Stability of Initial Boundary Value Problems for Hyperbolic Systems

1989 ◽  
pp. 698-707
Author(s):  
M. Ziółko
Keyword(s):  
Boundary Value Problems ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

Generic types and transitions in hyperbolic initial–boundary-value problems

2002 ◽  
Vol 132 (5) ◽  
pp. 1073-1104 ◽  
Author(s):  
Sylvie Benzoni-Gavage ◽  
Frédéric Rousset ◽  
Denis Serre ◽  
K. Zumbrun
Keyword(s):  
Boundary Value Problems ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Stability Class ◽  
Weakly Stable ◽  
The Stability ◽  
Initial Boundary Value ◽  
Open Class

The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition.Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system.Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

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