Lack of regularity of the transport density in the monge problem

In this paper, we provide a family of counterexamples to the regularity of the transport density in the classical Monge-Kantorovich problem. We prove that the W^{1,p} regularity of the source and target measures f ^\pm does not imply that the transport density $\sigma$ is W^{1,p} , that the BV regularity of f ^\pm does not imply that $\sigma$ is BV and that f^\pm $\in$ C^\infty does not imply that $\sigma$ is W^{1,p} , for large p

Derivation and analysis of a new 2D Green-Naghdi system

We derive here a variant of the 2D Green-Naghdi equations that model the propagation of two-directional, nonlinear dispersive waves in shallow water. This new model has the same accuracy as the standard $2D$ Green-Naghdi equations. Its mathematical interest is that it allows a control of the rotational part of the (vertically averaged) horizontal velocity, which is not the case for the usual Green-Naghdi equations. Using this property, we show that the solution of these new equations can be constructed by a standard Picard iterative scheme so that there is no loss of regularity of the solution with respect to the initial condition. Finally, we prove that the new Green-Naghdi equations conserve the almost irrotationality of the vertically averaged horizontal component of the velocity