**Question:** Is it possible to bound the density gradient $\nabla \rho_\epsilon$ of strong solutions to the compressible NSE uniformly in $L^\gamma$?

**Preliminaries:** Consider a bounded connected domain $\Omega\subset \mathbb{R}^3$ of class $C^2$. Perforate it with holes of size $\epsilon^3$, where the holes sit on a mesh of size $\epsilon$ inside $\Omega$. We get a domain $\Omega_\epsilon=\Omega\setminus \bigcup_{k\in\mathbb{Z}^3} (\epsilon k+\epsilon^3 T)$ where $T\subset\subset B_1(0)$ is a model hole of class $C^2$ strictly contained in the unit ball. For simplicity, we remove just holes that do not meet the boundary of $\Omega$ to ensure smoothness.

Let $\rho_\epsilon$ be the density of a gas in $\Omega_\epsilon$ and $u_\epsilon$ be its velocity. Define for $\gamma>3$ the pressure $p_\epsilon=\rho_\epsilon^\gamma$ and consider the compressible stationary NSE

$div(\rho_\epsilon u_\epsilon\otimes u_\epsilon)-(\lambda+\mu)\nabla div(u_\epsilon)-\mu\Delta u_\epsilon+\nabla p_\epsilon=\rho_\epsilon f \text{ in } \Omega_\epsilon,\\ div(\rho_\epsilon u_\epsilon)=0 \text{ in } \Omega_\epsilon,\\ u_\epsilon\restriction_{\partial\Omega_\epsilon}=0$

where $f\in L^\infty(\Omega)$ (or as good as needed) and $\lambda,\mu$ are the usual viscosity coefficients.

These equations admit strong solutions for *fixed* $\epsilon$ in $(u_\epsilon,\rho_\epsilon)\in (W^{2,\gamma}\cap W_0^{1,\gamma})\times W^{1,\gamma}$. The bounds for $u_\epsilon$ in its space can be made independent of $\epsilon$, also $||\rho_\epsilon||_{L^{2\gamma}(\Omega_\epsilon)}\leq C$ for some $C>0$ independent of $\epsilon$ is possible.

**Problem:** To get appropriate convergence of a subsequence one needs to bound $||\nabla \rho_\epsilon||_{L^\gamma(\Omega_\epsilon)}$ uniformly in $\epsilon$. Till now, I have no idea how to achieve this. It's the only assumption missing in my current homogenization work.

For simplicity, you can also consider the Stokes equations

$\nabla p_\epsilon-\Delta u_\epsilon=\rho_\epsilon f \text{ in } \Omega_\epsilon,\\ div(\rho_\epsilon u_\epsilon)=0 \text{ in } \Omega_\epsilon,\\ u_\epsilon\restriction_{\partial\Omega_\epsilon}=0$

since I can reduce the full problem to this particular one.

I'm thankful for any hint.

do not meetthe boundary of $\Omega$ to ensure smoothness" should be changed in "For simplicity, we remove just holes thatmeetthe boundary of $\Omega$ to ensure smoothness". It seems to me that it is the presence of holes touching the boundaries of $\Omega$ which could eventually produce cusps, edges and irregularities, reducing thus its $C^2$ smoothness. $\endgroup$ – Daniele Tampieri Dec 5 '19 at 11:28