Keywords:
Differential equations, Partial.
;
Electronic books.
Description / Table of Contents:
This volume, derived from the 'PDEs in Fluid Mechanics' workshop held at the University of Warwick in 2016, serves to consolidate and advance work in mathematical fluid dynamics. Consisting of surveys and original research, it will be a valuable resource for both established researchers and graduate students seeking an overview of current developments.
Type of Medium:
Online Resource
Pages:
1 online resource (340 pages)
Edition:
1st ed.
ISBN:
9781316997031
Series Statement:
London Mathematical Society Lecture Note Series ; v.Series Number 452
URL:
https://ebookcentral.proquest.com/lib/geomar/detail.action?docID=5500403
DDC:
532
Language:
English
Note:
Cover -- Series information -- Title page -- Copyright information -- Table of contents -- List of contributors -- Preface -- 1 Remarks on recent advances concerning boundary effects and the vanishing viscosity limit of the Navier-Stokes equations -- Abstract -- 1.1 Introduction and uniform estimates -- 1.2 Kato criterion for convergence to the regular solution -- 1.3 Mathematical and physical interpretation of Theorem 1.3 -- 1.3.1 Recirculation -- 1.3.2 The Prandtl equations and the Stewartson triple-deck ansatz -- 1.3.3 Von Karman turbulent Layer -- 1.3.4 Energy limit and d'Alembert paradox -- 1.4 Kato's criterion, anomalous energy dissipation, and turbulence -- References -- 2 Time-periodic flow of a viscous liquid past a body -- Abstract -- 2.1 Introduction -- 2.2 Notation -- 2.3 Preliminaries -- 2.4 An Embedding Theorem -- 2.5 Linearized Problem -- 2.6 Fully Nonlinear Problem -- Acknowledgements -- References -- 3 The Rayleigh-Taylor instability in buoyancy-driven variable density turbulence -- Abstract -- 3.1 Background to the Rayleigh-Taylor instability -- 3.2 The 3D Cahn-Hilliard-Navier-Stokes equations -- 3.3 The variable density model for two incompressible miscible fluids -- 3.3.1 The mathematical model -- 3.3.2 The roles played by θ = ln ρ and ∇θ -- 3.3.3 Summary of the D[sub(m)]-method used for the Navier-Stokes equations -- 3.4 Some L[sup(2m)]-estimates on ∇θ and ω -- 3.4.1 Definitions -- 3.4.2 The evolution of D[sub(1,θ)] -- References -- 4 On localization and quantitative uniqueness for elliptic partial differential equations -- Abstract -- 4.1 Introduction -- 4.2 A lower bound for the decay of Δu = W∇u + V u -- 4.3 A construction of a localized solution -- 4.4 A construction of a solution vanishing of high order -- 4.5 The equation Δu = Vu -- Acknowledgments -- References -- 5 Quasi-invariance for the Navier-Stokes equations.
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5.1 Introduction -- 5.2 Navier-Stokes equations -- 5.3 Burgers equation -- 5.4 Use of critical dependent variables -- 5.5 Cole-Hopf transform and Feynman-Kac formula -- 5.6 Dynamic scaling transform -- 5.6.1 Change of probability measures -- 5.6.2 Leray equations -- 5.6.3 Navier-Stokes equations -- 5.7 Summary -- Appendix A Wiener process -- References -- 6 Leray's fundamental work on the Navier-Stokes equations: a modern review of "Sur le mouvement d'un liquide visqueux emplissant l'espace" -- Abstract -- 6.1 Introduction -- 6.1.1 Preliminaries -- 6.1.2 The Oseen kernel T -- 6.2 The Stokes equations -- 6.2.1 A general forcing F -- 6.2.2 A forcing of the form F = −(Y · ∇)Y -- Notes -- 6.3 Strong solutions of the Navier-Stokes equations -- 6.3.1 Properties of strong solutions -- 6.3.2 Local existence and uniqueness of strong solutions -- 6.3.3 Characterisation of singularities -- 6.3.4 Semi-strong solutions -- Notes -- 6.4 Weak solutions of the Navier-Stokes equations -- 6.4.1 Well-posedness for the regularised equations -- 6.4.2 Global existence of a weak solution -- 6.4.3 Structure of the weak solution -- Notes -- Acknowledgements -- 6.5 Appendix -- 6.5.1 The heat equation and the heat kernel -- 6.5.2 The extension of Young's inequality for convolutions -- 6.5.3 Decay estimates of P(x, t) -- 6.5.4 Properties of the Stokes equations -- 6.5.5 Integral inequalities -- 6.5.6 The Volterra equation -- 6.5.7 A proof of (6.85) without the use of the Lemma 6.4 -- 6.5.8 Smooth approximation of the forcing -- References -- 7 Stable mild Navier-Stokes solutions by iteration of linear singular Volterra integral equations -- Abstract -- 7.1 The initial-boundary value problem of the Navier-Stokes equations -- 7.2 Results on stability of Navier-Stokes solutions -- 7.3 Bounds on P(u · ∇v) and on e[sup(−tA)].
