On hyperbolicity and Gevrey well-posedness. Part three: a model of weakly hyperbolic systems

2021 ◽  
Vol 70 (2) ◽  
pp. 743-780
Author(s):  
Baptiste Morisse
Keyword(s):  
Hyperbolic Systems ◽  
Well Posedness ◽  
Weakly Hyperbolic Systems

scholarly journals CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY

2000 ◽  
Vol 09 (01) ◽  
pp. 13-34 ◽  
Author(s):  
GEN YONEDA ◽  
HISA-AKI SHINKAI
Keyword(s):  
General Relativity ◽  
Hyperbolic System ◽  
Hyperbolic Systems ◽  
Equations Of Motion ◽  
Well Posedness ◽  
Symmetric Hyperbolic System ◽  
The Cauchy Problem ◽  
Long Time ◽  
Vacuum Spacetime ◽  
Gauge Conditions

Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.

scholarly journals Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness

2018 ◽  
Vol 372 (3-4) ◽  
pp. 1597-1629
Author(s):  
Claudia Garetto ◽  
Christian Jäh ◽  
Michael Ruzhansky
Keyword(s):  
Principal Part ◽  
Hyperbolic Systems ◽  
Well Posedness

scholarly journals The Well-Posedness Issue in Sobolev Spaces for Hyperbolic Systems with Zygmund-Type Coefficients

2015 ◽  
Vol 40 (11) ◽  
pp. 2082-2121 ◽  
Author(s):  
Ferruccio Colombini ◽  
Daniele Del Santo ◽  
Francesco Fanelli ◽  
Guy Métivier
Keyword(s):  
Sobolev Spaces ◽  
Hyperbolic Systems ◽  
Well Posedness

scholarly journals Linear Stability of Hyperbolic Moment Models for Boltzmann Equation

2017 ◽  
Vol 10 (2) ◽  
pp. 255-277 ◽  
Author(s):  
Yana Di ◽  
Yuwei Fan ◽  
Ruo Li ◽  
Lingchao Zheng
Keyword(s):  
Boltzmann Equation ◽  
Stability Condition ◽  
Hyperbolic Systems ◽  
Local Equilibrium ◽  
Cauchy Data ◽  
Binary Collision ◽  
Well Posedness ◽  
Convection Term ◽  
Collision Model ◽  
Moment Models

AbstractGrad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collision terms. It is proved that the regularized models are linearly stable at the local equilibrium and satisfy Yong's first stability condition with commonly used approximate collision terms, and particularly with Boltzmann's binary collision model.

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