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 where is a ring-shaped domain, a and μ are given positive constants, is the Heaviside maximal monotone graph: if 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 are n-dimensional hypersurfaces in the (x, t)-space and show that the dynamics of Γ _μ 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 Γ _μ 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.