Download or read book Nonlinear Wave Equations Formation of Singularities written by Fritz John and published by American Mathematical Soc.. This book was released on 1990-07-01 with total page 74 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second volume in the University Lecture Series, designed to make more widely available some of the outstanding lectures presented in various institutions around the country. Each year at Lehigh University, a distinguished mathematical scientist presents the Pitcher Lectures in the Mathematical Sciences. This volume contains the Pitcher lectures presented by Fritz John in April 1989. The lectures deal with existence in the large of solutions of initial value problems for nonlinear hyperbolic partial differential equations. As is typical with nonlinear problems, there are many results and few general conclusions in this extensive subject, so the author restricts himself to a small portion of the field, in which it is possible to discern some general patterns. Presenting an exposition of recent research in this area, the author examines the way in which solutions can, even with small and very smooth initial data, ``blow up'' after a finite time. For various types of quasi-linear equations, this time depends strongly on the number of dimensions and the ``size'' of the data. Of particular interest is the formation of singularities for nonlinear wave equations in three space dimensions.
Download or read book Remarks on the formation of singularities for nonlinear wave equations in bounded domains written by Reinhard Racke and published by . This book was released on 1991 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Remarks on the Formation of Singularities for Nonlinear Wave Equations in Bounded Domains written by R. Racke and published by . This book was released on 1991 with total page 18 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Nonlinear Wave Equations written by Satyanad Kichenassamy and published by CRC Press. This book was released on 2021-05-30 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work examines the mathematical aspects of nonlinear wave propagation, emphasizing nonlinear hyperbolic problems. It introduces the tools that are most effective for exploring the problems of local and global existence, singularity formation, and large-time behaviour of solutions, and for the study of perturbation methods.
Download or read book Nonlinear Wave Equations written by Tatsien Li and published by Springer. This book was released on 2017-11-23 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on nonlinear wave equations, which are of considerable significance from both physical and theoretical perspectives. It also presents complete results on the lower bound estimates of lifespan (including the global existence), which are established for classical solutions to the Cauchy problem of nonlinear wave equations with small initial data in all possible space dimensions and with all possible integer powers of nonlinear terms. Further, the book proposes the global iteration method, which offers a unified and straightforward approach for treating these kinds of problems. Purely based on the properties of solut ions to the corresponding linear problems, the method simply applies the contraction mapping principle.
Download or read book Lectures on Nonlinear Evolution Equations written by Reinhard Racke and published by Birkhäuser. This book was released on 2015-08-31 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behaviour of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial boundary value problems and for open questions are provided. In this second edition, initial-boundary value problems in waveguides are additionally considered.
Download or read book Studies in Recurrence and Singularity Formation in Nonlinear Dispersive Wave Equations written by Juan-Ming Yuan and published by . This book was released on 2001 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book On the Existence and Stability of Self similar Blowup in Nonlinear Wave Equations written by Irfan Glogic and published by . This book was released on 2018 with total page 97 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the existence and stability of singularity formation in nonlinear wave equations.
Download or read book Nonlinear Wave Equations written by Yan Guo and published by American Mathematical Soc.. This book was released on 2000 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents original research papers and expository articles from the conference in honour of Walter A. Strauss's 60th birthday, held at Brown University in Providence, Rhode Island. The book offers a collection of original papers and expository articles mainly devoted to the study of nonlinear wave equations. The articles cover a wide range of topics, including scattering theory, dispersive waves, classical field theory, mathematical fluid dynamics, kinetic theory, stability theory, and variational methods. The book offers a cross-section of current trends and research directions in the study of nonlinear wave equations and related topics.
Download or read book Blowup for Nonlinear Hyperbolic Equations written by Serge Alinhac and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: Solutions to partial differential equations or systems often, over specific time periods, exhibit smooth behaviour. Given sufficient time, however, they almost invariably undergo a brutal change in behaviour, and this phenomenon has become known as blowup. In this book, the author provides an overview of what is known about this situation and discusses many of the open problems concerning it.
