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A uniqueness result for the continuity equation in two dimensions
Journal
Journal of the European Mathematical Society
Volume
16
Number
2
Pages / Article-Number
201-234
Keywords
Continuity equation, transport equation, uniqueness of weak solutions, weak Sard property, disintegration of measures, coarea formula
Abstract
We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation partial derivative(t)u + div(bu) = 0 admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential f associated to b. As a corollary we obtain uniqueness under the assumption that the curl of b is a measure. This result can be extended to certain nonautonomous vector fields b with bounded divergence.
