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Linear and nonlinear parabolic forward-backward problems

Anne-Laure Dalibard, Frédéric Marbach, Jean Rax

TL;DR

The paper develops a rigorous framework for well-posedness of linear and nonlinear parabolic forward-backward PDEs with sign-changing solutions, centering on the Kolmogorov-type equation $y\partial_x u - \partial_{yy} u = f$ and extending to nonlinear perturbations including VPFP, Burgers-type, and Prandtl recirculation models. A core technique is to straighten the zero-set curve via a nonlinear change of variables, enabling a linear-like operator whose forward-backward structure mirrors the Kolmogorov model; this underpins a decomposition of solutions into singular profiles and a regular remainder. Regularity results require orthogonality conditions that define finite-codimension data manifolds (codimension 2 for Burgers-like problems, and codimensions 1–3 for Prandtl variants and VPFP), with dual profiles encoding singular behavior near $(x_i,0)$. An abstract nonlinear scheme, together with interpolation in Pagani-type anisotropic spaces, yields existence and uniqueness results for perturbative data within these manifolds, and a robust strategy to handle reconstruction and boundary data in the recirculation context. The work advances the mathematical understanding of forward-backward parabolic PDEs in fluid-mechanics settings, offering a systematic approach that could adapt to broader degenerate-elliptic problems with similar orthogonality structures.

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.

Linear and nonlinear parabolic forward-backward problems

TL;DR

The paper develops a rigorous framework for well-posedness of linear and nonlinear parabolic forward-backward PDEs with sign-changing solutions, centering on the Kolmogorov-type equation and extending to nonlinear perturbations including VPFP, Burgers-type, and Prandtl recirculation models. A core technique is to straighten the zero-set curve via a nonlinear change of variables, enabling a linear-like operator whose forward-backward structure mirrors the Kolmogorov model; this underpins a decomposition of solutions into singular profiles and a regular remainder. Regularity results require orthogonality conditions that define finite-codimension data manifolds (codimension 2 for Burgers-like problems, and codimensions 1–3 for Prandtl variants and VPFP), with dual profiles encoding singular behavior near . An abstract nonlinear scheme, together with interpolation in Pagani-type anisotropic spaces, yields existence and uniqueness results for perturbative data within these manifolds, and a robust strategy to handle reconstruction and boundary data in the recirculation context. The work advances the mathematical understanding of forward-backward parabolic PDEs in fluid-mechanics settings, offering a systematic approach that could adapt to broader degenerate-elliptic problems with similar orthogonality structures.

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 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 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 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 . 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.
Paper Structure (80 sections, 86 theorems, 457 equations, 3 figures)

This paper contains 80 sections, 86 theorems, 457 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Forward-backward nature
  3. Orthogonality conditions
  4. Statement of the main results
  5. Linear theory
  6. A nonlinear toy model from kinetic theory
  7. The viscous Burgers system
  8. The Prandtl system
  9. Comments and previous results
  10. On weak solutions for the linear problem.
  11. On strong solutions for the linear problem.
  12. On orthogonality conditions for higher regularity.
  13. On orthogonality conditions for nonlinear problems.
  14. On entropy solutions.
  15. On the choice of the linear shear flow for equation \ref{['eq:eq0-uux']}.
  16. ...and 65 more sections

Key Result

Theorem 1

There exists a vector subspace $\mathcal{X}^\perp_{B,\operatorname{sg}} \subset \mathcal{X}_B$ of codimension 2 such that, for each $(f,\delta_0, \delta_1) \in \mathcal{X}_B$, there exists a solution $u \in {Q^1}$ to the problem if and only if $(f,\delta_0,\delta_1) \in \mathcal{X}^\perp_{B,\operatorname{sg}}$. Such a solution is unique and satisfies

Figures (3)

  • Figure 1: Fluid domain $\Omega$ and inflow boundaries $\Sigma_0 \cup \Sigma_1$
  • Figure 2: Plot of $t \mapsto \Lambda_{0}(t)$ for $t \in (-7,7)$, highlighting the main properties: $\Lambda_{0}$ is a smooth, monotone decreasing function on $\mathbb{R}$, such that $\Lambda_{0}(-\infty)=1$ and $\Lambda_{0}(+\infty) = 0$
  • Figure 3: Fluid domain $\Omega_P$ defined in \ref{['eq:def-OmP']} with free top and bottom boundaries $\Gamma_t$ and $\Gamma_b$, and fixed inflow boundaries $\Sigma^P_0$ and $\Sigma^P_1$.

Theorems & Definitions (202)

  • Theorem 1: Orthogonality conditions for linear forward-backward parabolic equations
  • Theorem 2: Decomposition of solutions as a sum of singular profiles and a smooth remainder
  • Theorem 3: Existence and uniqueness of strong solutions to \ref{['eq:eq0-uux']} under orthogonality conditions
  • Proposition 1.1: Necessity of the orthogonality conditions
  • Remark 1.2
  • Theorem 4
  • Remark 1.4
  • Remark 1.5
  • Lemma 1.6
  • Proposition 1.7
  • ...and 192 more