Local existence and uniqueness in the largest critical space for a surface growth model

TL;DR

This paper proves local and global existence and uniqueness of solutions for a surface growth model in the largest critical space, extending previous work by considering scaling-critical spaces and addressing weak solutions.

Contribution

It introduces the largest critical space framework for the surface growth equation, establishing existence and uniqueness results in this setting.

Findings

Existence of solutions in critical spaces

Uniqueness of solutions for small initial data

Open problem of uniqueness for large data

Abstract

We show the existence and uniqueness of solutions (either local or global for small data) for an equation arising in different aspects of surface growth. Following the work of Koch and Tataru we consider spaces critical with respect to scaling and we prove our results in the largest possible critical space such that weak solutions are defined. The uniqueness of global weak solutions remains unfortunately open, unless the initial conditions are sufficiently small.