Abstract
We consider weakly coupled systems of nonlinear hyperbolic conservation laws on moving surfaces. As in the Euclidean space, see, for example, (Levy, Commun Partial Differ Equ 17(3–4):657–698, 1992, [9], Rohde, Weakly coupled systems of hyperbolic conservation laws. PhD thesis, Mathematische Fakultät der Albert-Ludwigs-Universität Freiburg 1996, [16]), the coupling is realized by a source term, which only depends on (x, t) and the unknown function u(x, t) but not on its derivatives. Scalar conservation laws on moving surfaces were considered in Dziuk et al., Interfaces Free Boundaries 15:202–236, 2013, [4]), Lengeler and Müller (J Differ Equ 254(4):1705–1727, 2013, [10]). The velocity of the surface is given by a smooth function, and we assume the surface to be compact. We prove the existence for an entropy solution. First, we consider the regularized parabolic problem with viscosity parameter \(\varepsilon \) and show that there exists a weak solution by decoupling and linearizing the problem. Then, we prove the boundedness of this solution in \(L^\infty (G_T)\), use standard regularity results to prove that this solution is a solution in the classical sense, and show uniform boundedness in \(L^\infty (G_T)\) and \(W^{1,1}(G_T)\) with respect to \(\varepsilon \).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
P. Amorim, M. Ben-Artzi, P.G. LeFloch et al., Hyperbolic conservation laws on manifolds: total variation estimates and the finite volume method. Methods Appl. Anal. 12(3), 291–324 (2005)
M. Ben-Artzi, P. G. Le Floch, Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds. Ann. Inst. H. Poincaré Anal. Nonlinéaire 24(6), 989–1008 (2007)
G. Dziuk, C.M. Elliott, Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262–292 (2007)
G. Dziuk, D. Kröner, T. Müller, Scalar conservation laws on moving hypersurfaces. Interfaces Free Boundaries 15, 202–236 (2013)
D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics (Springer, Berlin, 2015)
T. Hillen, C. Rohde, F. Lutscher, Existence of weak solutions for a hyperbolic model of chemosensitive movement. J. Math. Anal. Appl. 260(1), 173–199 (2001)
A. Korsch, Weakly Coupled Systems of Conservation Laws on Moving Surfaces. Ph.D. thesis, Mathematische Fakultät der Albert-Ludwigs-Universität Freiburg (2016)
O. Ladyz̆enskaja, V. Solonnikov, N. Ural’ceva, Linear and Quasilinear Equations of Parablic Type, vol. 23. Translations of Mathematical Monographs, Rhode Island (1968)
A. Levy, On Majda’s model for dynamic combustion. Commun. Partial Differ. Equ. 17(3–4), 657–698 (1992)
D. Lengeler, T. Müller, Scalar conservation laws on constant and time-dependent riemannian manifolds. J. Differ. Equ. 254(4), 1705–1727 (2013)
A. Majda, A qualitative model for dynamic combustion. SIAM J. Appl. Math. 41(1), 70–93 (1981)
T. Müller, Scalar Conservation Laws on Time-Dependent Riemannian Manifolds. Ph.D. thesis, Albert-Ludwigs-Universität Freiburg (2014)
J. Malek, J. Necas, M. Rokyta, M. Ruzicka. Weak and measure-valued solutions to evolutionary PDEs, volume 13 of Applied Mathematics and Mathematical Computation. Chapman & Hall (1996)
M. Ohlberger, C. Rohde, Adaptive finite volume approximations for weakly coupled convection dominated parabolic systems. IMA J. Numer. Anal. 22(2), 253–280 (2002)
E.Y. Panov, On the Dirichlet problem for first order quasilinear equations on a manifold. Trans. Am. Math. Soc. 363, 2393–2446 (2011)
C. Rohde, Weakly Coupled Systems of Hyperbolic Conservation Laws. Ph.D. thesis, Mathematische Fakultät der Albert-Ludwigs-Universität Freiburg (1996)
C. Rohde, Entropy solutions for weakly coupled hyperbolic systems in several space dimensions. Zeitschrift für angewandte Mathematik und Physik ZAMP 49(3), 470–499 (1998)
C. Rohde, W.-A. Yong, The nonrelativistic limit in radiation hydrodynamics: I. Weak entropy solutions for a model problem. J. Differ. Equ. 234(1), 91–109 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Korsch, A. (2018). Weakly Coupled Systems of Conservation Laws on Moving Surfaces. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 237. Springer, Cham. https://doi.org/10.1007/978-3-319-91548-7_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-91548-7_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91547-0
Online ISBN: 978-3-319-91548-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)