On the Motion of Non-dissipative Inhomogeneous Fluid-like Bodies

We study a system of equation governing evolution of inhomogeneous fluid-like bodies (granular fluid) disregarding effects of viscosity. A local well-posedness theory is developed on a bounded smooth domain with no-slip boundary condition on velocity and vanishing gradient of density. The cases of open space and periodic box are also considered, where the local existence and uniqueness of solutions is shown in Sobolev spaces up to the critical smoothness n2 + 1. 1. PROBLEM FORMULATION We consider a system of partial differential equations: (1) divxv = 0, (2) ∂t%+ v · ∇x% = 0, (3) % ( ∂tv+ divx(v⊗v) ) +βdivx(∇x%⊗∇x%) = −∇xp+ β 3 ∇x|∇x%|, where β > 0 is a given constant, the unknowns % = %(t, x), v = v(t, x), (and p = p(t, x)) are functions of the time t ∈ (0, T ) and the spatial coordinate x ∈ Ω a bounded regular domain in the Euclidean space R. Problem (1) (3) is supplemented with the boundary conditions (4) v · n|∂Ω =