A fragmentation model for sprays and L² stability estimates for shockes solutions of scalar conservation laws using the relative entropy method

We present a mathematical study of two conservative systems in fluid mechanics. First, we study a fragmentation model for sprays. The model takes into account the break-up of spray droplets due to drag forces. In particular, we establish the existence of global weak solutions to a system of incompressible Navier-Stokes equations coupled with a Boltzmann-like kinetic equation. We assume the particles initially have bounded radii and bounded velocities relative to the gas, and we show that those bounds remain as the system evolves. One interesting feature of the model is the apparent accumulation of particles with arbitrarily small radii. As a result, there can be no nontrivial hydrodynamical equilibrium for this system. Next, with an interest in understanding hydrodynamical limits in discontinuous regimes, we study a classical model for shock waves. Specifically, we consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension, and we investigate the behavior of bounded L² perturbations of shock wave solutions to the Riemann problem using the relative entropy method. We show that up to a time-dependent translation of the shock, the L² norm of a perturbed solution relative to the shock wave is bounded above by the L² norm of the initial perturbation. Finally, we include some preliminary relative entropy estimates which are suitable for a study of shock wave solutions to n x n systems of conservation laws having a convex entropy.