On the string equation with a singular weight belonging to the space of multipliers in Sobolev spaces with negative index of smoothness

2016 ◽  
Vol 80 (6) ◽  
pp. 1242-1256
Author(s):  
Yu V Tikhonov ◽  
I A Sheipak
Keyword(s):  
Sobolev Spaces ◽  
String Equation ◽  
Negative Index ◽  
Singular Weight

scholarly journals GLOBAL WELL-POSEDNESS OF THE 1D DIRAC–KLEIN–GORDON SYSTEM IN SOBOLEV SPACES OF NEGATIVE INDEX

2009 ◽  
Vol 06 (03) ◽  
pp. 631-661 ◽  
Author(s):  
ACHENEF TESFAHUN
Keyword(s):  
Sobolev Spaces ◽  
Well Posedness ◽  
Dirac Spinor ◽  
Negative Index ◽  
Main Ingredient ◽  
Time Estimates ◽  
The Cauchy Problem ◽  
Klein Gordon ◽  
Positive Index ◽  
Well Posed

We prove that the Cauchy problem for the Dirac–Klein–Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in the proof is the theory of "almost conservation law" and "I-method" introduced by Colliander, Keel, Staffilani, Takaoka, and Tao. Our proof also relies on the null structure in the system, and bilinear space–time estimates of Klainerman–Machedon type.

THE RELLICH LEMMA ON COMPACT ABELIAN GROUPS AND EQUATIONS OF INFINITE ORDER

2012 ◽  
Vol 10 (02) ◽  
pp. 1220030 ◽  
Author(s):  
PRZEMYSŁAW GÓRKA ◽  
TOMASZ KOSTRZEWA ◽  
ENRIQUE G. REYES
Keyword(s):  
Abelian Group ◽  
String Theory ◽  
Sobolev Spaces ◽  
Compact Abelian Group ◽  
Infinite Order ◽  
String Equation ◽  
Continuous Solutions ◽  
Locally Compact ◽  
Compact Abelian Groups ◽  
Lca Groups

Partially motivated by a class of nonlinear equations of interest for string theory, cosmology and p-adic analysis, we continue our research on Sobolev spaces on arbitrary locally compact abelian (LCA) groups. In this paper we focus on an analog for LCA groups of the classical Rellich compactness theorem. As an application, we prove the existence of continuous solutions to a generalized Euclidean bosonic string equation posed on a compact abelian group.

SOBOLEV SPACES ON LOCALLY COMPACT ABELIAN GROUPS AND THE BOSONIC STRING EQUATION

2014 ◽  
Vol 98 (1) ◽  
pp. 39-53 ◽  
Author(s):  
PRZEMYSŁAW GÓRKA ◽  
ENRIQUE G. REYES
Keyword(s):  
Sobolev Spaces ◽  
Abelian Groups ◽  
Linear Equations ◽  
Bosonic String ◽  
Sobolev Embedding ◽  
String Equation ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Compactness Theorems

AbstractMotivated by a class of nonlinear nonlocal equations of interest for string theory, we introduce Sobolev spaces on arbitrary locally compact abelian groups and we examine some of their properties. Specifically, we focus on analogs of the Sobolev embedding and Rellich–Kondrachov compactness theorems. As an application, we prove the existence of continuous solutions to a generalized bosonic string equation posed on an arbitrary compact abelian group, and we also remark that our approach allows us to solve very general linear equations in a $p$-adic context.

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