Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Global smooth solution to a hyperbolic system on an interval with dynamic boundary conditions


Authors: Gilbert Peralta and Georg Propst
Journal: Quart. Appl. Math. 74 (2016), 539-570
MSC (2010): Primary 35L40, 35F61
Published electronically: June 16, 2016
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Abstract: 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.


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Gilbert Peralta
Affiliation: Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, Baguio City, 2600, Philippines – and – Institut für Mathematik und Wissenschaftliches Rechnen, NAWI Graz, Karl-Franzens-Universität Graz, Heinrichstrasse 36, A-8010 Graz, Austria
Email: grperalta@upb.edu.ph

Georg Propst
Affiliation: Institut für Mathematik und Wissenschaftliches Rechnen, NAWI Graz, Karl-Franzens-Universität Graz, Heinrichstrasse 36, A-8010 Graz, Austria
Email: georg.propst@uni-graz.at

DOI: https://doi.org/10.1090/qam/1432
Received by editor(s): July 6, 2015
Published electronically: June 16, 2016
Additional Notes: The first author was supported by the grant Technologiestipendien Südostasien in the frame of ASEA-Uninet granted by the Austrian Agency for International Cooperation in Education and Research (OeAD-GmbH) and financed by the Austrian Federal Ministry for Science and Research (BMWF)
Article copyright: © Copyright 2016 Brown University


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