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Author: Rocca, Elisabetta × Subject: weak solutions ×

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Now showing 1 - 3 of 3
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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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    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.
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    A model for resistance welding including phase transitions and Joule heating
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Hömberg, Dietmar; Rocca, Elisabetta
    In this paper we introduce a new model for solid-liquid phase transitions triggered by Joule heating as they arise in the case of resistance welding of metal parts. The main novelties of the paper are the coupling of the thermistor problem with a phase field model and the consideration of phase dependent physical parameters through a mixture ansatz. The PDE system resulting from our modelling approach couples a strongly nonlinear heat equation, a non-smooth equation for the the phase parameter (standing for the local proportion of one of the two phases) with quasistatic electric charge conservation law. We prove existence of weak solutions in the 3D case, while the regularity result and the uniqueness of solution is stated only in the 2D case. Indeed, uniqueness for the three dimensional system is still an open problem.