We describe a constructive algorithm for obtaining smooth solutions of a nonlinear, nonhyperbolic pair of balance laws modeling incompressible two-phase flow in one space dimension and time. Solutions are found as stationary solutions of a related hyperbolic system, based on the introduction of an artificial time variable. As may be expected for such nonhyperbolic systems, in general the solutions obtained do not satisfy both components of the given initial data. This deficiency may be overcome, however, by introducing an alternative “solution” satisfying both components of the initial data and an approximate form of a corresponding linearized system.

Classification: 35M99, 35Q35, 76T10

Keywords: nonhyperbolic balance laws, incompressible two-fluid flow

@article{M2AN_2005__39_1_37_0, author = {Sever, Michael}, title = {Solutions of a nonhyperbolic pair of balance laws}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {39}, number = {1}, year = {2005}, pages = {37-58}, doi = {10.1051/m2an:2005003}, zbl = {1080.35092}, mrnumber = {2136199}, language = {en}, url = {http://www.numdam.org/item/M2AN_2005__39_1_37_0} }

Sever, Michael. Solutions of a nonhyperbolic pair of balance laws. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 1, pp. 37-58. doi : 10.1051/m2an:2005003. http://www.numdam.org/item/M2AN_2005__39_1_37_0/

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