Energy equality and renormalization of Navier-Stokes-Vlasov equations

We study weak solutions of the incompressible and the compressible Navier--Stokes--Vlasov systems on a bounded domain, where a viscous fluid is coupled to a dispersed particle phase through a drag force; the fluid obeys the no-slip boundary condition and the particles are subject to specular reflection. For solutions with Besov regularity, we show that the Vlasov equation is renormalizable in the sense of DiPerna--Lions, and we give regularity thresholds for the validity of the global energy equality. If the distribution function f belongs to Bₚα,∞ in space and velocity with time integrability p, and the fluid velocity u belongs to Bqβ,∞ jointly in space and time with integrability q, then the energy equality holds provided $α+β>1$ and $β$ exceeds an explicit bound depending on p, q and the dimension, under a natural integrability assumption on the pressure; an analogous criterion holds for the compressible system when the density is bounded from above and below. The condition $αp>1$ imposed on f is not merely technical: it is exactly the range in which the macroscopic momentum flux possesses a boundary trace, so that the specular reflection condition passes to the macroscopic energy balance. When the particle phase is absent, our thresholds coincide with the criteria β>12+1q and β>3q obtained by Yu for the incompressible Navier--Stokes equations alone. The proofs combine a boundary cutoff argument with commutator estimates based on a scale-localized Besov modulus, together with a boundary-strip analysis.