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Mechanical models for tumor growth have been used extensively in recent years for the analysis of medical observations and for the prediction of cancer evolution based on image analysis. This work deals with the numerical approximation of a mechanical model for tumor growth and the analysis of its dynamics. The system under investigation is given by a multi-phase flow model: The densities of the different cells are governed by a transport equation for the evolution of tumor cells, whereas the velocity field is given by a Brinkman regularization of the classical Darcy’s law. An efficient finite difference scheme is proposed and shown to converge to a weak solution of the system. Our approach relies on convergence and compactness arguments in the spirit of Lions [P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 2. Vol. 10 of Oxford Lecture Series Math. Appl. The Clarendon Press, Oxford University Press, New York (1998)]. [ABSTRACT FROM AUTHOR]

We establish the global existence of weak solutions to a class of kinetic flocking equations. The models under consideration include the kinetic Cucker--Smale equation [Cucker and Smale, IEEE Trans. Automat. Control, 52 (2007), pp. 852-862, Cucker and Smale, Japan. J. Math., 2 (2007), pp. 197-227] with possibly nonsymmetric flocking potential, the Cucker-Smale equation with additional strong local alignment, and a newly proposed model by Motsch and Tadmor [J. Statist. Phys., 141 (2011), pp. 923-947]. The main tools employed in the analysis are the velocity-averaging lemma and the Schauder fixed-point theorem along with various integral bounds. [ABSTRACT FROM AUTHOR]