Hyperbolic systems well posed in all Gevrey classes

In this paper, the author considers the motion of a relativistic perfect fluid with self-interaction mediated by Nordström's scalar theory of gravity. The evolution of the fluid is determined by a quasilinear hyperbolic system of PDEs, and a cosmological constant is introduced in order to ensure the existence of nonzero constant solutions. Accordingly, the initial value problem for a compact perturbation of an infinitely extended quiet fluid is studied. Although the system is neither symmetric hyperbolic nor strictly hyperbolic, Christodoulou's constructive results on the existence of energy currents for equations derivable from a Lagrangian can be adapted to provide energy currents that can be used in place of the standard energy principle available for first-order symmetric hyperbolic systems. After providing such energy currents, the author uses them to prove that the Euler–Nordström system with a cosmological constant is well-posed in a suitable Sobolev space.

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Can hyperbolic systems of conservation laws be well-posed in BV(R;RN)?

Lecture Notes in Mathematics - Nonlinear Hyperbolic Problems ◽

10.1007/bfb0078333 ◽

1987 ◽

pp. 253-264

Author(s):

Michelle Schatzman

Keyword(s):

Conservation Laws ◽

Hyperbolic Systems ◽

Systems Of Conservation Laws ◽

Well Posed

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ON BOUNDARY CONDITIONS FOR FIRST-ORDER SYMMETRIC HYPERBOLIC SYSTEMS WITH CONSTRAINTS

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891613500264 ◽

2013 ◽

Vol 10 (04) ◽

pp. 725-734 ◽

Cited By ~ 2

Author(s):

NICOLAE TARFULEA

Keyword(s):

Boundary Conditions ◽

Wave Equations ◽

Hyperbolic Systems ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

First Order ◽

The Cauchy Problem ◽

Initial Boundary Value ◽

Well Posed

The Cauchy problem for many first-order symmetric hyperbolic (FOSH) systems is constraint preserving, i.e. the solution satisfies certain spatial differential constraints whenever the initial data does. Frequently, artificial space cut-offs are performed for such evolution systems, usually out of the necessity for finite computational domains. However, it may easily happen that boundary conditions at the artificial boundary for such a system lead to an initial boundary value problem which, while well-posed, does not preserve the constraints. Here we consider the problem of finding constraint-preserving boundary conditions for constrained FOSH systems in the well-posed class of maximal non-negative boundary conditions. Based on a characterization of maximal non-negative boundary conditions, we discuss a systematic technique for finding such boundary conditions that preserve the constraints, pending that the constraints satisfy a FOSH system themselves. We exemplify this technique by analyzing a system of wave equations in a first-order formulation subject to divergence constraints.

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