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Title: Weak Solutions in Nonlinear Poroelasticity with Incompressible Constituents
Speaker: Boris Muha
University of Zagreb, Croatia
Abstract: We consider quasi-static poroelastic systems with incompressible constituents, nonlinear permeability dependent on solid dilation, and physical types of boundary conditions (Dirichlet, Neumann, and mixed) for the fluid pressure, motivated by applications in bio mechanics and in particular tissue perfusion. These systems fall in the category of implicit, degenerate nonlinear evolution problems. We provide a straightforward fixed point map strategy for proving existence of weak solutions, made possible due to a novel result on uniqueness of weak solution to the associated linear poroelasticity system with given permeability as a function of space and time. The uniqueness proof is based on obtaining energy estimates for all weak solutions, rather than just the constructed (as limits of approximations) solutions. The results of this work provide a foundation to addresss trong solutions and uniqueness of weak solutions for the nonlinear porous media system.This is joint work with L. Bociu and J. Webster.