Linear and nonlinear parabolic forward-backward problems

TL;DR

This paper studies the well-posedness of linear and nonlinear parabolic forward-backward equations, revealing singular solutions, regularity conditions, and extending the theory to complex systems like Vlasov--Poisson--Fokker--Planck and Prandtl equations.

Contribution

It provides a detailed analysis of singular solutions and regularity conditions for parabolic forward-backward problems, extending existing theory to nonlinear and complex systems.

Findings

Finite number of singular solutions for the Kolmogorov equation.

Solutions are regular if source term satisfies orthogonality conditions.

Existence and uniqueness of regular solutions in nonlinear systems under certain conditions.

Abstract

The purpose of this paper is to investigate the well-posedness of several linear and nonlinear equations with a parabolic forward-backward structure, and to highlight the similarities and differences between them. The epitomal linear example will be the stationary Kolmogorov equation $y\partial_x u -\partial_{yy} u=f$ in a rectangle. We first prove that this equation admits a finite number of singular solutions, of which we provide an explicit construction. Hence, the solutions to the Kolmogorov equation associated with a smooth source term are regular if and only if $f$ satisfies a finite number of orthogonality conditions. This is similar to well-known phenomena in elliptic problems in polygonal domains. We then extend this theory to a Vlasov--Poisson--Fokker--Planck system, and to two quasilinear equations: the Burgers type equation $u \partial_x u - \partial_{yy} u = f$ in the vicinity of the linear shear flow, and the Prandtl system in the vicinity of a recirculating solution, close to the line where the horizontal velocity changes sign. We therefore revisit part of a recent work by Iyer and Masmoudi. For the two latter quasilinear equations, we introduce a geometric change of variables which simplifies the analysis. In these new variables, the linear differential operator is very close to the Kolmogorov operator $y\partial_x -\partial_{yy}$. Stepping on the linear theory, we prove existence and uniqueness of regular solutions for data within a manifold of finite codimension, corresponding to some nonlinear orthogonality conditions.