Scaling regimes of the 1D Kuramoto-Sivashinsky equation

Résumé :

The Kuramoto-Sivashinsky (KS) equation describes various phenomena exhibiting chaos and instabilities, including flames, reaction-diffusion processes, and thin liquid film flows. It resembles the Kardar-Parisi-Zhang (KPZ) equation, with the key difference being the negative « viscosity » in the KS equation. The KS equation is known to belong to the KPZ universality class: at large scales, its statistical properties are governed by the KPZ fixed point, characterized by the dynamical exponent $z=3/2$ and a known PDF of fluctuations. In this talk, I will analyze the stochastic KS equation using the functional renormalization group (FRG). In this framework, attractive fixed points of the FRG equation determine large-scale universal behavior, while unstable fixed points govern small-scale universal behavior. I will show that the two-point correlation function exhibits, besides the KPZ scaling at large scale, another universal scaling regime at small scale, with $z=1$. This scaling is very robust as it is intrinsic to the KS dynamics. It arises from the vanishing of the effective viscosity when evolving from its microscopic negative KS value, to its macroscopic effective positive KPZ value. I will also present results from direct numerical simulations, which confirm these findings. The talk is based on the following articles: 1 L. Gosteva, D. Roy, N. Wschebor, and L. Canet, arXiv:2605.23364 (2026). 2 L. Gosteva, N. Wschebor, and L. Canet, arXiv:2607.15784 (2026).

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Contact : pierre.nataf@lpmmc.cnrs.fr