---
tag: 7X9N9H3RK6
kind: concept
name: Quasilinear PDE
msc: [35A99, 35J62, 35K59]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Nonlinear_partial_differential_equation
---

## Summary

A quasilinear PDE is a [[partial differential equation|GC5BDZ6QBC]] that is linear in the highest-order derivatives of the unknown.
Its coefficients may depend on the unknown and its lower-order derivatives.
This class contains the equations of fluid dynamics, the [[minimal surface equation|3GV2E2YTZX]] and the [[p-Laplace equation|XJMYGY0ZZF]].
It sits between [[semilinear|RCNY89A4NY]] and [[fully nonlinear|2D3N1ZJWBW]] equations.
See the [[definition|DV92N2YWZJ]].

## Explanations

### L1 human

In a quasilinear equation the highest derivatives still enter one at a time, multiplied by coefficients.
But these coefficients may depend on the solution itself.
In [[Burgers' equation|NEGHXN8RE1]] $\partial_t u + u \, \partial_x u = 0$, the speed at which a wave moves is the value $u$ of the wave.
So high parts of the wave overtake low parts, and the solution can form a shock in finite time.