Two-velocity hydrodynamics in fluid mechanics: global existence for 2D case

In this paper, we consider a compressible-incompressible two-velocity hydrodynamic system studied by Bresch et al (2015 J. Math. Pure Appl. 104 762-800) and Lions (1996 Mathematical Topics in Fluid Mechanics (Oxford: OUP)). When the density rho is a small perturbation of a constant, we establish a priori estimates by using some delicate structure of the nonlinear terms and Hardy space. By using these a priori estimates, we prove the existence of global strong solutions and weak solutions. Our results do not require any constraint between the viscosity and the conductivity and improve the results of Bresch et al (2015) and Lions (1996) in the two-dimensional case. As an application of our results we also establish global strong solutions and weak solutions for a model of gaseous mixture and the ghost effect system.