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

Cheverry, Christophe
Farhat, Shahnaz
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Texas State University, Department of Mathematics
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.
Nonlinear differential equations, Normal forms, Integrability, Blow-up procedure, Geometrical optics, WKB analysis, Multiple-scale analysis, Hamilton-Jacobi equations, Microstructures, Turbulent flows
Cheverry, C., & Farhat, S. (2023). Paradigm for the creation of scales and phases in nonlinear evolution equations. <i>Electronic Journal of Differential Equations, 2023</i>(09), pp. 1-59.