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

Peralta, Gilbert; Propst, Georg

doi:10.1090/qam/1432

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
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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

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, DOI 10.1007/s00205-010-0321-y
Alfio Borzí and Georg Propst, Numerical investigation of the Liebau phenomenon, Z. Angew. Math. Phys. 54 (2003), no. 6, 1050–1072. MR 2022157, 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, 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. MR 2255190, 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, 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
Miguel Ángel Fernández, Vuk Milišić, and Alfio Quarteroni, Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and hyperbolic PDEs, Multiscale Model. Simul. 4 (2005), no. 1, 215–236. MR 2164715, DOI 10.1137/030602010
B. Hanouzet and R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal. 169 (2003), no. 2, 89–117. MR 2005637, DOI 10.1007/s00205-003-0257-6
Shuichi Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 169–194. MR 899951, DOI 10.1017/S0308210500018308
Ta Tsien Li, Global classical solutions for quasilinear hyperbolic systems, RAM: Research in Applied Mathematics, vol. 32, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. MR 1291392
Tatsien Li and Libin Wang, Global propagation of regular nonlinear hyperbolic waves, Progress in Nonlinear Differential Equations and their Applications, vol. 76, Birkhäuser Boston, Ltd., Boston, MA, 2009. MR 2524566
P.-L. Lions, B. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys. 163 (1994), no. 2, 415–431. MR 1284790
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, 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
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, 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, DOI 10.1007/978-1-4612-5561-1
Gilbert Peralta and Georg Propst, Local well-posedness of a class of hyperbolic PDE-ODE systems on a bounded interval, J. Hyperbolic Differ. Equ. 11 (2014), no. 4, 705–747. MR 3312050, DOI 10.1142/S0219891614500222
Gilbert Peralta and Georg Propst, Stability and boundary controllability of a linearized model of flow in an elastic tube, ESAIM Control Optim. Calc. Var. 21 (2015), no. 2, 583–601. MR 3348413, DOI 10.1051/cocv/2014039
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, 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, DOI 10.14492/hokmj/1381757663
Michael E. Taylor, Partial differential equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Nonlinear equations; Corrected reprint of the 1996 original. MR 1477408
Wen-An Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal. 172 (2004), no. 2, 247–266. MR 2058165, DOI 10.1007/s00205-003-0304-3