Asymptotic Behavior of the Steady Prandtl Equation

TL;DR

This paper investigates the long-term behavior of solutions to the steady Prandtl equation with a specific outer flow, providing explicit decay estimates and extending previous results to more general initial data using maximum principle techniques.

Contribution

It establishes explicit decay estimates for the difference between the solution and the Blasius profile for general initial data with exponential decay, including derivative estimates under concavity assumptions.

Findings

Proves decay estimates of the solution difference in L-infinity norm.

Extends decay results to general initial data with exponential decay.

Provides derivative decay estimates under concavity assumptions.

Abstract

We study the asymptotic behavior of the Oleinik's solution to the steady Prandtl equation when the outer flow $U(x)=1$. Serrin proved that the Oleinik's solution converges to the famous Blasius solution $\bar u$ in $L^\infty_y$ sense as $x\rightarrow+\infty$. The explicit decay estimates of $u-\bar u$ and its derivatives were proved by Iyer[ARMA 237(2020)] when the initial data is a small localized perturbation of the Blasius profile. In this paper, we prove the explicit decay estimate of $\|u(x,y)-\bar{u}(x,y)\|_{L^\infty_y}$ for general initial data with exponential decay. We also prove the decay estimates of its derivatives when the data has an additional concave assumption. Our proof is based on the maximum principle technique. The key ingredient is to find a series of barrier functions.