Scalar conservation law ($u_t+{\partial }_{x}f(u)=0$)

A scalar conservation law in one space dimension has the form $u_t+{\partial }_{x}f(u)=0$. While a classical solution exists, it obeys the quasilinear transport equation $u_t+f^{\prime }(u)u_x=0$.

Across a discontinuity $x=\sigma t$ of a weak solution to $u_t+f(u{)}_x=0$, with left and right states $u_{-}$ and $u_{+}$, conservation requires

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.

Characteristic crossing occurs when the map from an initial point to its position at time $t$ loses injectivity. For $u_t+f(u{)}_x=0$, its Jacobian is $1+t{f}^{\prime \prime }(u_0)u_0^{\prime }$ in the autonomous case, so a negative value of $f^{\prime \prime }(u_0)u_0^{\prime }$ causes finite-time crossing.

Gradient blow-up means that a solution stays bounded while the norm of a spatial derivative tends to infinity. In a scalar conservation law this happens as characteristics meet, because the derivative contains the reciprocal of the characteristic-map Jacobian.