the Lax-Oleinik formula
/ lahks oh-LAY-nik /
It would be wonderful to have, for a nonlinear conservation law, an honest formula that writes down the entropy solution directly from the initial data — shocks, rarefactions and all — without having to track characteristics by hand. For a scalar law with a convex flux, the Lax-Oleinik formula is exactly that: a single minimisation that produces the unique physical solution.
Here is the idea for u_t + f(u)_x = 0 with convex f. Pass to the integral (potential) by writing u = w_x; the conservation law becomes a Hamilton-Jacobi equation w_t + f(w_x) = 0, and its viscosity solution has the Hopf-Lax representation w(x, t) = min over y of [ W_0(y) + t L((x - y)/t) ], where W_0 is the initial potential and L is the Legendre transform of f (the convex 'dual' of the flux). Differentiate and you get the Lax-Oleinik formula for u itself: u(x, t) = (f')^(-1)( (x - y*(x,t)) / t ), where y*(x,t) is the minimising point. The minimiser y* is the foot of the characteristic that 'wins' at (x, t); when two characteristics tie (two minimisers), that is exactly a shock, and the formula automatically chooses the entropy-admissible one. No separate jump or entropy condition has to be imposed — the min does it all.
Why it matters: it is a rare closed-form handle on a genuinely nonlinear, shock-forming PDE. It proves existence and uniqueness of the entropy solution for convex scalar laws, shows the solution depends on the data through a stable minimisation (so it is well-posed), and reveals two beautiful regularising effects: the solution is automatically of bounded variation and satisfies a one-sided Lipschitz bound u_x <= 1/t (Oleinik's inequality) for any initial data, however rough. The price is generality: it needs the flux convex and the equation scalar — for systems no such formula exists.
For Burgers' equation f(u) = u^2/2 the dual is L(v) = v^2/2 and (f')^(-1)(v) = v, so the formula reads u(x, t) = (x - y*)/t where y* minimises W_0(y) + (x - y)^2/(2t). The instant two minimisers appear, the solution jumps — that is the shock — and the one-sided bound u_x <= 1/t holds for all t > 0 no matter how wild u_0 was.
A minimisation over initial points yields the entropy solution directly — shocks appear when the minimiser jumps.
The formula requires f to be convex (or uniformly convex); for nonconvex fluxes the simple minimisation no longer captures the wave structure and the more general Oleinik construction is needed. And it is strictly a scalar-equation tool — there is no Lax-Oleinik formula for systems of conservation laws.