# Density gradient of Navier-Stokes equations in perforated domain

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.

• Hi: perhaps the sentence "For simplicity, we remove just holes that do not meet the boundary of $\Omega$ to ensure smoothness" should be changed in "For simplicity, we remove just holes that meet the 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. – Daniele Tampieri Dec 5 '19 at 11:28
• Hi, I think it should be ok since I want to remove all holes that are entirely included in $\Omega$. I edit the task to $\Omega$ be connected. – FluidFlow Dec 5 '19 at 11:34