scholarly journals POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 81-141 ◽  
Author(s):  
PIOTR T. CHRUŚCIEL ◽  
SZYMON ŁȨSKI
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Space Time ◽  
Einstein Equations ◽  
Nonlinear Wave Equations ◽  
Linear Wave ◽  
Cauchy Problems ◽  
Time Dimension ◽  
Wave Maps ◽  
Corner Conditions

The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

scholarly journals On Classical Global Solutions of Nonlinear Wave Equations with Large Data

2017 ◽  
Vol 2019 (19) ◽  
pp. 5859-5913 ◽  
Author(s):  
Shuang Miao ◽  
Long Pei ◽  
Pin Yu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Wave Equations ◽  
Global Solutions ◽  
Large Data ◽  
Einstein Equations ◽  
Nonlinear Wave Equations ◽  
Classical Solutions ◽  
Linear Wave ◽  
The Cauchy Problem

Abstract This article studies the Cauchy problem for systems of semi-linear wave equations on $\mathbb{R}^{3+1}$ with nonlinear terms satisfying the null conditions. We construct future global-in-time classical solutions with arbitrarily large initial energy. The choice of the large Cauchy initial data is inspired by Christodoulou's characteristic initial data in his work [2] on formation of black holes. The main innovation of the current work is that we discovered a relaxed energy ansatz which allows us to prove decay-in-time-estimate. Therefore, the new estimates can also be applied in studying the Cauchy problem for Einstein equations.

scholarly journals On space-time properties of solutions for nonlinear evolutionary equations with random initial data

2021 ◽  
Vol 3 (1) ◽  
pp. 11-20
Author(s):  
Kyrill I. Vaninsky
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Space Time ◽  
Nonlinear Wave Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Random Initial Data ◽  
Evolutionary Equations ◽  
Hölder Exponents ◽  
Kdv Type Equations

We consider space-time properties of periodic solutions of nonlinear wave equations, nonlinear Schrödinger equations and KdV-type equations with initial data from the support of the Gibbs’ measure. For the wave and Schrödinger equations we establish the best Hölder exponents. We also discuss KdV-type equations which are more difficult due to a presence of the derivative in the nonlinearity.

scholarly journals Global existence and decay estimates for nonlinear wave equations with space-time dependent dissipative term

2015 ◽  
Vol 12 (02) ◽  
pp. 249-276
Author(s):  
Tomonari Watanabe
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Space Time ◽  
Time Dependent ◽  
Nonlinear Wave Equations ◽  
Energy Estimates ◽  
Space Region ◽  
Decay Estimates ◽  
Dissipative Term

We study the global existence and the derivation of decay estimates for nonlinear wave equations with a space-time dependent dissipative term posed in an exterior domain. The linear dissipative effect may vanish in a compact space region and, moreover, the nonlinear terms need not be in a divergence form. In order to establish higher-order energy estimates, we introduce an argument based on a suitable rescaling. The proposed method is useful to control certain derivatives of the dissipation coefficient.

LOCAL EXISTENCE FOR SEMILINEAR WAVE EQUATIONS AND APPLICATIONS TO YANG–MILLS EQUATIONS

2005 ◽  
Vol 02 (01) ◽  
pp. 61-76
Author(s):  
YUNG-FU FANG
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Nonlinear Term ◽  
A Priori ◽  
Local Existence ◽  
A Priori Estimate ◽  
Linear Wave ◽  
Yang Mills ◽  
Linear Wave Equations ◽  
Local Result

In this work we are concerned with a local existence of certain semi-linear wave equations for which the initial data has minimal regularity. Assuming the initial data are in H1+∊ and H∊ for any ∊ > 0, we prove a local result by using a fixed point argument, the main ingredient being an a priori estimate for the quadratic nonlinear term uDu. The technique applies to the Yang–Mills equations in the Lorentz gauge.

scholarly journals SOLUTIONS OF QUASI-LINEAR WAVE EQUATIONS POLYHOMOGENEOUS AT NULL INFINITY IN HIGH DIMENSIONS

2011 ◽  
Vol 08 (02) ◽  
pp. 269-346 ◽  
Author(s):  
PIOTR T. CHRUŚCIEL ◽  
ROGER TAGNE WAFO
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Maxwell Equations ◽  
Wave Equations ◽  
Hyperbolic Systems ◽  
Small Data ◽  
Linear Wave ◽  
High Dimensions ◽  
Null Infinity ◽  
Linear Wave Equations

