable weak solutions of nonstationary three-dimensional Navier–Stokes equations is discussed. The corresponding results are stated in terms of functionals, which are invariant with respect to the scaling of the Navier–Stokes equations. Bibliography: 27 titles.     

On partial regularity of suitable weak solutions to the stationary fractional Navier–Stokes equations in dimension four and five

In this paper, we investigate the partial regularity of suitable weak solutions to the multidimensional stationary Navier Stokes equations with fractional power of the Laplacian (-△)~α 1 and α≠ 1/2). It is shown that the n + 2-6α(3 ≤ n ≤ 5) dimensional Hausdorff measure of the set of the possible singular points of suitable weak solutions to the system is zero, which extends a recent result of Tang and Yu [19] to four and five dimension. Moreover, the pressure in e-regularity criteria is an improvement of corresponding results in [1, 13, 18, 20].     

On Landau’s solutions of the Navier–Stokes equations

A special class of solutions of the n-dimensional steady-state Navier–Stokes equations is considered. Bibliography: 23 titles.     

On partial regularity of suitable weak solutions to the Navier–Stokes equations in unbounded domains

Consider the nonstationary Navier–Stokes equations in Ω × (0, T), where Ω is a general unbounded domain with non-compact boundary in R ³. We prove the regularity of suitable weak solutions for large |x|. It should be noted that our result also holds near the boundary. Our result extends the previous ones by Caffarelli–Kohn–Nirenberg in R ³ and Sohr-von Wahl in exterior domains to general domains.     

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.     

Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

Existence of weak solutions for the generalized Navier–Stokes equations with damping

In this work we consider the generalized Navier–Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier–Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any ${q > \frac{2N}{N+2}}$ and any σ > 1, where q is the exponent of the diffusion term and σ is the exponent which characterizes the damping term.     

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.     

Local regularity for suitable weak solutions of Navier–Stokes equations near the boundary

A class of sufficient conditions for the local boundary regularity of suitable weak solutions of nonstationary three-dimensional Navier–Stokes equations is discussed. The corresponding results are stated in terms of functionals invariant with respect to the scaling of Navier–Stokes equations. Bibliography: 26 titles.     

Regularity of weak solutions to 3D incompressible Navier–Stokes equations

In this paper, we establish some new local and global regularity properties for weak solutions of 3D non-stationary Navier–Stokes equations in the class of L ^r (0, T ; L ³(Ω)) with ${r \in [1, \infty)}In this paper, we establish some new local and global regularity properties for weak solutions of 3D non-stationary Navier–Stokes equations in the class of L ^r (0, T ; L ³(Ω)) with r ? [1, ￥){r \in [1, \infty)} , which are beyond Serrin’s condition.     

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.