# A diffused interface with the advection term in a Sobolev space

## TL;DR

This paper investigates the asymptotic behavior of diffused surface energy with advection in a Sobolev space, showing the limit interface as an integral varifold influenced by advection, and applies this to solve a prescribed mean curvature problem.

## Contribution

It introduces a new analysis of the van der Waals--Cahn--Hilliard theory with advection, establishing the limit interface as an integral varifold with mean curvature affected by advection, and solves a prescribed mean curvature problem.

## Key findings

- Limit interface is an integral varifold.
- Generalized mean curvature vector is determined by advection.
- Application to prescribed mean curvature problem using min-max method.

## Abstract

We study the asymptotic limit of diffused surface energy in the van der Waals--Cahn--Hillard theory when an advection term is added and the energy is uniformly bounded. We prove that the limit interface is an integral varifold and the generalized mean curvature vector is determined by the advection term. As the application, a prescribed mean curvature problem is solved using the min-max method.