We prove propagation of weighted Sobolev regularity for solutions of the hyperboloidal Cauchy problem for a class of quasi-linear symmetric hyperbolic systems, under structure conditions compatible with the Einstein–Maxwell equations in space-time dimensions n + 1 ≥ 7. Similarly we prove propagation of polyhomogeneity in dimensions n + 1 ≥ 9. As a byproduct we obtain, in those last dimensions, polyhomogeneity at null infinity of small data solutions of vacuum Einstein, or Einstein–Maxwell equations evolving out of initial data which are stationary outside of a ball.

scholarly journals Growth of Solutions with Positive Initial Energy to Systems of Nonlinear Wave Equations with Damping and Source Terms

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Erhan Pişkin
Keyword(s):  
Initial Data ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Initial Boundary ◽  
Initial Energy ◽  
Nonlinear Wave Equations ◽  
Source Terms ◽  
Positive Initial Energy ◽  
Damping And Source Terms ◽  
Growth Of Solutions

We consider initial-boundary conditions for coupled nonlinear wave equations with damping and source terms. We prove that the solutions of the problem are unbounded when the initial data are large enough in some sense.

scholarly journals ASYMPTOTICAL ANALYSIS OF SOME COUPLED NONLINEAR WAVE EQUATIONS

2011 ◽  
Vol 16 (1) ◽  
pp. 97-108 ◽  
Author(s):  
Aleksandras Krylovas ◽  
Rima Kriauzienė
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Linear Equations ◽  
Travelling Wave ◽  
Nonlinear Wave Equations ◽  
Linear Wave ◽  
Resonance Case ◽  
Non Linear ◽  
Asymptotical Analysis

We consider coupled nonlinear equations modelling a family of travelling wave solutions. The goal of our work is to show that the method of internal averaging along characteristics can be used for wide classes of coupled non-linear wave equations such as Korteweg-de Vries, Klein – Gordon, Hirota – Satsuma, etc. The asymptotical analysis reduces a system of coupled non-linear equations to a system of integro – differential averaged equations. The averaged system with the periodical initial conditions disintegrates into independent equations in non-resonance case. These equations describe simple weakly non-linear travelling waves in the non-resonance case. In the resonance case the integro – differential averaged systems describe interaction of waves and give a good asymptotical approximation for exact solutions.

Continuous-Valued Cellular Automata in Two Dimensions

2003 ◽  
Author(s):  
Rudy Rucker
Keyword(s):  
Cellular Automata ◽  
Wave Equations ◽  
Reaction Diffusion ◽  
Space Time ◽  
Two Dimensions ◽  
Nonlinear Wave Equations ◽  
Flow Of Time ◽  
Action At A Distance ◽  
Necessary Evil ◽  
Update Process

We explore a variety of two-dimensional continuous-valued cellular automata (CAs). We discuss how to derive CA schemes from differential equations and look at CAs based on several kinds of nonlinear wave equations. In addition we cast some of Hans Meinhardt’s activator-inhibitor reaction-diffusion rules into two dimensions. Some illustrative runs of CAPOW, a. CA simulator, are presented. A cellular automaton, or CA, is a computation made up of finite elements called cells. Each cell contains the same type of state. The cells are updated in parallel, using a rule which is homogeneous, and local. In slightly different words, a CA is a computation based upon a grid of cells, with each cell containing an object called a state. The states are updated in discrete steps, with all the cells being effectively updated at the same time. Each cell uses the same algorithm for its update rule. The update algorithm computes a cell’s new state by using information about the states of the cell’s nearby space-time neighbors, that is, using the state of the cell itself, using the states of the cell’s nearby neighbors, and using the recent prior states of the cell and its neighbors. The states do not necessarily need to be single numbers, they can also be data structures built up from numbers. A CA is said to be discrete valued if its states are built from integers, and a CA is continuous valued if its states are built from real numbers. As Norman Margolus and Tommaso Toffoli have pointed out, CAs are well suited for modeling nature [7]. The parallelism of the CA update process mirrors the uniform flow of time. The homogeneity of the CA update rule across all the cells corresponds to the universality of natural law. And the locality of CAs reflect the fact that nature seems to forbid action at a distance. The use of finite space-time elements for CAs are a necessary evil so that we can compute at all. But one might argue that the use of discrete states is an unnecessary evil.

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