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We study a new model for the evolution of a liquid-liquid dispersion. The droplets of the dispersed phase are supposed to move due to diffusion and to undergo coalescence and breakage. The main feature of the model is the inclusion of a maximal droplet size. This requires a consistent mechanism opposing the increase of droplets due to coalescence. The resulting system of uncountably many coupled reaction-diffusion equations is interpreted as a vector-valued Cauchy problem. We prove existence and uniqueness of nonnegative and mass-preserving solutions. Furthermore, we give sufficient conditions for global existence.
We prove null controllability results for the degenerate one-dimensional heat equation $$ u_t - (x^\alpha u_x)_x = f \chi _\omega , \quad x\in (0,1), \ \ t\in (0,T) .$$ As a consequence, we obtain null controllability results for a Crocco-type equation that describes the velocity field of a laminar flow on a flat plate.
We consider the Dirichlet problem for positive solutions of the equation $ -\Delta_m (u) = f(u) \; $ in a bounded, smooth domain $\, \Omega $, with $f$ positive and locally Lipschitz continuous. We prove a weak maximum principle in small domains for the linearized operator that we exploit to prove a weak maximum principle for the linearized operator. We then consider the case $f(s)=s^q$ and prove a nondegeneracy result in weighted Sobolev spaces.