= Inviscid Burgers equation
{c}
{title2=$u_t+uu_x=0$}
{wiki}
The <Inviscid Burgers equation> is the <scalar conservation law>
$$
u_t+\left(\frac{u^2}{2}\right)_x=0.
$$
For a <classical solution>, the <chain rule> makes this equivalent to $u_t+uu_x=0$, a nonlinear <transport equation> with <characteristic speed> $u$. The conservative form remains meaningful for a discontinuous <weak solution>, whereas the product $uu_x$ cannot simply be evaluated pointwise at a jump. Its discontinuities obey the <Rankine-Hugoniot condition> $s=(u_L+u_R)/2$; this condition alone permits inadmissible expansion jumps. For bounded <initial conditions> on the infinite line, imposing the <Kruzhkov entropy inequalities> and the initial trace selects a unique <entropy solution>. For step data, incoming <characteristic curves> select $u_L>s>u_R$, while $u_L<u_R$ produces a <rarefaction wave>.<Riemann Problems and Jupyter Solutions by David I. Ketcheson, Randall J. LeVeque, and Mauricio J. del Razo/Burgers' equation>[Shock solution; Rarefaction wave; The entropy condition]{ref}
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