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
DOI:
https://doi.org/10.1090/qam/1432
Published electronically:
June 16, 2016
MathSciNet review:
3518227
Full-text PDF Free Access
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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.
References
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References
- Karine Beauchard and Enrique Zuazua, Large time asymptotics for partially dissipative hyperbolic systems, Arch. Ration. Mech. Anal. 199 (2011), no. 1, 177–227. MR 2754341 (2012e:35146), DOI 10.1007/s00205-010-0321-y
- Alfio Borzì and Georg Propst, Numerical investigation of the Liebau phenomenon, Z. Angew. Math. Phys. 54 (2003), no. 6, 1050–1072. MR 2022157 (2004k:76138), DOI 10.1007/s00033-003-1108-x
- Sunčica Čanić and Eun Heui Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math. Methods Appl. Sci. 26 (2003), no. 14, 1161–1186. MR 2002976 (2004g:76139), DOI 10.1002/mma.407
- Jean-François Coulombel and Thierry Goudon, The strong relaxation limit of the multidimensional isothermal Euler equations, Trans. Amer. Math. Soc. 359 (2007), no. 2, 637–648 (electronic). MR 2255190 (2007h:35215), DOI 10.1090/S0002-9947-06-04028-1
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377 (2011i:35150), DOI 10.1007/978-3-642-04048-1
- Klemens Fellner and Gaël Raoul, Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Modelling 53 (2011), no. 7-8, 1436–1450. MR 2782822, DOI 10.1016/j.mcm.2010.03.021
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- Hailiang Liu, Jinghua Wang, and Tong Yang, Stability of a relaxation model with a nonconvex flux, SIAM J. Math. Anal. 29 (1998), no. 1, 18–29. MR 1617172 (99k:39055), DOI 10.1137/S003614109629903X
- Takaaki Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publications Mathématiques d’Orsay, No. 78-02, Département de Mathématique, Université de Paris-Sud, Orsay, 1978. MR 0481578 (58 \#1690)
- J. T. Ottesen, Valveless pumping in a fluid-filled closed elastic tube-system: one-dimensional theory with experimental validation, J. Math. Biol. 46 (2003), no. 4, 309–332. MR 1980846 (2004d:76044), DOI 10.1007/s00285-002-0179-1
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486 (85g:47061), DOI 10.1007/978-1-4612-5561-1
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- H. J. Rath and I. Teipel, Der Fördereffekt in ventillosen, elastischen Leitungen, Z. Angew. Math. Phys. 29 (1978), 123–133.
- Weihua Ruan, M. E. Clark, Meide Zhao, and Anthony Curcio, Global solution to a hyperbolic problem arising in the modeling of blood flow in circulatory systems, J. Math. Anal. Appl. 331 (2007), no. 2, 1068–1092. MR 2313701 (2008c:35175), DOI 10.1016/j.jmaa.2006.09.034
- Yasushi Shizuta and Shuichi Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), no. 2, 249–275. MR 798756 (86k:35107), DOI 10.14492/hokmj/1381757663
- Michael E. Taylor, Partial differential equations. III, Nonlinear equations. Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. MR 1477408 (98k:35001)
- Wen-An Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal. 172 (2004), no. 2, 247–266. MR 2058165 (2005c:35195), DOI 10.1007/s00205-003-0304-3
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Additional Information
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
MR Author ID:
142285
Email:
georg.propst@uni-graz.at
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