Giulio Schimperna (Università di Pavia)
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We will present some mathematical results for a new model coupling the
Cahn-Hilllard system with an evolutionary equation
describing the active (chemotactic) transport of a chemical species influencing the phase separation process. 
Specifically, the model may arise in connection with tumor growth processes;
mathematically speaking, it may be interesting
in itself as it provides a new coupling between a Keller-Segel-like relation (the equation describing the 
evolution of the concentration of the chemical substance) and a fourth order
(rather than a second order as in most
models for chemotaxis) evolutionary system. Our main result will be devoted to proving existence of weak solutions 
in the case when the chemotaxis sensitivity function has a controlled growth at infinity; a particular emphasis
will be given to discussing the occurrence of critical exponents and to
presenting a regularization scheme compatible with the a-priori estimates.
Moreover, we will discuss an extension of the model where the effects of a
macroscopic velocity flow of
Brinkman type are taken into account and analyze the Darcy limit
regime. Finally, referring to the
(more difficult) case of linear chemotactic
sensitivity we will shortly present some work in progress, in collaboration with
Elisabetta Rocca (Pavia)
and Robert Lasarzik (WIAS), related to the existence of very weak solutions as well as weak-strong uniqueness.