English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper, we study a tumor growth model where the growth is driven bynutrient availability and the tumor expands according to Darcy's law with amechanical pressure resulting from the incompressibility of the cells. Ourfocus is on the free boundary regularity of the tumor patch that holds beyondtopological changes. A crucial element in our analysis is establishing theregularity of the hitting time T, which records the first time the tumor patchreaches a given point. We achieve this by introducing a novelHamilton-Jacobi-Bellman (HJB) interpretation of the pressure, which is ofindependent interest. The HJB structure is obtained by viewing the model as alimit of the Porous Media Equation (PME) and building upon a new variant of theAB estimate. Using the HJB structure, we establish a new Hopf-Lax type formulafor the pressure variable. Combined with barrier arguments, the formula allowsus to show that T is C^{\alpha}, where \alpha depends only on the dimension,which translates into a mild nondegeneracy of the tumor patch evolution.Building on this and obstacle problem theory, we show that the tumor patchboundary is regular in spacetime except on a set of Hausdorff dimension at most$d-\alpha$. On the set of regular points, we further show that the tumor patchis locally $C^{1,\alpha}$ in space-time. This conclusively establishes thatinstabilities in the boundary evolution do not amplify arbitrarily highfrequencies.

Free boundary regularity for tumor growth with nutrients and diffusion