Transport equation ($u_t+b\cdot \nabla u=0$)

A transport equation carries a quantity along the flow of a velocity field $b$. For a smooth solution of $u_t+b\cdot \nabla u=0$, the value of $u$ is constant along each characteristic curve.

For the transport field $b(t,x)$, a characteristic curve solvesThe chain rule converts differentiation of $u(t,X(t))$ into the transport operator applied to $u$.

The characteristic flow map $Z_{s,t}(x)$ is the point reached at time $t$ by the characteristic curve that passes through $x$ at time $s$. Whenever the flow is invertible, transport solutions pull their initial data back by $Z_{0,t}^{-1}$.

The method of characteristics solves a first-order partial differential equation by reducing it to ordinary differential equations along characteristic curves.

A linear transport equation has a prescribed velocity field and is linear in the transported unknown.

An adjoint transport equation transfers the transport operator from a weak solution to a test function. Solving the terminal-value adjoint equation allows arbitrary compactly supported source terms to be used in uniqueness proofs.

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.