Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology

We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation ut −∆ u ∈ a H(u − μ) in Q = Ω × (0, T ], whereΩ ⊂ Rn is a ring-shaped domain, a andμ are given positive constants,H(·) is the Heaviside maximal monotone graph:H(s) = 1 if s > 0,H(0) = [0, 1],H(s) = 0 if s < 0. Such equations arise in climatology (the so-called Budyko energy balance model), as well as in other contexts such as combustion. We show that under certain conditions on the initial data the level sets Γμ = {(x, t) :u(x, t) = μ} are n-dimensional hypersurfaces in the (x, t)-space and show that the dynamics of Γμ is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This differential equation expresses the velocity of advancement of the level surface Γμ through spatial derivatives of the solution u. Our approach is based on the introduction of a local set of Lagrangian coordinates: the equation is formally considered as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows us to watch the trajectory of each of the fluid particles.