Asymptotics for a Class of Singular Integrals of Quotients with Highly Degenerate Denominators

Abstract

In a rigorous study of stochastic models for the wave turbulence theory and Peierls’ kinetic theory for the thermal conductivity in solids, analysis of integrals of the form \(\int_{ \mathcal{M} }\frac{F\omega_ \mathcal{M} }{\Omega^2 + \nu^2\Gamma^2}\) and \(\int_{ \mathcal{M} }\frac{F\cos(\nu^{-1}\Omega)\omega_ \mathcal{M} }{\Omega^2 + \nu^2\Gamma^2}\) plays a crucial role, where \(\nu>0\) is a small parameter, \( \mathcal{M} \) is a closed Riemannian manifold with volume form \(\omega_ \mathcal{M} \), and the functions \(\Gamma > 0\), \(F\), \(\Omega\) are sufficiently smooth. We investigate the asymptotic behavior of the integrals in the limit as \(\nu\rightarrow 0\). This paper continues the studies S. B. Kuksin, J. Math. Phys. Anal. Geom., 2018, vol. 14, no. 4, pp. 510тАУ518 and A. V. Dymov, Theoret. and Math. Phys., 2023, vol. 214, no. 2, pp. 153тАУ169, in which the authors considered similar integrals for the case in which \( \mathcal{M} =\mathbb{R}^d\) and the function \(\Omega\) is Morse. We significantly weaken the latter assumption, which played an important role in the aforementioned papers. This makes the results thus obtained applicable to the problem of rigorous justification of Peierls’ kinetic theory.