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Stability of the Favorable Falkner-Skan Profiles for the Stationary Prandtl Equations

Sameer Iyer

Abstract

The (favorable) Falkner-Skan boundary layer profiles are a one parameter ($β\in [0,2]$) family of self-similar solutions to the stationary Prandtl system which describes the flow over a wedge with angle $β\fracπ{2}$. The most famous member of this family is the endpoint Blasius profile, $β= 0$, which exhibits pressureless flow over a flat plate. In contrast, the $β> 0$ profiles are physically expected to exhibit a \textit{favorable pressure gradient}, a common adage in the physics literature. In this work, we prove quantitative scattering estimates as $x \rightarrow \infty$ which precisely captures the effect of this favorable gradient through the presence of new ``CK" (Cauchy-Kovalevskaya) terms that appear in a quasilinear energy cascade.

Stability of the Favorable Falkner-Skan Profiles for the Stationary Prandtl Equations

Abstract

The (favorable) Falkner-Skan boundary layer profiles are a one parameter () family of self-similar solutions to the stationary Prandtl system which describes the flow over a wedge with angle . The most famous member of this family is the endpoint Blasius profile, , which exhibits pressureless flow over a flat plate. In contrast, the profiles are physically expected to exhibit a \textit{favorable pressure gradient}, a common adage in the physics literature. In this work, we prove quantitative scattering estimates as which precisely captures the effect of this favorable gradient through the presence of new ``CK" (Cauchy-Kovalevskaya) terms that appear in a quasilinear energy cascade.
Paper Structure (25 sections, 28 theorems, 305 equations, 2 figures)

This paper contains 25 sections, 28 theorems, 305 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Favorable Falkner-Skan Profiles
  3. Main Ideas
  4. Derivation of Canonical Systems
  5. Commutation Properties
  6. Linear Good Variables & Algebraic Structure
  7. Nonlinear Good Variables & Algebraic Structure
  8. Cauchy Data: Iteration Scheme & Compatibility Conditions
  9. Norms & Embedding Theorems
  10. Energy-CK-Dissipation Functionals
  11. $L^2$ Bounds on Good Variables
  12. $L^p_x L^q_y$ Bounds on Original Variables
  13. Quasilinearized Estimates
  14. Linear Estimates
  15. Quasilinear Energy Estimates
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $m \ge 0$, and set $u_E(x) = x^m$. Let $f_{FS}$ be the corresponding solution to fs:Form:1 -- fs:Form:2, with $\beta = \frac{2m}{m+1}$. Let $u_P$ be any solution to Pr:intro:1. Then the following scattering estimate holds Of course, in comparison with our results below, it is more natural to mul

Figures (2)

  • Figure 1: Flow over a wedge, Schlichting
  • Figure 2: Background Euler Flow

Theorems & Definitions (71)

  • Theorem 1.1: Serrin, Serrin
  • Theorem 1.2: Iyer, MR4097332, informal statement
  • Theorem 1.3: Scattering Estimates, $m \ge 0$
  • Theorem 1.4: Enhanced Scattering, $m \ge 0$
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 61 more