A nonlinear forward-backward problem

Anne-Laure Dalibard (1, 2, 3, 4, 5) , Frédéric Marbach (6, 2, 7) , Jean Rax (1, 2, 3)

1 SU - Sorbonne Université
2 CNRS - Centre National de la Recherche Scientifique
3 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
4 ENS-PSL - École normale supérieure - Paris
5 DMA - Département de Mathématiques et Applications - ENS Paris
6 UR - Université de Rennes
7 IRMAR - Institut de Recherche Mathématique de Rennes

We prove existence and uniqueness of strong solutions of the equation u u_x − u_{yy} = f in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we consider have opposite signs in the lower and upper half of the domain, this is a forward-backward parabolic problem, which changes type across a critical curved line within the domain. In particular, lateral boundary conditions can be imposed only where the characteristics are inwards. There are several difficulties associated with this problem. First, the forward-backward geometry depends on the solution itself. This requires to be quite careful with the approximation procedure used to construct solutions. Second, and maybe more importantly, the linearized equations solved at each step of the iterative scheme admit a finite number of singular solutions. This is similar to well-known phenomena in elliptic problems in nonsmooth domains. Hence, the solutions of the equation are regular if and only if the source terms satisfy a finite number of orthogonality conditions. A key difficulty of this work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow.

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Anne-Laure Dalibard, Frédéric Marbach, Jean Rax. A nonlinear forward-backward problem. 2022. ⟨hal-03812909v1⟩