Uniform convergence for complex diffusion equation via a probabilistic approach and application

Abstract

Motivated by the well-posedness theory in fluid mechanics, we investigate the question of uniform convergence of solutions to the following heat equation perturbed with a complex-valued potential and a small positive parameter: We introduce a class of potentials allowing a full representation of uε by the so-called Feynman–Kac integral in the Wiener space. We show that this ansatz provides an asymptotic expansion of uε as ε tends to zero as well as a uniform control of \documentclass{article}\usepackage{amssymb}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$\|\nabla u^{\varepsilon}(t,\cdot)\|_{L^{2}({\mathbb{R}}^{d})}$\end{document} for small times. The proof is carried out by the stationary phase method for the Wiener measure introduced by Ben Arous in (Stochastics 1988; 25:125–153) and essentially applied in the context of the semi-classical approximation of solutions to the Schrödinger equation (J. Funct. Anal. 1996; 137:243–280). At last, we prove as an application a uniform local existence result for inhomogeneous rotating fluid problem in the frame of anisotropic Sobolev spaces. Copyright © 2010 John Wiley & Sons, Ltd.