{"title":"Stable mild Navier–Stokes solutions by iteration of linear singular Volterra integral equations","authors":"","doi":"10.1017/9781108610575.008","DOIUrl":"https://doi.org/10.1017/9781108610575.008","url":null,"abstract":"","PeriodicalId":328236,"journal":{"name":"Partial Differential Equations in Fluid Mechanics","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128016001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Energy conservation in the 3D Euler equations on T2 × R+","authors":"","doi":"10.1017/9781108610575.009","DOIUrl":"https://doi.org/10.1017/9781108610575.009","url":null,"abstract":"","PeriodicalId":328236,"journal":{"name":"Partial Differential Equations in Fluid Mechanics","volume":"381 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115993854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Rayleigh–Taylor instability in buoyancy-driven variable density turbulence","authors":"","doi":"10.1017/9781108610575.004","DOIUrl":"https://doi.org/10.1017/9781108610575.004","url":null,"abstract":"","PeriodicalId":328236,"journal":{"name":"Partial Differential Equations in Fluid Mechanics","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117144535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A direct approach to Gevrey regularity on the half-space","authors":"I. Kukavica, V. Vicol","doi":"10.1017/9781108610575.011","DOIUrl":"https://doi.org/10.1017/9781108610575.011","url":null,"abstract":"We consider the inhomogeneous heat and Stokes equations on the half space and prove an instantaneous space-time analytic regularization result, uniformly up to the boundary of the half space. February 19, 2018","PeriodicalId":328236,"journal":{"name":"Partial Differential Equations in Fluid Mechanics","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133071033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Remarks on recent advances concerning boundary effects and the vanishing viscosity limit of the Navier–Stokes equations","authors":"","doi":"10.1017/9781108610575.002","DOIUrl":"https://doi.org/10.1017/9781108610575.002","url":null,"abstract":"","PeriodicalId":328236,"journal":{"name":"Partial Differential Equations in Fluid Mechanics","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124099109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quasi-invariance for the Navier–Stokes equations","authors":"K. Ohkitani","doi":"10.1017/9781108610575.006","DOIUrl":"https://doi.org/10.1017/9781108610575.006","url":null,"abstract":"In this contribution we focus on a few results regarding the study of the three-dimensional Navier-Stokes equations with use of the vector potentials. These dependent variables are critical in the sense that they are scale-invariant. By surveying recent results utilising criticality of various norms, we emphasise the advantages of working with scale-invariant variables. The Navier-Stokes equations, which are invariant under static scaling transforms, are not invariant under dynamic scaling transforms. Using the vector potential, we introduce scale-invariance in a weaker form, that is, invariance under dynamic scaling modulo a martingale (Maruyama-Girsanov density) when the equations are cast into Wiener path-integrals. We briefly discuss the implications of this quasi-invariance on the basic issues of the Navier-Stokes equations.","PeriodicalId":328236,"journal":{"name":"Partial Differential Equations in Fluid Mechanics","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126445762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Regularity of Navier–Stokes flows with bounds for the velocity gradient along streamlines and an effective pressure","authors":"","doi":"10.1017/9781108610575.010","DOIUrl":"https://doi.org/10.1017/9781108610575.010","url":null,"abstract":"","PeriodicalId":328236,"journal":{"name":"Partial Differential Equations in Fluid Mechanics","volume":"63 5-6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132285967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Leray’s fundamental work on the Navier–Stokes equations: a modern review of “Sur le mouvement d’un liquide visqueux emplissant l’espace”","authors":"W. S. O.za'nski, Benjamin C. Pooley","doi":"10.1017/9781108610575.007","DOIUrl":"https://doi.org/10.1017/9781108610575.007","url":null,"abstract":"This article offers a modern perspective which exposes the many contributions of Leray in his celebrated work on the Navier--Stokes equations from 1934. Although the importance of his work is widely acknowledged, the precise contents of his paper are perhaps less well known. The purpose of this article is to fill this gap. We follow Leray's results in detail: we prove local existence of strong solutions starting from divergence-free initial data that is either smooth, or belongs to $H^1$, $L^2cap L^p$ (with $pin(3,infty]$), as well as lower bounds on the norms $| nabla u (t) |_2$, $| u(t) |_p$ ($pin(3,infty]$) as $t$ approaches a putative blow-up time. We show global existence of a weak solution and weak-strong uniqueness. We present Leray's characterisation of the set of singular times for the weak solution, from which we deduce that its upper box-counting dimension is at most $tfrac{1}{2}$. Throughout the text we provide additional details and clarifications for the modern reader and we expand on all ideas left implicit in the original work, some of which we have not found in the literature. We use some modern mathematical tools to bypass some technical details in Leray's work, and thus expose the elegance of his approach.","PeriodicalId":328236,"journal":{"name":"Partial Differential Equations in Fluid Mechanics","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125122619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Weak-Strong Uniqueness in Fluid Dynamics","authors":"E. Wiedemann","doi":"10.1017/9781108610575.012","DOIUrl":"https://doi.org/10.1017/9781108610575.012","url":null,"abstract":"We give a survey of recent results on weak-strong uniqueness for compressible and incompressible Euler and Navier-Stokes equations, and also make some new observations. The importance of the weak-strong uniqueness principle stems, on the one hand, from the instances of non-uniqueness for the Euler equations exhibited in the past years; and on the other hand from the question of convergence of singular limits, for which weak-strong uniqueness represents an elegant tool.","PeriodicalId":328236,"journal":{"name":"Partial Differential Equations in Fluid Mechanics","volume":"212 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129433418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}