Regularity Of Solutions To Nonlinear Equations Of Schrodinger Type
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| Author | : Michael Ruzhansky |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 327 |
| Release | : 2012-08-04 |
| Genre | : Mathematics |
| ISBN | : 3034804547 |
Evolution equations of hyperbolic or more general p-evolution type form an active field of current research. This volume aims to collect some recent advances in the area in order to allow a quick overview of ongoing research. The contributors are first rate mathematicians. This collection of research papers is centred around parametrix constructions and microlocal analysis; asymptotic constructions of solutions; energy and dispersive estimates; and associated spectral transforms. Applications concerning elasticity and general relativity complement the volume. The book gives an overview of a variety of ongoing current research in the field and, therefore, allows researchers as well as students to grasp new aspects and broaden their understanding of the area.
| Author | : Luis A. Caffarelli |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 114 |
| Release | : 1995 |
| Genre | : Mathematics |
| ISBN | : 0821804375 |
The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov-Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients.
| Author | : Serge Levendorskii |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 442 |
| Release | : 2013-11-11 |
| Genre | : Mathematics |
| ISBN | : 9401712158 |
This volume is the first to be devoted to the study of various properties of wide classes of degenerate elliptic operators of arbitrary order and pseudo-differential operators with multiple characteristics. Conditions for operators to be Fredholm in appropriate weighted Sobolev spaces are given, a priori estimates of solutions are derived, inequalities of the Grding type are proved, and the principal term of the spectral asymptotics for self-adjoint operators is computed. A generalization of the classical Weyl formula is proposed. Some results are new, even for operators of the second order. In addition, an analogue of the Boutet de Monvel calculus is developed and the index is computed. For postgraduate and research mathematicians, physicists and engineers whose work involves the solution of partial differential equations.
| Author | : Thierry Cazenave |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 346 |
| Release | : 2003 |
| Genre | : Mathematics |
| ISBN | : 0821833995 |
The nonlinear Schrodinger equation has received a great deal of attention from mathematicians, particularly because of its applications to nonlinear optics. This book presents various mathematical aspects of the nonlinear Schrodinger equation. It studies both problems of local nature and problems of global nature.
| Author | : |
| Publisher | : |
| Total Pages | : 666 |
| Release | : 1998 |
| Genre | : Mathematics |
| ISBN | : |
| Author | : Andrei D. Polyanin |
| Publisher | : CRC Press |
| Total Pages | : 835 |
| Release | : 2004-06-02 |
| Genre | : Mathematics |
| ISBN | : 1135440816 |
The Handbook of Nonlinear Partial Differential Equations is the latest in a series of acclaimed handbooks by these authors and presents exact solutions of more than 1600 nonlinear equations encountered in science and engineering--many more than any other book available. The equations include those of parabolic, hyperbolic, elliptic and other types, and the authors pay special attention to equations of general form that involve arbitrary functions. A supplement at the end of the book discusses the classical and new methods for constructing exact solutions to nonlinear equations. To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology, outline some of the methods in a schematic, simplified manner, and arrange the equations in increasing order of complexity. Highlights of the Handbook:
| Author | : Terence Tao |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 394 |
| Release | : 2006 |
| Genre | : Mathematics |
| ISBN | : 0821841432 |
"Starting only with a basic knowledge of graduate real analysis and Fourier analysis, the text first presents basic nonlinear tools such as the bootstrap method and perturbation theory in the simpler context of nonlinear ODE, then introduces the harmonic analysis and geometric tools used to control linear dispersive PDE. These methods are then combined to study four model nonlinear dispersive equations. Through extensive exercises, diagrams, and informal discussion, the book gives a rigorous theoretical treatment of the material, the real-world intuition and heuristics that underlie the subject, as well as mentioning connections with other areas of PDE, harmonic analysis, and dynamical systems.".
| Author | : Jean Bourgain |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 193 |
| Release | : 1999 |
| Genre | : Mathematics |
| ISBN | : 0821819194 |
This volume presents recent progress in the theory of nonlinear dispersive equations, primarily the nonlinear Schrodinger (NLS) equation. The Cauchy problem for defocusing NLS with critical nonlinearity is discussed. New techniques and results are described on global existence and properties of solutions with Large Cauchy data. Current research in harmonic analysis around Strichartz's inequalities and its relevance to nonlinear PDE is presented and several topics in NLS theory on bounded domains are reviewed. Using the NLS as an example, the book offers comprehensive insight on current research related to dispersive equations and Hamiltonian PDEs.
| Author | : Ernst Hairer |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 526 |
| Release | : 2013-03-09 |
| Genre | : Mathematics |
| ISBN | : 3662050188 |
This book deals with numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by numerous figures, treats applications from physics and astronomy, and contains many numerical experiments and comparisons of different approaches.
| Author | : Nakao Hayashi |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 570 |
| Release | : 2006-04-21 |
| Genre | : Mathematics |
| ISBN | : 3540320598 |
Many of problems of the natural sciences lead to nonlinear partial differential equations. However, only a few of them have succeeded in being solved explicitly. Therefore different methods of qualitative analysis such as the asymptotic methods play a very important role. This is the first book in the world literature giving a systematic development of a general asymptotic theory for nonlinear partial differential equations with dissipation. Many typical well-known equations are considered as examples, such as: nonlinear heat equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev type equations, systems of equations of Boussinesq, Navier-Stokes and others.
