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Author: Frigeri, Sergio × Subject: weak solutions ×

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    On a diffuse interface model of tumor growth
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Frigeri, Sergio; Grasselli, Maurizio; Rocca, Elisabetta
    We consider a diffuse interface model of tumor growth proposed by A. Hawkins-Daruud et al. This model consists of the Cahn-Hilliard equation for the tumor cell fraction φ nonlinearly coupled with a reaction-diffusion equation for ψ which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation functionp(φ) multiplied by the differences of the chemical potentials for φ and ψ. The system is equipped with no-flux boundary conditions which entails the conservation of the total mass, that is, the spatial average of φ+ψ. Here we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.
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    On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Frigeri, Sergio; Gal, Cipian G.; Grasselli, Maurizio
    We consider a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. This model consists of the Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. Several results were already proven by two of the present authors. However, in the two-dimensional case, the uniqueness of weak solutions was still open. Here we establish such a result even in the case of degenerate mobility and singular potential. Moreover, we show the strong-weak uniqueness in the case of viscosity depending on the order parameter, provided that the mobility is constant and the potential is regular. In the case of constant viscosity, on account of the uniqueness results we can deduce the connectedness of the global attractor whose existence was obtained in a previous paper. The uniqueness technique can be adapted to show the validity of a smoothing property for the difference of two trajectories which is crucial to establish the existence of an exponential attractor.
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    A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Frigeri, Sergio; Grasselli, Maurizio; Rocca, Elisabetta
    We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Using a similar technique, we show that there is a global attractor for the convective nonlocal Cahn-Hilliard equation with degenerate mobility and singular potential in dimension three.