Calculus of Variations and Geometric Measure Theory
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M. Colombo - G. Crippa - S. Spirito

Renormalized solutions to the continuity equation with an integrable damping term

created by colombom on 03 Nov 2014
modified on 17 Jun 2015


Accepted Paper

Inserted: 3 nov 2014
Last Updated: 17 jun 2015

Journal: Calc. Var. PDE
Year: 2015


We consider the continuity equation with a nonsmooth vector field and a damping term. In their fundamental paper, DiPerna and Lions proved that, when the damping term is bounded in space and time, the equation is well posed in the class of distributional solutions and the solution is transported by suitable characteristics of the vector field. In this paper, we prove existence and uniqueness of renormalized solutions in the case of an integrable damping term, employing a new logarithmic estimate inspired by analogous ideas of Ambrosio, Lecumberry, and Maniglia, Crippa and De Lellis in the Lagrangian case.

Keywords: Continuity Equation, Well-Posedeness, Lagrangian Flow, Renormalized Solution


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