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

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:\documentclass{article}\usepackage{amssymb}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$${\partial}_{t}u^{\varepsilon}-\Delta u^{\varepsilon}+\frac{i}{\varepsilon}V(x)u^{\varepsilon}=0\quad \mbox{in}\ {\mathbb{R}}_{t}^{+}\times{\mathbb{R}}^{d}_x.$$\end{document}We introduce a class of potentials allowing a full representation of u ε by the so‐called Feynman–Kac integral in the Wiener space. We show that this ansatz provides an asymptotic expansion of u ε 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.