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7.4 The approximation schemes of Fujita-Kato and Giga-Miyakawa: sketch of the proof in Fujita & -- Kato (1964) for the uniform bound of the approximations -- 7.5 Stable mild Navier-Stokes solutions Theorems 7.4-7.6 -- 7.6 Basic results on linear singular Volterra integral equations -- 7.7 Proof of the theorems -- References -- 8 Energy conservation in the 3D Eulerequations on T[sup(2)] × R[sub(+)] -- Abstract -- 8.1 Introduction -- 8.2 Energy conservation without boundaries -- 8.2.1 Weak solutions of the Euler equations -- 8.2.2 Using u[sub(ε)] as a test function -- 8.2.3 'Mollifying the equation' -- 8.2.4 Energy Conservation -- 8.3 Two spatial conditions for energy conservation in the absence of boundaries -- 8.4 Energy Balance on T[sup(2)] × R[sub(+)] -- 8.4.1 Weak solutions of the Euler equations on D[sub(+)] := T[sup(2)] × R[sub(+)] -- 8.4.2 Half-plane reflection map -- 8.5 Energy Conservation on D[sub(+)] -- 8.6 Conclusion -- 8.7 Afterward: the result of Bardos & -- Titi on a general bounded domain -- Acknowledgements -- References -- 9 Regularity of Navier-Stokes flows with bounds for the velocity gradient along streamlines and an effective pressure -- Abstract -- 9.1 Introduction -- 9.2 Preliminaries -- 9.3 Results -- 9.4 Conclusion -- References -- 10 A direct approach to Gevrey regularity on the half-space -- Abstract -- 10.1 Introduction -- 10.2 Preliminaries -- 10.3 Derivative reduction -- 10.3.1 Normal derivative reduction -- 10.3.2 Tangential derivative reduction -- 10.3.3 The time-derivative reduction -- 10.4 Proof of Theorem 10.1 -- 10.4.1 The S[sub(1)] term -- 10.4.2 The S[sub(2)] term -- 10.4.3 The S[sub(3)] term -- 10.4.4 The S[sub(4)] term -- 10.4.5 The S[sub(5)] term -- 10.4.6 The S[sub(6)] term -- 10.4.7 Conclusion of the proof -- 10.5 Derivative reduction for the Stokes problem and the proof of Theorem 10.2.
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10.5.1 Normal derivative reduction for the Stokes operator -- 10.5.2 Tangential derivative reduction for the Stokes operator -- 10.5.3 Time derivative reduction for the Stokes operator -- Acknowledgments -- References -- 11 Weak-Strong Uniqueness in Fluid Dynamics -- Abstract -- 11.1 Introduction -- 11.2 The Relative Energy Method -- 11.3 Dissipative Measure-Valued Solutions -- 11.4 Dealing with Viscosity -- 11.4.1 Incompressible Navier-Stokes Equations -- 11.4.2 Compressible Navier-Stokes Equations -- 11.5 Physical Boundaries in the Inviscid Situation -- 11.6 An Alternative Approach -- References.
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