Leray Weak Solutions Research Articles

Intermittency in solutions of the three-dimensional Navier–Stokes equations

Dissipation-range intermittency was first observed by Batchelor & Townsend (1949) in high Reynolds number turbulent flows. It typically manifests itself in spatio-temporal binary behaviour which is characterized by long, quiescent periods in the signal which are interrupted by short, active ‘events’ during which there are large excursions away from the average. It is shown that Leray's weak solutions of the three-dimensional incompressible Navier–Stokes equations can have this binary character in time. An estimate is given for the widths of the short, active time intervals, which decreases with the Reynolds number. In these ‘bad’ intervals singularities are still possible. However, the average width of a ‘good’ interval, where no singularities are possible, increases with the Reynolds number relative to the average width of a bad interval.

Blow–up rate estimates for weak solutions of the Navier—Stokes equations

The interior regularity problem for the Leray weak solutions u of the Navier–Stokes equations in a domain Ω ⊂ Rn with n ⩾ 3 is investigated. It is shown that u is regular in a neighbourhood of a point ( x , t ) ∈ Omega; × (0, T ) if there exist constants 0 ⩽ θ < 1 and small ∈ 0 such that with Q 1/ k i ( x , t ) = { x ∈ R n ; | x – x < 1/ k } ( t –1/ k 2, t + 1/ k 2). If ( x , t ) is an irregular point of u , there exists a sequence of non–zero measure sets E ki ⊂ Q 1/ ki ( x , t ) for i = 1,2,..., such that the blow–up rate estimate holds.

Navier-Stokes equations and area of interfaces

We present new a priori estimates for the vorticity of solutions of the three dimensional Navier-Stokes equations. These estimates imply that theL 1 norm of the vorticity is a priori bounded in time and that the time average of the 4/(3+e) power of theL 4/(3+e) spatial norm of the gradient of the vorticity is a priori bounded. Using these bounds we construct global Leray weak solutions of the Navier-Stokes equations which satisfy these inequalities. In particular it follows that vortex sheet, vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the Navier-Stokes equations. Next we apply these inequalities in conjunction with geometric measure theoretical arguments to study the two dimensional Hausdorff measure of level sets of the vorticity magnitude. We obtain a priori bounds on an average such measure, . When expressed in terms of the Reynolds number and the Kolmogorov dissipation length η, these bounds are $$\left\langle \mu \right\rangle \leqq \frac{{L^3 }}{\eta }\left( {1 + \operatorname{Re} ^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,$$ The right-hand side of this inequality has a simple geometrical interpretation: it represents the area of a union of non-overlapping spheres of radii η which fill a fraction of the spatial domain. As the Reynolds number increases, this fraction decreases. We study also the area of level sets of scalars and in particular isotherms in Rayleigh-Benard convection. We define a quantity, 〈μ〉 r, t (x 0) describing an average value of the area of a portion of a level set contained in a small ball of radiusr about the pointx 0. We obtain the inequality $$\left\langle \mu \right\rangle _{r,t} \left( {x_0 } \right) \leqq C_\kappa ,^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}_r {5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}} \left\langle {v\left( {x_0 } \right)} \right\rangle ^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ,$$ wherek is the diffusivity constant and is the velocity. The inequality is valid forr small but larger than a small scale λ=k −1. In the case of turbulent velocities this scale is smaller than the smallest significant physical scale, suggesting that 2.5 is a lower bound for the fractal dimension of an (ensemble) average interface in turbulent flow, a fact which agrees with the experimental lower bound of 2.35. We define a similar quantity 〈μ〉 δ, t representing an average value of the area of a portion of a level set contained in the region of spaceD δ={x∈D; dist (x, δD)>δ}, whereD is the domain of aspect ratio of order one and diameterL, where the convection takes place. We obtain the inequality $$\left\langle \mu \right\rangle _{\delta ,t} \leqq C\left( {L^3 \delta ^{ - 1} + L^{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-\nulldelimiterspace} 2}} \delta ^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} Ra^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right),$$ where Ra is the Rayleigh number.