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

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 setti...

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Bibliographic Details
Published in:Journal of hyperbolic differential equations 2014-12, Vol.11 (4), p.705-747
Main Authors: Peralta, Gilbert, Propst, Georg
Format: Article
Language:English
Subjects:
Blood flow
Boundary conditions
Computational fluid dynamics
Criteria
Derivatives
Estimates
Fluid flow
Fluid mechanics
Hyperbolic differential equations
Hyperbolic systems
Intervals
Partial differential equations
Sobolev space
Well posed problems
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Summary: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.
ISSN:0219-8916
1793-6993
DOI:10.1142/S0219891614500222