Electron. J. Differential Equations, Vol. 2023 (2023), No. 09, pp. 1-59.

Paradigm for the creation of scales and phases in nonlinear evolution equations

Christophe Cheverry, Shahnaz Farhat

The transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly under the impact of a forcing term. This phenomenon is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. Under integrability conditions, we show that the blow-up procedure (a type of normal form method) and the Wentzel-Kramers-Brillouin approximation (of supercritical type) introduced in [7,8] still apply. This allows to obtain the existence of solutions during long times, as well as asymptotic descriptions and reduced models. Then, by exploiting transparency conditions (coming from the integrability conditions), by implementing the Hadamard's global inverse function theorem and by involving some specific WKB analysis, we can justify in the context of Hamilton-Jacobi equations the onset of smaller scales and new phases.

Submitted December 28, 2022. Published January 25, 2023.
Math Subject Classifications: 35B35, 35B40, 35K57, 35Q92, 92C17.
Key Words: Nonlinear differential equations; normal forms; integrability; blow-up procedure; geometrical optics; WKB analysis; Multiple-scale analysis; Hamilton-Jacobi equations; microstructures; turbulent flows.
DOI: https://doi.org/10.58997/ejde.2023.09

Show me the PDF file (639 KB), TEX file for this article.

Christophe Cheverry
Univ Rennes, CNRS, IRMAR - UMR 6625
F-35000 Rennes, France
email: christophe.cheverry@univ-rennes1.fr
Shahnaz Farhat
Univ Rennes, CNRS, IRMAR - UMR 6625
F-35000 Rennes, France
email: shahnaz.farhat@univ-rennes1.fr

Return to the EJDE web page