Conservation law

Remarks

The name comes from the integral form. Let $u$ be a $C^1$ solution, and let $V \subseteq \R^n$ be a bounded region with smooth boundary and outer unit normal $\nu$. Then $$\frac{d}{dt} \int_V u \, dx = - \int_{\partial V} \sum_{j=1}^n F_j(u)\, \nu_j \, dS,$$ by the divergence theorem, so the total amount of $u$ in $V$ changes only through the flux across $\partial V$. The integral form, or the equivalent weak form with test functions, still makes sense for discontinuous $u$, and it is the basis of weak solutions.

Example: the inviscid Burgers' equation $u_t + (u^2/2)_x = 0$ is a scalar conservation law with $n = m = 1$ and $F(u) = u^2/2$. The Euler equations of gas dynamics are a system with $m = n + 2$ conserved quantities: mass, momentum and energy.

Non-example: the equation $u_t + x\, u_x = 0$ is not of conservation form with a flux depending on $u$ only. Indeed it equals $u_t + (x u)_x = u$, so the amount of $u$ is not conserved: $\int_\R u(x, t)\, dx$ grows like $e^{t}$ for integrable solutions, as $u(x, t) = g(x e^{-t})$ shows.