Partial Differential Equations in Fluid Mechanics

doi:10.1017/9781108610575

The Euler and Navier–Stokes equations are the fundamental mathematical models of fluid mechanics, and their study remains central in the modern theory of partial differential equations. This volume of articles, derived from the workshop 'PDEs in Fluid Mechanics' held at the University of Warwick in 2016, serves to consolidate, survey and further advance research in this area. It contains reviews of recent progress and classical results, as well as cutting-edge research articles. Topics include Onsager's conjecture for energy conservation in the Euler equations, weak-strong uniqueness in fluid models and several chapters address the Navier–Stokes equations directly; in particular, a retelling of Leray's formative 1934 paper in modern mathematical language. The book also covers more general PDE methods with applications in fluid mechanics and beyond. This collection will serve as a helpful overview of current research for graduate students new to the area and for more established researchers.

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Partial Differential Equations in Fluid Mechanics

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Contributors
pp vii-x

Preface
pp xi-xii

1 - Remarks on recent advances concerning boundary effects and the vanishing viscosity limit of the Navier–Stokes equations
pp 1-19

2 - Time-periodic flow of a viscous liquid past a body
pp 20-49

3 - The Rayleigh–Taylor instability in buoyancy-driven variable density turbulence
pp 50-67

4 - On localization and quantitative uniqueness for elliptic partial differential equations
pp 68-96

5 - Quasi-invariance for the Navier–Stokes equations
pp 97-112

6 - Leray’s fundamental work on the Navier–Stokes equations: a modern review of “Sur le mouvement d’un liquide visqueux emplissant l’espace”
pp 113-203

7 - Stable mild Navier–Stokes solutions by iteration of linear singular Volterra integral equations
pp 204-223

8 - Energy conservation in the 3D Euler equations on T2 × R+
pp 224-251

9 - Regularity of Navier–Stokes flows with bounds for the velocity gradient along streamlines and an effective pressure
pp 252-267

10 - A direct approach to Gevrey regularity on the half-space
pp 268-288

11 - Weak-Strong Uniqueness in Fluid Dynamics
pp 289-326

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