On the regularity of the pressure field of relaxed solutions to Euler equations with variable density

In this paper we improve the regularity in time of the gradient of the pressure field in the solution of relaxed version of variational formulation proposed by V. I. Arnold and by Y. Brenier, for the incompressible Euler equations with variable density. We obtain that the pressure field is not only a measure, but a function in L l o c 2 ( ( 0 , T ) ; B V l o c ( D ) ) as an extension of the work of Ambrosio and Figalli (2008) in  [1] to the variable density case.