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We prove global stability results of DiPerna-Lionsrenormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann,...
We study molecular motor-induced microtubule self-organization in dilute and semi-dilute
filament solutions. In the dilute case, we use a probabilistic model of microtubule
interaction via molecular motors to investigate microtubule bundle dynamics. Microtubules
are modeled as polar rods interacting through fully inelastic, binary collisions. Our
model indicates that initially disordered systems of interacting rods exhibit an
orientational instability...
This paper discusses two new directions in velocity averaging. One is an improvement of the known velocity averaging results for functions. The other shows how to adapt some of the ideas of velocity averaging to a situation that is essentially a new formulation of the Vlasov-Maxwell system.
Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusion
should be nonlinear if there exist non-diagonal terms. The vast variety of nonlinear
multicomponent diffusion equations should be ordered and special tools are needed to
provide the systematic construction of the nonlinear diffusion equations for
multicomponent mixtures with significant interaction between components. We develop an
approach to nonlinear multicomponent...
This short course explains how the usual mean-field evolution PDEs in Statistical Physics - such as the Vlasov-Poisson, Schrödinger-Poisson or time-dependent Hartree-Fock equations - are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution of large, interacting particle systems.
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