Linear and Nonlinear Parabolic Forward-Backward Problems

von Anne-Laure Dalibard, Frédéric Marbach, Jean Rax

Hardcover

Kurzbeschreibung des Verlags

The purpose of this memoir 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.

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.

• Reihe: Memoirs of the European Mathematical Society
• ISBN: 9783985470983
• Sprache: Englisch
• Erscheinungsdatum: 01.10.2025
• Genre: Mathematik
• Verlag: EMS Press