# Strong averaging principle for slow-fast stochastic partial differential equations with locally monotone coefficients

## TL;DR

This paper establishes a strong averaging principle for coupled slow-fast stochastic partial differential equations with locally monotone coefficients, applicable to various nonlinear SPDEs like porous medium and Navier-Stokes equations.

## Contribution

It proves the strong averaging principle for a broad class of SPDEs with locally monotone coefficients, using time discretization and variational methods.

## Key findings

- Validates the averaging principle for stochastic porous medium and p-Laplace equations.
- Extends the theory to stochastic Burgers and 2D Navier-Stokes equations.
- Provides a framework for analyzing slow-fast SPDE systems with monotone coefficients.

## Abstract

This paper is devoted to proving the strong averaging principle for slow-fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations (SPDEs) with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic $p$-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier-Stokes equation, which are the nonlinear stochastic partial differential equations. The main techniques are based on time discretization and the variational approach to SPDEs.