---
tag: SJZX81A783
kind: concept
name: Weak solution
msc: [35D30, 35A15]
sources:
  - url: https://en.wikipedia.org/wiki/Weak_solution
same_as:
  wikidata: Q7977966
  wikipedia: Weak solution
---

## Summary

A weak solution of a [[partial differential equation|GC5BDZ6QBC]] is a function that satisfies the equation after multiplication by test functions and integration by parts.
In this form the derivatives fall on the test functions.
It need not be differentiable, and so it can describe shocks, corners and other rough behaviour.
Existence theory usually first produces a weak solution, often in a [[Sobolev space|M4CEZ5K9HP]].
Then regularity theory shows that it is smoother.

## Explanations

### L1 human

A classical solution must have all the derivatives that appear in the equation.
A weak solution only has to satisfy the equation "on average" against every smooth bump function.
For the transport equation $u_t + u_x = 0$, a moving step $u(x, t) = 1$ for $x > t$ and $0$ for $x < t$ is a weak solution.
It is not differentiable along the line $x = t$, but it describes a profile moving to the right with speed $1$.