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Existence theory for the kinetic-fluid coupling when small droplets are treated as part of the fluid. (English) Zbl 1293.35020
The authors consider in this paper a spray constituted of an incompressible viscous gas and small droplets which can break up. This spray is modeled by the coupling of the incompressible Navier-Stokes equation and the Vlasov-Boltzmann equation, together with a fragmentation kernel. First, the authors show at the formal level that if the droplets are very small after the breakup, then the solutions of this system converge towards the solution of a simplified system in which the small droplets produced by the breakup are treated as part of the fluid. Then, existence of global weak solutions for this last system is shown to hold, thanks to the use of the DiPerna-Lions theory for singular transport equations.

MSC:
35B25 Singular perturbations in context of PDEs
76T10 Liquid-gas two-phase flows, bubbly flows
35Q30 Navier-Stokes equations
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