Kružkov’s doubling of variables: how far does it go?

par Nicolas Masson (LMO)

mercredi 21 oct. 2026, 10:00 → 12:00 Europe/Paris

Scalar conservation laws typically admit several weak solutions for the same initial datum. To single out the physically relevant one, the equation must be supplemented with a family of differential inequalities, called the entropy inequalities. In his seminal 1970 paper, Kružkov proved that entropy solutions are unique: his argument relies on a one-parameter family of entropies, |u-k|, now called Kružkov entropies. The key idea is to replace the constant k in the entropy inequalities satisfied by one solution u_1 by a second entropy solution u_2 using a method known as doubling of variables. This yields the Kato inequality, from which it follows that ||u_1(t)-u_2(t)||_{L^1} is non-increasing in time, and therefore uniqueness. Textbooks usually present this proof under restrictive assumptions, but Kružkov’s original argument covers a much more general setting: multi-dimensional equations, fluxes depending on (t,x), and source terms. The goal of this talk will be to present Kružkov’s proof, identify the structural ingredients that make it work, and discuss how and when the strategy can be extended to other evolution equations, as well as where it breaks down.