We consider conservation laws on moving hypersurfaces. In this work the velocity of the surface is prescribed. But one may think of the velocity to be given by PDEs in the bulk phase. We prove existence and uniqueness for a scalar conservation law on the moving surface. This is done via a parabolic regularization of the hyperbolic PDE. We then prove suitable estimates for the solution of the regularized PDE, that are independent of the regularization parameter. We introduce the concept of an entropy solution for a scalar conservation law on a moving hypersurface. We also present some numerical experiments. As in the Euclidean case we expect discontinuous solutions, in particular shocks. It turns out that in addition to the “Euclidean shocks” geometrically induced shocks may appear.
Gerhard Dziuk
1
;
Dietmar Kröner
1
;
Thomas Müller
1
1
Universität Freiburg, Germany
Gerhard Dziuk; Dietmar Kröner; Thomas Müller. Scalar conservation laws on moving hypersurfaces. Interfaces and free boundaries, Tome 15 (2013) no. 2, pp. 203-236. doi: 10.4171/ifb/301
@article{10_4171_ifb_301,
author = {Gerhard Dziuk and Dietmar Kr\"oner and Thomas M\"uller},
title = {Scalar conservation laws on moving hypersurfaces},
journal = {Interfaces and free boundaries},
pages = {203--236},
year = {2013},
volume = {15},
number = {2},
doi = {10.4171/ifb/301},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/301/}
}
TY - JOUR
AU - Gerhard Dziuk
AU - Dietmar Kröner
AU - Thomas Müller
TI - Scalar conservation laws on moving hypersurfaces
JO - Interfaces and free boundaries
PY - 2013
SP - 203
EP - 236
VL - 15
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/301/
DO - 10.4171/ifb/301
ID - 10_4171_ifb_301
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%A Dietmar Kröner
%A Thomas Müller
%T Scalar conservation laws on moving hypersurfaces
%J Interfaces and free boundaries
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%P 203-236
%V 15
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4171/ifb/301/
%R 10.4171/ifb/301
%F 10_4171_ifb_301
