Free boundary limit of a tumor growth model with nutrient

Abstract

Both compressible and incompressible porous medium models are used in the literature to describe the mechanical properties of living tissues. These two classes of models can be related using a stiff pressure law. In the incompressible limit, the compressible model generates a free boundary problem of Hele-Shaw type where incompressibility holds in the saturated phase.Here we consider the case with a nutrient. Then, a badly coupled system of equations describes the cell density number and the nutrient concentration. For that reason, the derivation of the free boundary (incompressible) limit was an open problem, in particular a difficulty is to establish the so-called complementarity relation which allows to recover the pressure using an elliptic equation. To establish the limit, we use two new ideas. The first idea, also used recently for related problems, is to extend the usual Aronson-Bénilan estimates in L ∞ to an L 2 setting. The second idea is to derive a sharp uniform L 4 estimate on the pressure gradient, independently of space dimension.

An asymptotic preserving scheme for a tumor growth model of porous medium type

“…As shown in [16], one can also pass to the limit in the equation for the pressure, which leads to the Hele-Shaw problem…”

“…At finite time the empty bubble closes up and the topological change of the support generates a singularity of the pressure gradient. In [16], the authors show that the pressure gradient is uniformly bounded with respect to γ in L 4 (R d × (0, T )). Then, they prove the sharpness of this uniform bound using the focusing solution as counterexample.…”

“…Due to the degeneracy of the diffusion, the inner hole closes up in finite time and singularities occur due to this topological change. In particular, we perform numerical tests to study the blow-up of the L p -norms of the pressure gradient, which are uniformly (with respect to γ) bounded for p ≤ 4, as recently proved in [16]. This regularity is actually optimal, and the focusing solution represents the limiting case since the L p -norms of its gradient blow up for p > 4 as γ → ∞.…”

“…This regularity is actually optimal, and the focusing solution represents the limiting case since the L p -norms of its gradient blow up for p > 4 as γ → ∞. Our aim is to obtain a numerical verification of the study of the limiting exponents from [16].…”

“…In [16], the authors prove that the L 4 -norm of ∇p is uniformly bounded with respect to γ. Then, they show that this uniform estimate is optimal using the focusing solution as a counterexample.…”