Local well-posedness for quasilinear problems: a primer

TL;DR

This paper provides a comprehensive overview of methods and ideas for establishing local well-posedness in quasilinear PDEs, clarifying misconceptions and offering a adaptable framework for researchers.

Contribution

It compiles classical and recent techniques into a cohesive guide for proving local well-posedness in quasilinear problems, addressing common misconceptions.

Findings

Clarifies universal and problem-specific difficulties in proving well-posedness.

Summarizes classical and modern approaches in a unified framework.

Provides adaptable methods for various quasilinear PDEs.

Abstract

Proving local well-posedness for quasilinear problems in pde's presents a number of difficulties, some of which are universal and others of which are more problem specific. While a common standard, going back to Hadamard, has existed for a long time, there are by now both many variations and many misconceptions in the subject. The aim of these notes is to collect a number of both classical and more recent ideas in this direction, and to assemble them into a cohesive road map that can be then adapted to the reader's problem of choice.