Local in time solvability of a nonstandard free boundary problem in stratigraphy: A Lagrangian approach

We study a nonstandard free boundary problem arising in stratigraphy in the modeling of formation of sedimentary basins. The problem consists in finding the pair of functions (u,λ)(u,λ) such that ut−div(λ∇u)=0ut−div(λ∇u)=0 and λ∈H(ut+E)λ∈H(ut+E), where HH denotes the Heaviside graph. The discontinuous coefficient λλ represents the restriction on the admissible erosion rate, expressed by the constraint ut+E≥0ut+E≥0, E=const>0E=const>0, everywhere in the problem domain. We show that under suitable conditions on the data the problem admits a local in time solution and that there exists a smooth hypersurface ΓΓ that divides the problem domain into two regions where ut+Eut+E is strictly positive or equals zero. We prove that the motion of ΓΓ is governed by a differential equation which generalizes the Darcy law in filtration theory.