×

Nonlinear and linear hyperbolic systems with dynamic boundary conditions. (English) Zbl 1356.35129

Summary: We consider first order hyperbolic systems on an interval with dynamic boundary conditions. The well-posedness for linear systems is established by using a variational method. The linear theory is used to analyze the local-in-timewell-posedness for nonlinear systems. The results are applied to a model describing the flow of an incompressible fluid inside an elastic tube whose ends are attached to tanks. Global existence and stability for data that are smooth enough and close to the steady state are obtained by using energy and entropy methods.

MSC:

35L50 Initial-boundary value problems for first-order hyperbolic systems
35F61 Initial-boundary value problems for systems of nonlinear first-order PDEs
35B35 Stability in context of PDEs
35L60 First-order nonlinear hyperbolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] N. Antonic and K. Burazin. Graph spaces of first-order linear partial differential operators. Math. Commun., 14 (2009), 135-155. · Zbl 1183.35079
[2] S. Benzoni-Gavage and D. Serre. MultidimensionalHyperbolic PartialDifferential Equations, Oxford University Press (2007). · Zbl 1113.35001
[3] A. Borzì and G. Propst. Numerical investigation of the Liebau phenomenon. Z. Angew. Math. Phys., 54 (2003), 1050-1072. · Zbl 1047.76128 · doi:10.1007/s00033-003-1108-x
[4] J. Chazarain and A. Piriou. Introductionto the Theory of Linear Partial Differential Equations, North-Holland Publishing Co., Amsterdam (1982). · Zbl 0487.35002
[5] J.-F. Coulombel. Stabilité multidimensionnelle d’interfaces dynamiques; application aux transitions de phase liquide-vapeur, PhD thesis, École normale supérieure de Lyon (2002).
[6] K. Fellner and G. Raoul. Stability of stationary states of non-local interaction equations. Mathematical and Computer Modelling, 53 (2011), 2267-2291. · Zbl 1213.35079 · doi:10.1016/j.mcm.2010.03.021
[7] M. Fernández, V. Milisic and A. Quarteroni. Analysis of a geometrical multiscale blood flowmodel based on the coupling ofODEs and hyperbolic PDEs. Multiscale Model. Simul., 4 (2005), 215-236. · Zbl 1085.35095 · doi:10.1137/030602010
[8] K. O. Friedrichs. Symmetric positive linear differential equations. Comm. Pure Appl. Math., 11 (1958), 333-418. · Zbl 0083.31802 · doi:10.1002/cpa.3160110306
[9] M. Jensen. Discontinuous Galerkin methods for Friedrichs systems with irregular solutions, PhD thesis, University of Oxford (2004).
[10] T.-T. Li. Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson, Paris (1994). · Zbl 0841.35064
[11] G. Métivier. Stability of multidimensional shocks. Advances in the Theory of ShockWaves, ed. H. Freistühler and A. Szepessy, pp. 25-103, Birkhäuser, Boston (2001). · Zbl 1017.35075
[12] J. T. Ottesen. Valveless pumping in a fluid-filled closed elastic tube-system: onedimensional theory with experimental validation. J. Math. Biol., 46 (2003), 309-332. · Zbl 1039.92015 · doi:10.1007/s00285-002-0179-1
[13] G. Peralta and G. Propst. Local well-posedness of a class of hyperbolic PDE-ODE systems on a bounded interval. J. Hyperbolic Differ. Eq., 11 (2014), 705-747. · Zbl 1316.35176 · doi:10.1142/S0219891614500222
[14] G. Peralta and G. Propst. Stability and boundary controllability of a linearized model of flow in an elastic tube. ESAIM: Control, Optimisation and Calculus of Variations, 21 (2015), 583-601. · Zbl 1330.35033 · doi:10.1051/cocv/2014039
[15] G. Peralta and G. Propst. Well-Posedness and regularity of linear hyperbolic systems with dynamic boundary conditions, to appear in Proc. Roy. Soc. Edinburgh Sect. A. · Zbl 1432.35137
[16] G. Peralta and G. Propst. Global smooth solution to a hyperbolic system on an interval with dynamic boundary conditions, to appear in Q. Appl. Math. · Zbl 1347.35152
[17] H. J. Rath and I. Teipel. Der Fördereffekt in ventillosen, elastischen Leitungen. Z. Angew. Math. Phys., 29 (1978), 123-133. · doi:10.1007/BF01797309
[18] J. B. Rauch and F. J. Massey. Differentiability of solutions to hyperbolic initialboundary value problems. Trans. Amer. Math. Soc., 189 (1974), 303-318. · Zbl 0282.35014
[19] W. H. Ruan, M. E. Clark, M. Zhao and A. Curcio. Global solution to a hyperbolic problemarising in themodelingof blood flow in circulatory systems. J. Math. Anal. Appl., 331 (2007), 1068-1092. · Zbl 1113.35118 · doi:10.1016/j.jmaa.2006.09.034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.