  We investigate a compressible two-fluid Navier-Stokes type system with a
single velocity field and algebraic closure for the pressure law. The
constitutive relation involves densities of both fluids through an implicit
function. We are interested in regular solutions in a $L_p-L_q$ maximal
regularity setting. We show that such solutions exists locally in time and,
under additional smallness assumptions on the initial data, also globally. Our
proof rely on appropriate transformation of the original problem, application
of Lagrangian coordinates and maximal regularity estimates for associated
linear problem.
