arXiv subjects

Jean Rax

Publications and source records attributed to Jean Rax.

3 recordsLinked to original sources

arXiv2025

Nonlinear Forward-Backward Problems

We prove the existence and uniqueness of strong solutions to 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 quasilinear 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 more importantly, the linearized equations solved at each step of the iterative scheme admit a finite number of singular solutions, of which we provide an explicit construction. This is similar to well-known phenomena in elliptic problems in nonsmooth domains. Hence, the solutions to 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. To tackle this deceivingly simple problem, we develop a methodology which we believe to be both quite natural and adaptable to other situations in which one wishes to prove the existence of regular solutions to a nonlinear problem for suitable data despite the existence of singular solutions at the linear level. This paper is a shorter version of [3].

math.AP↗

arXiv2022

Linear and nonlinear parabolic forward-backward problems

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.

math.AP↗

arXiv2018

Mathematical study of the equatorial Ekman boundary layer

In this paper we study the well-posedness of a simple model of boundary layer for rotating fluids between two concentric spheres near the equator. We show that this model can be seen as a degenerate elliptic equation , for which we prove an existence result thanks to a Lax-Milgram type lemma. We also prove uniqueness under an additional integrability assumption and present a transparent boundary condition for such layers.

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