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The authors consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of the authors' analysis is to develop a sufficiently robust method (the "quasilinear I-method") which allows the authors to deal with strong singularities arising from time resonances in the applications of the normal form method (the so-called "division problem"). As a result, they are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions. Part of the authors' analysis relies on a new treatment of the Dirichlet-Neumann operator in dimension two which is of independent interest. As a consequence, the results in this paper are self-contained.

Waves --- Water waves --- Mathematical models.

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This memoir is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, the authors consider solutions such that the curvature of the initial free surface does not belong to L^2. The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and Hölder estimates. The authors first prove tame estimates in Sobolev spaces depending linearly on Hölder norms and then use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these Hölder norms.

Water waves --- Waves --- Streamflow velocity --- Inequalities (Mathematics) --- Mathematical models.

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Water waves --- Streamflow velocity. --- Inequalities (Mathematics) --- Cauchy problem. --- Cauchy, Problème de. --- Inégalités (mathématiques) --- Écoulement (hydrologie) --- Vagues --- Mathematical models. --- Vitesse. --- Modèles mathématiques. --- Waves --- Streamflow velocity --- Cauchy problem --- Differential equations, Partial --- Processes, Infinite --- Velocity of streamflow --- Rivers --- Cycles --- Hydrodynamics --- Benjamin-Feir instability --- Mathematical models --- Ondes --- Écoulement (Hydrologie) --- Inégalités (Mathématiques) --- Problème de Cauchy --- Modèles mathématiques --- Vitesse

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This book commemorates the 70th birthday of Eugene Morozov, the noted Russian observational oceanographer. It contains many contributions reflecting his fields of interest, including but not limited to tidal internal waves, ocean circulation, deep ocean currents, and Arctic oceanography. Special attention is paid to studies on internal waves and especially those on tidal internal waves in the Global Ocean. These papers describe the most important open problems concerning experimental studies of internal waves and their theoretical, numerical, and laboratory modeling. Further contributions investigate the physics of surface waves and their interaction with internal waves.  Here, the focus is on describing interaction processes between internal waves and deep currents in the ocean, especially currents of Antarctic Bottom Water in abyssal fractures. They also touch on the problem of oceanic circulation and related processes in fjords, including those occurring under sea ice. Given its breadth of coverage, the book will appeal to anyone interested in a survey of ocean dynamics, ranging from historic perspectives to modern research topics.

Ocean waves. --- Ocean currents. --- Wave-motion, Theory of. --- Earth sciences. --- Oceanography. --- Physical geography. --- Fluids. --- Earth Sciences. --- Fluid- and Aerodynamics. --- Earth System Sciences. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Hydraulics --- Mechanics --- Physics --- Hydrostatics --- Permeability --- Geography --- Oceanography, Physical --- Oceanology --- Physical oceanography --- Thalassography --- Earth sciences --- Marine sciences --- Ocean --- Geosciences --- Environmental sciences --- Physical sciences --- Undulatory theory --- Breakers --- Sea waves --- Surf --- Swell --- Oceanography --- Water waves --- Currents, Oceanic --- Ocean circulation --- Water currents --- Ocean surface topography --- Ocean circulation. --- Morozov, E.G. --- Career in oceanography. --- Statistical physics. --- Mathematical statistics --- Statistical methods