# On weak closure of some diffusion equations

## TL;DR

This paper investigates the weak closure properties of certain diffusion equations under different convergence modes, revealing that the closure under weak $H^1$-convergence aligns with the original equations, while under strong $L^2$-convergence it can be significantly larger.

## Contribution

It provides a detailed description of the weak $H^1$-closure of diffusion equations and highlights differences with the $L^2$-closure, offering new insights into their convergence behavior.

## Key findings

- Weak $H^1$-closure coincides with the original diffusion equations for parabolic cases.
- Strong $L^2$-closure can be much larger than the original equations.
- The closure properties depend critically on the mode of convergence used.

## Abstract

We study the closure of approximating sequences of some diffusion equations under certain weak convergence. A specific description of the closure under weak $H^1$-convergence is given, which reduces to the original equation when the equation is parabolic. However, the closure under strong $L^2$-convergence may be much larger, even for parabolic equations.