Download or read book Global Regularity for 2D Water Waves with Surface Tension written by Alexandru D. Ionescu and published by American Mathematical Soc.. This book was released on 2019-01-08 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Almost Global Solutions of Capillary Gravity Water Waves Equations on the Circle written by Massimiliano Berti and published by Springer. This book was released on 2018-11-02 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this monograph is to prove that any solution of the Cauchy problem for the capillary-gravity water waves equations, in one space dimension, with periodic, even in space, small and smooth enough initial data, is almost globally defined in time on Sobolev spaces, provided the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, a normal forms-based procedure is used, eliminating those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations form a quasi-linear system, the usual normal forms approaches would face the well-known problem of losses of derivatives in the unbounded transformations. To overcome this, after a paralinearization of the capillary-gravity water waves equations, we perform several paradifferential reductions to obtain a diagonal system with constant coefficient symbols, up to smoothing remainders. Then we start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization. The reversible structure of the water waves equations, and the fact that we seek solutions even in space, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.
Download or read book Free Boundary Problems in Fluid Dynamics written by Albert Ai and published by Springer Nature. This book was released on with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Quasi Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity written by Roberto Feola and published by American Mathematical Society. This book was released on 2024-04-17 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
Download or read book The Einstein Klein Gordon Coupled System written by Alexandru D. Ionescu and published by Princeton University Press. This book was released on 2022-03-15 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: A definitive proof of global nonlinear stability of Minkowski space-time as a solution of the Einstein-Klein-Gordon equations This book provides a definitive proof of global nonlinear stability of Minkowski space-time as a solution of the Einstein-Klein-Gordon equations of general relativity. Along the way, a novel robust analytical framework is developed, which extends to more general matter models. Alexandru Ionescu and Benoît Pausader prove global regularity at an appropriate level of generality of the initial data, and then prove several important asymptotic properties of the resulting space-time, such as future geodesic completeness, peeling estimates of the Riemann curvature tensor, conservation laws for the ADM tensor, and Bondi energy identities and inequalities. The book is self-contained, providing complete proofs and precise statements, which develop a refined theory for solutions of quasilinear Klein-Gordon and wave equations, including novel linear and bilinear estimates. Only mild decay assumptions are made on the scalar field and the initial metric is allowed to have nonisotropic decay consistent with the positive mass theorem. The framework incorporates analysis both in physical and Fourier space, and is compatible with previous results on other physical models such as water waves and plasma physics.
Download or read book A Paralinearization of the 2d and 3d Gravity Water Wave System in Infinite Depth written by Stanley Paul Palasek and published by . This book was released on 2017 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the 2d and 3d water waves system with gravity and no surface tension in infinite depth. Loss of derivatives from the Dirichlet-to-Neumann operator make studying solutions for long times difficult and until recently only local results were available. Several authors have since made use of paradifferential calculus to overcome these difficulties and prove global regularity in 2d and 3d with and without surface tension. The purpose of this thesis is to formulate the paralinearization of the system based on the Weyl quantization due to Deng-Ionescu-Pausader-Pusateri but with several key modifications. Namely, we work in lower regularity L2-based Sobolev spaces and do not include surface tension. This makes the problem more difficult by reducing the regularity on the surface elevation. We flatten the interface to arrive at a paralinearization of the Dirichlet-to-Neumann operator. As a result we are able to paralinearize and symmetrize the entire system and derive a single equation for a single complex unknown. The result is suited for obtaining energy estimates that would be useful, for example, when proving rigorous modulation approximations to the water waves in various regimes.
Download or read book Quiver Grassmannians of Extended Dynkin Type D Part I Schubert Systems and Decompositions into Affine Spaces written by Oliver Lorscheid and published by American Mathematical Soc.. This book was released on 2019-12-02 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let Q be a quiver of extended Dynkin type D˜n. In this first of two papers, the authors show that the quiver Grassmannian Gre–(M) has a decomposition into affine spaces for every dimension vector e– and every indecomposable representation M of defect −1 and defect 0, with the exception of the non-Schurian representations in homogeneous tubes. The authors characterize the affine spaces in terms of the combinatorics of a fixed coefficient quiver for M. The method of proof is to exhibit explicit equations for the Schubert cells of Gre–(M) and to solve this system of equations successively in linear terms. This leads to an intricate combinatorial problem, for whose solution the authors develop the theory of Schubert systems. In Part 2 of this pair of papers, they extend the result of this paper to all indecomposable representations M of Q and determine explicit formulae for the F-polynomial of M.
Download or read book One Dimensional Empirical Measures Order Statistics and Kantorovich Transport Distances written by Sergey Bobkov and published by American Mathematical Soc.. This book was released on 2019-12-02 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work is devoted to the study of rates of convergence of the empirical measures μn=1n∑nk=1δXk, n≥1, over a sample (Xk)k≥1 of independent identically distributed real-valued random variables towards the common distribution μ in Kantorovich transport distances Wp. The focus is on finite range bounds on the expected Kantorovich distances E(Wp(μn,μ)) or [E(Wpp(μn,μ))]1/p in terms of moments and analytic conditions on the measure μ and its distribution function. The study describes a variety of rates, from the standard one 1n√ to slower rates, and both lower and upper-bounds on E(Wp(μn,μ)) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.
Download or read book Automorphisms ofTwo Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane written by William Goldman and published by American Mathematical Soc.. This book was released on 2019-06-10 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt: The automorphisms of a two-generator free group F acting on the space of orientation-preserving isometric actions of F on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action of a group on by polynomial automorphisms preserving the cubic polynomial and an area form on the level surfaces .
