Linear and Nonlinear Parabolic Forward-Backward Problems
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Beschreibung
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.
Autor*in
Anne-Laure Dalibard
Sorbonne Université, Université Paris Cité, CNRS, INRIA, Paris, France
Themen in »Linear and Nonlinear Parabolic Forward-Backward Problems«
mixed-type boundary value problems forward-backward equations nonlinear orthogonality conditions free boundary problems in fluid mechanics recirculation in boundary layers