Singularities on the boundary of the hyperbolicity region

1990 ◽  
Vol 52 (4) ◽  
pp. 3326-3337
Author(s):  
A. D. Vainshtein ◽  
B. Z. Shapiro
Keyword(s):  
Hyperbolicity Region

Local well-posedness of a class of hyperbolic PDE–ODE systems on a bounded interval

2014 ◽  
Vol 11 (04) ◽  
pp. 705-747 ◽  
Author(s):  
Gilbert Peralta ◽  
Georg Propst
Keyword(s):  
Blow Up ◽  
Hyperbolic Systems ◽  
Energy Estimates ◽  
Well Posedness ◽  
Hyperbolic Pde ◽  
Flow Models ◽  
Bounded Interval ◽  
First Order ◽  
Hyperbolicity Region ◽  
Set Up

The well-posedness theory for hyperbolic systems of first-order quasilinear PDE's with ODE's boundary conditions (on a bounded interval) is discussed. Such systems occur in multi-scale blood flow models, as well as valveless pumping and fluid mechanics. The theory is presented in the setting of Sobolev spaces Hm (m ≥ 3 being an integer), which is an appropriate set-up when it comes to proving existence of smooth solutions using energy estimates. A blow-up criterion is also derived, stating that if the maximal time of existence is finite, then the state leaves every compact subset of the hyperbolicity region, or its first-order derivatives blow-up. Finally, we discuss physical examples which fit in the general framework presented.

scholarly journals On the connectivity of the hyperbolicity region of irreducible polynomials

2019 ◽  
Vol 19 (2) ◽  
pp. 231-233
Author(s):  
Mario Denis Kummer
Keyword(s):  
Hyperbolic Polynomial ◽  
Irreducible Polynomials ◽  
Hyperbolicity Region ◽  
Hyperbolicity Cones

Abstract We give a proof for the fact that an irreducible hyperbolic polynomial has only one pair of hyperbolicity cones. Apart from the use of Bertini’s Theorem the proof is elementary.

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