# Contents

## Idea

Burger’s equation is a partial differential equation that was originally proposed as a simplified model of turbulence as exhibited by the full-fledged Navier-Stokes equations.

In one spatial dimension it is

$\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = 0$

in a spatial region $\Omega$, with a real constant $\nu \gt 0$, with appropriate boundary and initial conditions.

It is possible to write Burgers’s equation in a conservation form (with a flow that is conserved):

$\frac{\partial u}{\partial t} + \frac{\partial F(u)}{\partial x} = 0$

with

$F(u) := \frac{1}{2} u^2 - \nu \frac{\partial u}{\partial x}$