Nonlinear electrodynamics as a symmetric hyperbolic system

In this paper, we study the dissipative structure of first-order linear symmetric hyperbolic system with general relaxation and provide the algebraic characterization for the uniform dissipativity up to order 1. Our result extends the classical Shizuta–Kawashima condition for the case of symmetric relaxation, with a full generality and optimality.

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CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY

International Journal of Modern Physics D ◽

10.1142/s0218271800000037 ◽

2000 ◽

Vol 09 (01) ◽

pp. 13-34 ◽

Cited By ~ 6

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.

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Characterization of the dissipative structure for the symmetric hyperbolic system with non-symmetric relaxation

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891621500053 ◽

2021 ◽

Vol 18 (01) ◽

pp. 195-219

Author(s):

Yoshihiro Ueda

Keyword(s):

Hyperbolic System ◽

Stability Condition ◽

Dissipative Structure ◽

General System ◽

Partial Result ◽

Symmetric Hyperbolic System ◽

Standard Type ◽

Classical Stability ◽

The Stability

This paper is concerned with the dissipative structure for the linear symmetric hyperbolic system with non-symmetric relaxation. If the relaxation matrix of the system has symmetric property, Shizuta and Kawashima in 1985 introduced the suitable stability condition called Classical Stability Condition in this paper, and Umeda, Kawashima and Shizuta in 1984 analyzed the dissipative structure of the standard type. On the other hand, Ueda, Duan and Kawashima in 2012 and 2018 focused on the system with non-symmetric relaxation, and got the partial result which is the extension of known results. Furthermore, they argued the new dissipative structure called the regularity-loss type. In this situation, our purpose of this paper is to extend the stability theory introduced by Shizuta and Kawashima in 1985 and Umeda, Kawashima and Shizuta in 1984 for our general system.

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