### An ill posed Cauchy problem for a hyperbolic system in two space dimensions

Alberto Bressan (2003)

Rendiconti del Seminario Matematico della Università di Padova

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Alberto Bressan (2003)

Rendiconti del Seminario Matematico della Università di Padova

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Luigi Ambrosio (2004)

Journées Équations aux dérivées partielles

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Michael Sever (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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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...

Michael Sever (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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Nohel, J. A.

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Francois Bouchut, Francois James, Simona Mancini (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction...