Pressure moderation and effective pressure in Navier–Stokes flows. (17th August 2016)

• Record Type: Journal Article
• Authors: Tran, Chuong V; Yu, Xinwei
• Is Part Of: Nonlinearity. Volume 29:Number 10(2016:Oct.)
• Page Start: 2990
• Page End: 3005
• Publication Date: 2016-08-17
• Subjects: Navier–Stokes equations -- pressure moderation -- effective pressure -- global regularity; 76D03 -- 76D05; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/10/2990
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: 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 ∥ L q, for q ⩾ 3 . A key idea behind this investigation is due to the fact that the pressure p in this term is determined up to a function of both space and | u |, say P ( x, | u | ), which may assume relatively broad forms. This allows us to use P as a pressure moderator in the evolution equation for ∥ u ∥ L q, whereby optimal regularity criteria can be sought by varying P 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 + P . A simple moderation scheme and the plausibility of the present approach to the problem of Navier–Stokes regularity are discussed.