Burgers' equation
= Burgers' equation
{c}
{title2=$u_t+uu_x=\nu u_{xx}$}
{wiki}
<Burgers' equation> is the one-dimensional <partial differential equation>
$$
u_t+uu_x=\nu u_{xx},\qquad \nu\geq0.
$$
The term $uu_x$ supplies nonlinear transport in a <transport equation>: the transported value $u$ is also its own <characteristic speed> when $\nu=0$. Larger values catch up with smaller values ahead of them, steepening a decreasing profile. The <diffusion> term $\nu u_{xx}$ opposes this steepening. Positive $\nu$ gives the <viscous Burgers equation>; setting $\nu=0$ gives the <Inviscid Burgers equation>.
This small equation makes three different behaviors accessible in one model: <Burgers characteristic breaking> from continuous decreasing data, an <entropy shock> for a downward step, and a <rarefaction wave> for an upward step. The <Burgers Riemann problem with positive flux> and the <Viscous Burgers travelling shock> below give exact examples. These are mathematical calculations with chosen data; the figures contain no experimental measurements.
Unless stated otherwise, take $x\in\mathbb R$, $t\geq0$, constant $\nu$, no <PDE source term>, and an <initial condition> $u(x,0)=u_0(x)$. There are no finite endpoints requiring <boundary conditions>. Where a profile has different limiting states at $x=\pm\infty$, those states are specified explicitly. On a finite <interval>, <boundary conditions> become part of the problem: positive <diffusion> normally requires a condition at each endpoint, whereas the <Inviscid Burgers equation> requires appropriate <inflow transport boundary conditions>. With dimensional variables, $u$ has units of <velocity> and $\nu$ of <kinematic viscosity>; all numerical examples below use <dimensionless quantities>.