On the hydrodynamic models with non-local forces: Critical thresholds and large-time behavior

Dr Ewelina Zatorska (UCL)

Frank Adams 1,

I will discuss the one-dimensional pressureless Euler-Poisson equations with a linear damping and non-local interaction forces. These equations are relevant for modelling  collective behavior in mathematical biology. We provide a sharp threshold between the supercritical region with finite-time breakdown and the subcritical region with global-in-time existence of the classical solution. We derive an explicit form of solution in Lagrangian coordinates which enables us to study the time-asymptotic behavior of classical solutions with the initial data in the subcritical region.  I will also mention more recent result for the multi-dimensional Navier-Stokes type system with similar nonlocal forces and a local repulsion modelled by the pressure. 
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