Convergence rate for the incompressible limit of nonlinear diffusion-advection equations

Abstract

The incompressible limit of nonlinear diffusion equations of porous medium type has attracted a lot of attention in recent years, due to its ability to link the weak formulation of cell-population models to free boundary problems of Hele-Shaw type. Although vast literature is available on this singular limit, little is known on the convergence rate of the solutions. In this work, we compute the convergence rate in a negative Sobolev norm and, upon interpolating with BV -uniform bounds, we deduce a convergence rate in appropriate Lebesgue spaces.

On the incompressible limit for a tumour growth model incorporating convective effects

“…Their work improves the results previously achieved in [1], extending the class of initial data from patches to any continuous and compactly supported function bounded between zero and one. In the absence of any growth dynamics, the rate of convergence as 𝛾 → ∞ in the Wasserstein distance was obtained in [1] and was recently improved (in an 𝐻 −1 sense) by [21] who also allow for growth dynamics.…”

Incompressible limits of Patlak-Keller-Segel model and its stationary state

“…In order to prove the uniqueness of the solution to the Hele-Shaw limit system (2.6)-(2.8), we use the lifting method in Ḣ−1 as in [4,9,25] rather than the duality method [27,62] or the entropy method [40]. The main new difficulty comes from the nonlocal interaction.…”

“…But it is difficult to extend these cases or Newtonian potential to more general attractive potential, because our proof of the time derivative estimate of pressure strongly depends on the structure of Newtonian potential. Among open problems, let us also mention the convergence rate with m → ∞, which has been obtained in few papers, [1,25]. Finally the case of systems is only treated without drift, see [15,55].…”