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Conservation laws are PDEs of the form u_t + f(u)_x = 0 expressing that some quantity (mass, momentum, energy) is locally conserved. Even with smooth initial data, nonlinear conservation laws develop discontinuities (shock waves) in finite time, as faster-moving parts of the solution overtake slower parts. Classical (smooth) solutions cease to exist, necessitating the concept of weak solutions—functions that satisfy the PDE in an integral sense rather than pointwise. Since weak solutions are generally non-unique, additional entropy conditions are required to select the physically relevant one.
Conservation laws are among the most important PDEs in applied mathematics, modeling fluid dynamics (Euler equations), traffic flow, gas dynamics, and many other phenomena. The simplest example is Burgers' equation u_t + uu_x = 0, where the flux function f(u) = u²/2 makes the wave speed depend on the solution itself. This nonlinearity is the source of the fundamental difficulty: characteristics carrying different values of u travel at different speeds, and when faster waves overtake slower ones, the characteristics cross and a smooth solution cannot persist.
The resolution is the concept of a weak solution: a bounded measurable function u that satisfies ∫∫[uφ_t + f(u)φ_x]dxdt + ∫u(x,0)φ(x,0)dx = 0 for all smooth test functions φ vanishing at large x and t. This integral formulation is equivalent to the PDE where u is smooth but also meaningful where u has jumps. At a jump discontinuity, the integral formulation yields the Rankine-Hugoniot condition relating the shock speed to the jumps in u and f(u). This condition is the mathematical expression of conservation across a discontinuity.
The central difficulty with weak solutions is non-uniqueness. Given initial data that develops a shock, there are infinitely many weak solutions—some physically reasonable (shock waves) and others not (rarefaction shocks that spontaneously create discontinuities). The entropy condition resolves this by imposing an additional selection principle. The Lax entropy condition requires that characteristics enter a shock from both sides (information is absorbed, not created). Equivalently, the viscosity criterion selects the weak solution that is the limit of solutions to u_t + f(u)_x = εu_xx as ε → 0, the solution obtained by adding a small amount of physical dissipation.
The theory of conservation laws extends to systems (such as the Euler equations of gas dynamics) where the situation becomes considerably richer and more difficult. Systems support multiple wave families—shocks, rarefactions, and contact discontinuities—and the interaction of these waves produces complex behavior. The Riemann problem (piecewise constant initial data) serves as the fundamental building block, and Glimm's existence theorem shows that solutions exist for systems with small data. Modern research continues on large-data existence, uniqueness, and the development of efficient numerical schemes (Godunov, WENO) that correctly capture shocks.