Kruzhkov's uniqueness theorem
/ KROOZH-kof /
The theory of conservation laws had a gap: smooth solutions break down at shocks, weak solutions exist but are not unique, and the entropy condition is supposed to fix uniqueness — but does it really, in any number of space dimensions, for any reasonable data? Kruzhkov's theorem is the clean, complete answer for a single scalar law: yes, there is exactly one entropy solution, and it depends stably on the data.
The statement, for u_t + div f(u) = 0 in any dimension with bounded measurable initial data: there exists a unique entropy solution, where 'entropy solution' means u satisfies the family of Kruzhkov entropy inequalities |u - k|_t + div [ sign(u - k)(f(u) - f(k)) ] <= 0 in the distributional sense for every constant k. Kruzhkov's celebrated proof technique is the 'doubling of variables': compare two entropy solutions u(x, t) and v(y, s) by treating them as functions of independent variables, use the entropy inequality for each with the other's value as the constant k, and add. The cross terms cancel beautifully and what is left is the L1 contraction estimate: the integral of |u(x,t) - v(x,t)| dx is non-increasing in time. Setting v = u with shifted data shows two solutions from the same data must coincide — uniqueness — and from different data they stay close, giving stability.
Why it matters: this is the capstone of scalar conservation-law theory. It puts existence, uniqueness, and continuous dependence on a rigorous footing in every dimension and for merely bounded data, vindicating the entropy condition as exactly the right admissibility criterion. The L1 contraction it yields is also the natural stability framework for proving that numerical schemes converge. The honest boundary of the result: it is fundamentally a SCALAR theorem — for systems of conservation laws in several space dimensions, an analogous global existence-and-uniqueness theory is still missing and is a major open problem.
Take two entropy solutions of Burgers' equation with initial data u_0 and v_0. Kruzhkov's theorem guarantees that for all t > 0, integral |u(x,t) - v(x,t)| dx <= integral |u_0(x) - v_0(x)| dx. In particular if u_0 = v_0 then u = v for all time: the entropy solution is unique. Without the entropy inequalities this fails — many weak solutions share the same data.
Entropy solutions contract in L1: distance between two solutions never grows.
The theorem is a triumph specifically for the SCALAR equation; do not over-extend it. Replacing the single equation by a system breaks the comparison/doubling argument (there is no order on vectors), and global well-posedness for multi-dimensional systems of conservation laws remains open.