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The rigorous mathematical theory of the Navier-Stokes and Euler equations has been a focus of intense activity in recent years. This volume, the product of a workshop in Venice in 2013, consolidates, surveys and further advances the study of these canonical equations. It consists of a number of reviews and a selection of more traditional research articles on topics that include classical solutions to the 2D Euler equation, modal dependency for the 3D Navier-Stokes equation, zero viscosity Boussinesq equations, global regularity and finite-time singularities, well-posedness for the diffusive Burgers equations, and probabilistic aspects of the Navier-Stokes equation. The result is an accessible summary of a wide range of active research topics written by leaders in their field, together with some exciting new results. The book serves both as a helpful overview for graduate students new to the area and as a useful resource for more established researchers.

Differential equations, Partial. --- Navier-Stokes equations. --- Lagrange equations. --- D'Alembert equation --- Equations, Euler-Lagrange --- Equations, Lagrange --- Euler-Lagrange equations --- Lagrangian equations --- Differential equations --- Equations of motion --- Equations, Navier-Stokes --- Differential equations, Partial --- Fluid dynamics --- Viscous flow --- Partial differential equations

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The principle of least action originates in the idea that, if nature has a purpose, it should follow a minimum or critical path. This simple principle, and its variants and generalizations, applies to optics, mechanics, electromagnetism, relativity, and quantum mechanics, and provides an essential guide to understanding the beauty of physics. This unique text provides an accessible introduction to the action principle across these various fields of physics, and examines its history and fundamental role in science. It includes - with varying levels of mathematical sophistication - explanations from historical sources, discussion of classic papers, and original worked examples. The result is a story that is understandable to those with a modest mathematical background, as well as to researchers and students in physics and the history of physics.

Least action. --- Variational principles. --- Mechanics. --- Lagrange equations. --- Hamilton-Jacobi equations. --- Equations, Hamilton-Jacobi --- Equations, Jacobi-Hamilton --- Jacobi-Hamilton equations --- Calculus of variations --- Differential equations, Partial --- Hamiltonian systems --- Mechanics --- D'Alembert equation --- Equations, Euler-Lagrange --- Equations, Lagrange --- Euler-Lagrange equations --- Lagrangian equations --- Differential equations --- Equations of motion --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Extremum principles --- Minimal principles --- Variation principles --- Variational principles

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Differential equations --- Classical mechanics. Field theory --- Differentiable dynamical systems --- Lagrange equations --- Variational principles --- Extremum principles --- Minimal principles --- Variation principles --- Calculus of variations --- D'Alembert equation --- Equations, Euler-Lagrange --- Equations, Lagrange --- Euler-Lagrange equations --- Lagrangian equations --- Equations of motion --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics --- Lagrange equations. --- Lagrange, Équations de --- Differentiable dynamical systems. --- Systèmes dynamiques --- Variational principles. --- Principes variationnels --- Lagrange, Équations de. --- Systèmes dynamiques. --- Principes variationnels.

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This book uses the Lagrangian approach, especially useful and convenient for studying large-scale transport and mixing in the ocean, to present a detailed view of ocean circulation. This approach focuses on simulations and on monitoring the trajectories of fluid particles, which are governed by advection equations. The first chapter of the book is devoted to dynamical systems theory methods, which provide the framework, methodology and key concepts for the Lagrangian approach. The book then moves on to an analysis of chaotic mixing and cross-stream transport in idealized models of oceanic meandering currents like the Gulfstream in the Atlantic, the Kuroshio in the Pacific, and Antarctic Circumpolar Current, after which the current state of physical oceanography is reviewed. The latter half of the book applies the techniques and methods already described in order to study eddies, currents, fronts and large-scale mixing and transport in the Far-Eastern seas and the north-western part of the Pacific Ocean. Finally, the book concludes with a discussion of Lagrangian simulation and monitoring of water contamination after the Fukushima disaster of 2011. The propagation of Fukushima-derived radionuclides, surface transport across the Kuroshio Extension current, and the role of mesoscale eddies in the transport of Fukushima-derived cesium isotopes in the ocean are examined, and a comparison of simulation results with actual measurements are presented. Written by some of the world leaders in the application of Lagrangian methods in oceanography, this title will be of benefit to the oceanographic community by presenting the necessary background of the Lagrangian approach in an accessible manner.

Physics. --- Oceanography. --- Fluids. --- Environmental monitoring. --- Environmental sciences. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Fluid- and Aerodynamics. --- Monitoring/Environmental Analysis. --- Math. Appl. in Environmental Science. --- Ocean currents --- Lagrange equations. --- Navier-Stokes equations. --- Mathematical models. --- Equations, Navier-Stokes --- D'Alembert equation --- Equations, Euler-Lagrange --- Equations, Lagrange --- Euler-Lagrange equations --- Lagrangian equations --- Currents, Oceanic --- Differential equations, Partial --- Fluid dynamics --- Viscous flow --- Differential equations --- Equations of motion --- Ocean circulation --- Water currents --- Ocean surface topography --- Environmental science --- Science --- Oceanography, Physical --- Oceanology --- Physical oceanography --- Thalassography --- Earth sciences --- Marine sciences --- Ocean --- Statistical physics. --- Biomonitoring (Ecology) --- Ecological monitoring --- Environmental quality --- Monitoring, Environmental --- Applied ecology --- Environmental engineering --- Pollution --- Hydraulics --- Mechanics --- Physics --- Hydrostatics --- Permeability --- Mathematical statistics --- Measurement --- Monitoring --- Statistical methods