Download or read book Fusion of Defects written by Arthur Bartels and published by American Mathematical Soc.. This book was released on 2019-04-10 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: Conformal nets provide a mathematical model for conformal field theory. The authors define a notion of defect between conformal nets, formalizing the idea of an interaction between two conformal field theories. They introduce an operation of fusion of defects, and prove that the fusion of two defects is again a defect, provided the fusion occurs over a conformal net of finite index. There is a notion of sector (or bimodule) between two defects, and operations of horizontal and vertical fusion of such sectors. The authors' most difficult technical result is that the horizontal fusion of the vacuum sectors of two defects is isomorphic to the vacuum sector of the fused defect. Equipped with this isomorphism, they construct the basic interchange isomorphism between the horizontal fusion of two vertical fusions and the vertical fusion of two horizontal fusions of sectors.
Download or read book Geometric Pressure for Multimodal Maps of the Interval written by Feliks Przytycki and published by American Mathematical Soc.. This book was released on 2019-06-10 with total page 81 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. The authors work in a setting of generalized multimodal maps, that is, smooth maps f of a finite union of compact intervals Iˆ in R into R with non-flat critical points, such that on its maximal forward invariant set K the map f is topologically transitive and has positive topological entropy. They prove that several notions of non-uniform hyperbolicity of f|K are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in K hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). They prove that several definitions of geometric pressure P(t), that is pressure for the map f|K and the potential −tlog|f′|, give the same value (including pressure on periodic orbits, “tree” pressure, variational pressures and conformal pressure). Finally they prove that, provided all periodic orbits in K are hyperbolic repelling, the function P(t) is real analytic for t between the “condensation” and “freezing” parameters and that for each such t there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.
Download or read book Crossed Products of Operator Algebras written by Elias G. Katsoulis and published by American Mathematical Soc.. This book was released on 2019-04-10 with total page 85 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. They develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. They complement their generic results with the detailed study of many important special cases. In particular they study crossed products of tensor algebras, triangular AF algebras and various associated C -algebras. They make contributions to the study of C -envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. They also answer questions from the pertinent literature.
Download or read book Time Changes of the Brownian Motion Poincar Inequality Heat Kernel Estimate and Protodistance written by Jun Kigami and published by American Mathematical Soc.. This book was released on 2019-06-10 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including n-dimensional cube [0,1]n are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on [0,1]n, density of the medium is homogeneous and represented by the Lebesgue measure. The author's study includes densities which are singular to the homogeneous one. He establishes a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on [0,1]2 and self-similar measures. The author shows the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, he obtains diagonal lower and upper estimates of the heat kernel as time tends to 0. In particular, to express the principal part of the lower diagonal heat kernel estimate, he introduces “protodistance” associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.
Download or read book Flat Rank Two Vector Bundles on Genus Two Curves written by Viktoria Heu and published by American Mathematical Soc.. This book was released on 2019-06-10 with total page 103 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors study the moduli space of trace-free irreducible rank 2 connections over a curve of genus 2 and the forgetful map towards the moduli space of underlying vector bundles (including unstable bundles), for which they compute a natural Lagrangian rational section. As a particularity of the genus case, connections as above are invariant under the hyperelliptic involution: they descend as rank logarithmic connections over the Riemann sphere. The authors establish explicit links between the well-known moduli space of the underlying parabolic bundles with the classical approaches by Narasimhan-Ramanan, Tyurin and Bertram. This allows the authors to explain a certain number of geometric phenomena in the considered moduli spaces such as the classical -configuration of the Kummer surface. The authors also recover a Poincaré family due to Bolognesi on a degree 2 cover of the Narasimhan-Ramanan moduli space. They explicitly compute the Hitchin integrable system on the moduli space of Higgs bundles and compare the Hitchin Hamiltonians with those found by van Geemen-Previato. They explicitly describe the isomonodromic foliation in the moduli space of vector bundles with -connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles.
Download or read book Moufang Sets and Structurable Division Algebras written by Lien Boelaert and published by American Mathematical Soc.. This book was released on 2019-06-10 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group. It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. The authors extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, they show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field k of characteristic different from 2 and 3 arises from a structurable division algebra. The authors also obtain explicit formulas for the root groups, the τ-map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups.
Download or read book Generalized Mercer Kernels and Reproducing Kernel Banach Spaces written by Yuesheng Xu and published by American Mathematical Soc.. This book was released on 2019-04-10 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: This article studies constructions of reproducing kernel Banach spaces (RKBSs) which may be viewed as a generalization of reproducing kernel Hilbert spaces (RKHSs). A key point is to endow Banach spaces with reproducing kernels such that machine learning in RKBSs can be well-posed and of easy implementation. First the authors verify many advanced properties of the general RKBSs such as density, continuity, separability, implicit representation, imbedding, compactness, representer theorem for learning methods, oracle inequality, and universal approximation. Then, they develop a new concept of generalized Mercer kernels to construct p-norm RKBSs for 1≤p≤∞ .
Download or read book Spinors on Singular Spaces and the Topology of Causal Fermion Systems written by Felix Finster and published by American Mathematical Soc.. This book was released on 2019-06-10 with total page 83 pages. Available in PDF, EPUB and Kindle. Book excerpt: Causal fermion systems and Riemannian fermion systems are proposed as a framework for describing non-smooth geometries. In particular, this framework provides a setting for spinors on singular spaces. The underlying topological structures are introduced and analyzed. The connection to the spin condition in differential topology is worked out. The constructions are illustrated by many simple examples such as the Euclidean plane, the two-dimensional Minkowski space, a conical singularity, a lattice system as well as the curvature singularity of the Schwarzschild space-time. As further examples, it is shown how complex and Kähler structures can be encoded in Riemannian fermion systems.
