Strong Solutions for the Stochastic Cahn-Hilliard Convective Brinkman-Forchheimer Model for Tumor Growth

Abstract

In this work, we analyze a diffuse-interface model for tumor growth, subject to multiplicative white noises, posed on a bounded domain $\mathcal{O}\subset {\mathbb{R}}^d$, $d=2,3$. The model couples a stochastic incompressible convective Brinkman-Forchheimer (CBF) equation or Navier-Stokes equation with damping $\eta |v{|}^{r-1}v$ for the averaged velocity field $v$, to a Cahn-Hilliard (CH) equation for the phase field variable $\varphi$ and to a stochastic reaction-diffusion equation governing the nutrient concentration $\sigma$. We establish the existence of local strong solutions , for $r\ge 1$ in $d=2$ and $r\in [1,3]$ in $d=3$. We prove the weak-strong uniqueness holds in both $d=2,3$. In addition, for $d=2$, the uniqueness of weak solutions is obtained for all $\eta ,\nu >0$, and $r\ge 1$, while it holds in $d=3$ for $r\ge 3$ and $\eta \nu \ge 1$ when $r=3$ under an assumption on $\sigma$. Moreover, for $d=2$ and $r\in [1,3]$, we obtain that the strong solution exists globally in time.