A uniqueness result for the continuity equation in dimension two
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Mon, 09/05/2011 17:00 |
Giovanni Alberti (Universita di Pisa) |
Partial Differential Equations Seminar  |
Gibson 1st Floor SR |
| We give a characterization of divergence-free vector fields on the plane such that the Cauchy problem for the associated continuity (or transport) equation has a unique bounded solution (in the sense of distribution). Unlike previous results in this directions (Di Perna-Lions, Ambrosio), the proof relies on a dimension-reduction argument, which can be regarded as a variant of the method of characteristics. Note that our characterization is not stated in terms of function spaces, but is based on a suitable weak formulation of the Sard property for the potential associated to the vector-field. This is a joint work with S. Bianchini (SISSA, Trieste) and Gianluca Crippa (Parma). | |||