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Sharp Asymptotic Behavior of the Steady Pressure-free Prandtl system

Chen Gao, Chuankai Zhao

TL;DR

This work analyzes the far-field behavior of steady, pressure-free Prandtl flows in a half-plane by transforming to a weighted vorticity variable $\omega$ via a modified von Mises map. It proves global convergence of solutions to the Blasius profile as $x\to\infty$, establishing sharp decay rates through a weighted energy framework and barrier/maximum-principle techniques, with a principal eigenvalue of the linearized operator equal to $1$. Two data regimes are treated: small localized perturbations yield optimal $(x+1)^{-1-i-j/2}$ decay for derivatives, and general decaying data yield $L^\infty$ rates of $(x+1)^{-1}$ (with higher derivatives decaying at rate $(x+1)^{-1-i-j/2}$ under regularity). The results are complemented by a matching optimality construction based on Blasius-based exact solutions, and by a detailed analysis of higher-order derivatives, providing a comprehensive sharp asymptotic theory for steady Prandtl flows.

Abstract

This paper investigates the asymptotic behavior of solutions to the steady pressure-free Prandtl system. By employing a modified von Mises transformation, we rigorously prove the far-field convergence of Prandtl solutions to Blasius flow. A weighted energy method is employed to establish the optimal convergence rate assuming that the initial data constitutes a perturbation of the Blasius profile. Furthermore, a sharp maximum principle technique is applied to derive the optimal convergence rate for concave initial data. The critical weights and comparison functions depend on the first eigenfunction of the linearized operator associated with the system.

Sharp Asymptotic Behavior of the Steady Pressure-free Prandtl system

TL;DR

This work analyzes the far-field behavior of steady, pressure-free Prandtl flows in a half-plane by transforming to a weighted vorticity variable via a modified von Mises map. It proves global convergence of solutions to the Blasius profile as , establishing sharp decay rates through a weighted energy framework and barrier/maximum-principle techniques, with a principal eigenvalue of the linearized operator equal to . Two data regimes are treated: small localized perturbations yield optimal decay for derivatives, and general decaying data yield rates of (with higher derivatives decaying at rate under regularity). The results are complemented by a matching optimality construction based on Blasius-based exact solutions, and by a detailed analysis of higher-order derivatives, providing a comprehensive sharp asymptotic theory for steady Prandtl flows.

Abstract

This paper investigates the asymptotic behavior of solutions to the steady pressure-free Prandtl system. By employing a modified von Mises transformation, we rigorously prove the far-field convergence of Prandtl solutions to Blasius flow. A weighted energy method is employed to establish the optimal convergence rate assuming that the initial data constitutes a perturbation of the Blasius profile. Furthermore, a sharp maximum principle technique is applied to derive the optimal convergence rate for concave initial data. The critical weights and comparison functions depend on the first eigenfunction of the linearized operator associated with the system.

Paper Structure

This paper contains 15 sections, 24 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Localized Perturbation Case:
  3. General Initial Data Case:
  4. Optimality Demonstration:
  5. Preliminary
  6. Proof of Theorem \ref{['energy1']}
  7. Stability Analysis
  8. Sharp Decay Estimates of $\tilde{\omega}$
  9. Sharp $L^\infty$ Decay Estimates of $u$
  10. Proof of Theorem \ref{['main1']}
  11. Proof of Theorem \ref{['main2']}
  12. Sharp Decay Estimates of $\omega_\xi$, $\omega_\psi$ and $\omega_{\psi\psi}$
  13. Sharp Decay Estimates of High-order Derivatives of $\omega$
  14. Sharp $L^\infty$ Decay Estimates of $u$
  15. The Principal Eigenvalue of $\mathcal{L}$

Key Result

Theorem 1.1

Assume that the initial data $u_0(y)$ satisfies for some $\alpha \in (0,1)$: Moreover, assume that for small $y$ the following compatibility condition is satisfied at the point $(0, 0)$: Then the Prandtl system prandtl-boundary admits a global classical solution $[u,v]$ in the streamwise direction with the following regularity for any fixed $X_0>0$:

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2: Serrin
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Example 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 34 more