A coupled system of elliptic/conservation law arising in cell self-organization

Tuesday, July 21, 2009 - 9:00am - 9:50am
EE/CS 3-180
Benoit Perthame (Université de Paris VI (Pierre et Marie Curie))
Several models have been proposed in order to describe cell communities self-organisation. One of them consists in coupling a multidimensional scalar conservation law with an elliptic equation which gradient determines the flux in the conservation law.

In dimension larger than 1, the model looses all nice properties of hyperbolic conservation laws: no contraction property, no BV bound, no regularizing effect. That is the reason why our approach for existence of solutions is based on the kinetic formulation. We recall how weak limits can be handled with this tool and strong convergence follows from uniqueness.

In the case at hand, the specific nonlinearity creates an additional defect measure. Fine analysis of properties of this measure provides us with the lacking information to prove uniqueness and deduce that the weak limit still satisfies the system.
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