Abstract
We prove global existence of a weak solution to the angiogenesis model proposed by A. Tosin, D. Ambrosi, L. Preziosi in Bull. Math. Biol. (2006) 7, 1819-1836. The model consists of compressible Navier-Stokes equations coupled with a reaction-diffusion equation describing the concentration of a chemical solution responsible of endothelial cells migration and blood vessels formation.
Proofs are based on the control of the entropy associated to the hyperbolic equation of conservation mass and the adaptation of the results of P.L. Lions dealing with compressible fluids which are inevitable for all models dealing with compressible Navier-Stokes equations.
We use the vanishing artificial viscosity method to prove existence of solutions, the main difficulty for passing to the limit is the lack of compactness due to hyperbolic equation which usually induces resonance phenomenon. This is overcome by using the concept of the compactness of effective viscous pressure combined with suitable renormalized solutions to the hyperbolic equation of mass conservation.
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References
Ambrosi, D., Gamba, A., Serini, G.: Cell directional and chemotaxis in vascular morphogenesis. Bull. Math. Biol. 66, 1851–1873 (2004)
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Tosin, A., Ambrosi, D., Preziosi, L.: Mechanics and chemotaxis in the morphogenesis of vascular networks. Bull. Math. Biol. 68(7), 1819–1836 (2006)
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Aïssa, N., Alexandre, R. (2016). Global Existence of Weak Solutions to an Angiogenesis Model. In: Russo, G., Capasso, V., Nicosia, G., Romano, V. (eds) Progress in Industrial Mathematics at ECMI 2014. ECMI 2014. Mathematics in Industry(), vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-23413-7_151
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DOI: https://doi.org/10.1007/978-3-319-23413-7_151
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