GLOBAL SMOOTH SOLUTION TO A HYPERBOLIC SYSTEM ON AN INTERVAL WITH DYNAMIC BOUNDARY CONDITIONS

We consider a hyperbolic two component system of partial differential equations in one space dimension with ODE boundary conditions describing the flow of an incompressible fluid in an elastic tube that is connected to a tank at each end. Using the local-existence theory together with entropy method...

Full description

Saved in:
Bibliographic Details
Published in:Quarterly of applied mathematics 2016-01, Vol.74 (3), p.539-570
Main Authors: PERALTA, GILBERT, PROPST, GEORG
Format: Article
Language:English
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We consider a hyperbolic two component system of partial differential equations in one space dimension with ODE boundary conditions describing the flow of an incompressible fluid in an elastic tube that is connected to a tank at each end. Using the local-existence theory together with entropy methods, the existence and uniqueness of a global-in-time smooth solution is established for smooth initial data sufficiently close to the equilibrium state. Energy estimates are derived using the relative entropy method for zero order estimates while constructing entropy-entropy flux pairs for the corresponding diagonal system of the shifted Riemann invariants to deal with higher order estimates. Finally, using the linear theory and interpolation estimates, we show that the solution converges exponentially to the equilibrium state.
ISSN:0033-569X
1552-4485
DOI:10.1090/qam/1432