Download or read book Uniqueness and Singularities of Weak Solutions to Some Nonlinear Wave Equations written by Qingtian Zhang and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Wave phenomenon is very common in physics. Understanding wave dynamics is very important and helpful to reveal physical principles. In this dissertation I consider three kinds of equations: Camassa-Holm equation, variational wave equation, and p-system. Camassa-Holm equation is an approximate model for the shallow water phenonmenon. Variational wave equation is derived from a physical model of liquid crystal, and p-system is for one-dimensional compressible gas dynamics. I mainly care about the analytical properties of weak solutions to these equations.In the first project, I proved the uniqueness of conservative weak solutions for the (two-component) Camassa- Holm equations and variational wave equation by the method of characteristics. They all have H1 energy, so the characteristic equations have H older continuous right-hand-side. In general, the ODEs with Ho lder continuous right-hand-side do not have unique solutions. We resolve this difficulty by using an additional balance law, related to energy, which is a change of coordinate based on the fact of energy concentration. And we proved uniqueness by rewriting the quasilinear PDE into a equivalent semilinear system with Lipschitz continuous right-hand- side. The main difference for the variational wave equation is that it is a second-order equation. It has two families of waves and they may interact with each other. We show that the total amount of interaction is bounded on bounded time intervals. Then by carefully choosing new characteristic variables, we can reduce the PDE to an equivalent first-order semilinear system with Lipschitz right-hand-side. In this way, we obtain uniqueness of conservative weak solutions.I also proved the global well-posedness of cubic Camassa-Holm equation.In the second project, we studied the generic singular behavior of these two equations. Here the generic behavior refers to the properties which are true for all solutions starting from an open dense set of initial data.In particular, we showed that a generic solution is piecewise smooth. The proof mainly relies on Thoms transversality Lemma from differential geometry. We also calculated the detailed local asymptotic expansion of solutions near the singular points and found that the Camassa-Holm equation and the two-component Camassa-Holm equation producevery different types of singularities.In the third project, we studied solutions to one-dimensional conservation lawswith large BV data. Specially, we considered the p-system, modeling the isentropic gas dynamics. We consider two cases: when the density is uniformly away from zero and when the density may approach zero. In both cases, we construct examples of front-tracking approximate solutions where the total variation becomes arbitrarily large.
Download or read book Numerical investigations of singularity formation in non linear wave equations in the adiabatic limit written by Jean-Marie Linhart and published by . This book was released on 1999 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Analysis Of Singularities For Partial Differential Equations written by Shuxing Chen and published by World Scientific. This book was released on 2010-09-24 with total page 207 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides a comprehensive overview on the theory on analysis of singularities for partial differential equations (PDEs). It contains a summarization of the formation, development and main results on this topic. Some of the author's discoveries and original contributions are also included, such as the propagation of singularities of solutions to nonlinear equations, singularity index and formation of shocks.
Download or read book Handbook of Differential Equations Evolutionary Equations written by C.M. Dafermos and published by Gulf Professional Publishing. This book was released on 2005-11-30 with total page 684 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains several introductory texts concerning the main directions in the theory of evolutionary partial differential equations. The main objective is to present clear, rigorous, and in depth surveys on the most important aspects of the present theory.
Download or read book Nonlinear Hyperbolic Equations Spectral Theory and Wavelet Transformations written by Sergio Albeverio and published by Birkhäuser. This book was released on 2012-12-06 with total page 444 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume focuses on recent developments in non-linear and hyperbolic equations. It will be a most valuable resource for researchers in applied mathematics, the theory of wavelets, and in mathematical and theoretical physics. Nine up-to-date contributions have been written on invitation by experts in the respective fields. The book is the third volume of the subseries "Advances in Partial Differential Equations".
Download or read book Nonlinear Waves written by Peter R. Popivanov and published by World Scientific. This book was released on 2011 with total page 179 pages. Available in PDF, EPUB and Kindle. Book excerpt: Big Nate is the star goalie of his school's soccer team, and he is tasked with defending his goal and saving the day against Jefferson Middle School, their archrival.
Download or read book Nonlinear Waves in Inhomogeneous and Hereditary Media written by Aleksandr Aleksandrovich Lokshin and published by Springer. This book was released on 1992 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt:
