Inviscid Burgers equation ($u_t+u{u}_x=0$)

The inviscid Burgers equation is the scalar conservation lawIts discontinuities obey the Rankine-Hugoniot condition $\sigma =(u_{-}+u_{+})/2$. Restricting to entropy solutions restores uniqueness for bounded initial data.

An entropy solution is a weak solution of a nonlinear scalar conservation law that also satisfies entropy inequalities selecting physically admissible shocks. For convex fluxes, this excludes expansion shocks and restores uniqueness for bounded initial data.