---
tag: P4RF8A1562
kind: definition
name: Burgers' equation
msc: [35Q35, 35L65, 35L67]
defines: NEGHXN8RE1
uses: [8RA7AMWSAF, 09EGC6F8PT]
sources:
  - url: https://en.wikipedia.org/wiki/Burgers%27_equation
---

## Definition

Let $\nu \ge 0$ be a constant, the *viscosity*.
*Burgers' equation* with viscosity $\nu$ is the equation
$$\partial_t u + u\, \partial_x u = \nu\, \partial_x^2 u$$
for an unknown function $u \colon \R \times (0, \infty) \to \R$, where $\partial_t$ and $\partial_x$ are [[partial derivatives|09EGC6F8PT]].
For $\nu > 0$ it is the *viscous* Burgers' equation.
For $\nu = 0$ it is the *inviscid* Burgers' equation, usually written in the conservation form
$$\partial_t u + \partial_x \left( \tfrac12 u^2 \right) = 0.$$

## Remarks

The inviscid equation is the scalar [[conservation law|8RA7AMWSAF]] with flux $F(u) = u^2/2$, which is uniformly convex.
For smooth $u$ the two forms agree, but for discontinuous weak solutions only the conservation form is meaningful, and it fixes the shock speed $(u_l + u_r)/2$.
By the method of characteristics, a smooth solution with initial value $g$ satisfies $u = g(x - ut)$; if $g$ is somewhere decreasing, the solution ceases to be smooth at time $1/\max(-g')$.

Example: $u(x, t) = x/(t + 1)$ solves the inviscid equation, and also the viscous one for every $\nu$.
Check: $\partial_t u = -x/(t + 1)^2$, $u\, \partial_x u = x/(t + 1)^2$, and $\partial_x^2 u = 0$.

Non-example: $u(x, t) = e^{-t} \sin x$ solves the linear heat equation $u_t = u_{xx}$ but not Burgers' equation with $\nu = 1$.
Indeed for it $\partial_t u - \partial_x^2 u = 0$, while $u\, \partial_x u = e^{-2t} \sin x \cos x$ is not identically zero.