Pressure moderation and effective pressure in Navier-Stokes flows

We study the Cauchy problem of the Navier–Stokes equations by both semi-analytic and classical energy methods. The former approach provides a physical picture of how viscous effects may or may not be able to suppress singularity development. In the latter approach, we examine the pressure term that drives the dynamics of the velocity norms ||u||Lq , for q ≥ 3. A key idea behind this investigation is due to the fact that the pressure p in this term is determined upto a function of both space and |u|, say Ƥ(x, |u|), which may assume relatively broad forms. This allows us to use Ƥ as a pressure moderator in the evolution equation for ||u||Lq , whereby optimal regularity criteria can be sought by varying Ƥ within its admissible classes. New regularity criteria are derived with and without making use of the moderator. The results obtained in the absence of the moderator feature some improvement over existing criteria in the literature. Several criteria are derived in terms of the moderated (effective) pressure p+Ƥ. A simple moderation scheme and the plausibility of the present approach to the problem of Navier–Stokes regularity are discussed.PostprintPeer reviewe

Velocity-pressure correlation in Navier-Stokes flows and the problem of global regularity

Funding: Yu is supported by an NSERC Discovery grant.Incompressible fluid flows are characterised by high correlations between low pressure and high velocity and vorticity. The velocity-pressure correlation is an immediate consequence of fluid acceleration towards low pressure regions. On the other hand, fluid converging to a low pressure centre is driven sideways by a resistance due to incompressibility, giving rise to the formation of a strong vortex, hence the vorticity-pressure correlation. Meanwhile, the formation of such a vortex effectively shields the low pressure centre from incoming energetic fluid. As a result, a local pressure minimum can usually be found at the centre of a vortex where the vorticity is greatest but the velocity is relatively low,hence the misalignment of local pressure minima and velocity maxima. For Navier--Stokes flows, this misalignment has profound implications on extreme momentum growth and maintenance of regularity. This study examines the role of the velocity-pressure correlation on the problem of Navier--Stokes global regularity. On the basis of estimates for flows locally satisfying the critical scaling of the Navier--Stokes system, a qualitative theory of this correlation is considered. The theory appears to be readily quantified, advanced and tested by theoretical, mathematical and numerical methods. Regularity criteria depending on the degree of the velocity-pressure correlation are presented and discussed in light of the above theory. The result suggests that as long as global pressure minimum (or minima) and velocity maximum (or maxima) are mutually exclusive, then regularity is likely to persist. This is the first result that makes use of an explicit measure of the velocity-pressure correlation as a key factor in the maintenance of regularity or development of singularity.PreprintPublisher PDFPeer reviewe

Regularity of Navier--Stokes flows with bounds for the pressure

This talk is concerned with global regularity of solutions of the Navier--Stokes equations. Let \Omega(t) \assign \{x:|u(x,t)| > c\norm{u}_{L^r}\}, for some $r\ge3$ and constant $c$ independent of $t$, with measure $|\Omega|$. It is shown that if \int_\Omega|p+\mathcal{P}|^{3/2}\mathd x becomes sufficiently small as $|\Omega|$ decreases, then \norm{u}_{L^{(r+6)/3}} decays and regularity is secured. Here $p$ is the physical pressure and $\mathcal{P}$ is a pressure moderator of relatively broad forms. The implications of the results are discussed and regularity criteria in terms of bounds for $|p+\mathcal{P}|$ within $\Omega$ are deduced. This is joint work with Xinwei Yu.Non UBCUnreviewedAuthor affiliation: University of St. AndrewsFacult

Regularity of Navier--Stokes flows with bounds for the pressure

This paper was presented at the Warwick EPSRC Symposium on PDEs in Fluid Mechanics, September 2016. Part of this research was carried out when CVT was visiting the University of Alberta, whose hospitality is gratefully acknowledged. XY was partially supported by NSERC Discovery grant RES0020476This study derives regularity criteria for solutions of the Navier–Stokes equations. Let Ω(t) := {x : |u(x, t)| > c ||u||Lr(R3) }, for some r ≥ 3 and constant c independent of t, with measure |Ω|. It is shown that if ||p + P||L3/2(Ω) becomes sufficiently small as |Ω| decreases, then||u||L(r+6)/3(R3) decays and regularity is secured. Here p is the physical pressure and P is a pressure moderator of relatively broad forms. The implications of the results are discussed and regularity criteria in terms of bounds for |p + P| within Ω are deduced.PostprintPeer reviewe