Existence and Uniqueness of Solutions for Steady Triple-Deck Equations

Abstract: Triple-deck structures arise in boundary-layer flows when small localized wall perturbations induce moderate mean-flow distortion. Most existing studies in this field focus on the derivations of the asymptotic and numerical solutions of this system, while rigorous mathematical results remain scarce. In this paper, we establish, for the first time, the existence and uniqueness of the solutions to the steady triple-deck equations. The steady triple-deck equations can be reduced to an Airy equation with frequency as a parameter via Fourier transformation. The main difficulty is that the Airy equation degenerates at zero frequency, hindering uniform estimates of the solution with respect to the frequency parameter. A key contribution is that we find a subtle function involving the distribution of zeros of the Bessel function to overcome the degeneracy by analyzing the structure of the Airy function. Additionally, we derive a higher-frequency estimate and modify an elliptic multiplier to obtain uniform estimates of the solution.