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A general theorem on Fuchsian systems proved there has since together with some small later modifications been a workhorse for investigating spacetime singularities in various classes of spacetimes. One of the highlights of this development was the proof by Lars Andersson and myself Commun.
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Poincare 3, we were able to generalize this considerably, in particular to the case of solutions of the vacuum Einstein equations in sufficiently high dimensions. For these results no symmetry assumptions were necessary. More recently these results were generalized in another direction by Mark Heinzle and Patrik Sandin arXiv Fuchsian systems have also been applied to the study of the late-time behaviour of cosmological models with positive cosmological constant.
In the meantime there are more satisfactory results on this question useing other methods see this post but this example does show that the Fuchsian method can be applied to problems in general relativity which have nothing to do with the big bang. In general this method is a kind of machine for turning heuristic calculations into theorems. The Fuchsian method works as follows. Suppose that a system of PDE is given and write it schematically as. I consider the case that the equation itself is regular and the aim is to find singular solutions.
Let be an explicit function which satisfies the equation up to a certain order in an expansion parameter and which is singular on a hypersurface defined by. Look for a solution of the form where is less singular than.
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The original equation can be rewritten as , where is singular. Now the aim is to show that there is a unique solution of the transformed equation which vanishes as. A theorem of this kind was proved in the analytic case in the paper mentioned above which I wrote with Kichenassamy. Results on the smooth case are harder to prove and there are less of them known. A variant of the procedure is to define as a solution not necessarily explicit of an equation which is a simplified version of the equation. In the cases of spacetime singularities which have been successfully handled the latter system is the so-called velocity-dominated system.
This entry was posted on July 21, at am and is filed under general relativity , partial differential equations. You can follow any responses to this entry through the RSS 2. For instance, we examine such eq- tions by analyzing sp. During the past decade, the mathematics of superconductivity has been the subject of intense activity. This book examines in detail the nonlinear Ginzburg—Landau functional, the model most commonly used in the study of superconductivity.
The present volume is a collection of papers mainly concerning Phase Space Analysis,alsoknownasMicrolocal Analysis,anditsapplicationstothetheory of Partial Di?
The basic idea behind this theory, at the crossing of harmonic a. In modern theoretical physics, gauge field theories are of great importance since they keep internal symmetries and account for phenomena such as spontaneous symmetry breaking, the quantum Hall effect, charge fractionalization, superconductivity a. Differential equations are a fast evolving branch of mathematics and one of the mathematical tools most used by scientists and engineers. This book gathers a collection of original articles and state-of-the-art contributions, written by highly dis.
Fuchsian Reduction: Applications to Geometry, Cosmology and Mathematical Physics
More than ten years have passed since the book of F. Bethuel, H. Brezis and F. Fuchsian reduction is a method for representing solutions of nonlinear PDEs near singularities. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Ann. ENS , 36 , Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Ann.
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