Publications [#347990] of Jian-Guo Liu

Papers Published

1. Liu, JG; Tang, M; Wang, L; Zhou, Z, Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics, Discrete and Continuous Dynamical Systems - Series B, vol. 24 no. 7 (July, 2019), pp. 3011-3035 [doi] [high impact paper]
   (last updated on 2024/04/23)

   Abstract:
   In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density n is governed by the Darcy law via the pressure p(n) = n γ . For finite γ, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As γ → ∞, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.