Asymptotic Behavior of the Steady Prandtl Equation

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.