Existence and analyticity of solutions of the Kuramoto-Sivashinsky equation with singular data

TL;DR

This paper establishes the existence and analyticity of solutions to the Kuramoto-Sivashinsky equation with low-regularity initial data across multiple spatial dimensions, expanding the understanding of solution regularity.

Contribution

It introduces new existence results for solutions with singular data in Wiener algebra and pseudomeasure spaces, and proves solutions become analytic over time.

Findings

Solutions exist with low-regularity data in various function spaces.

Solutions gain regularity and become analytic at positive times.

The results apply to multiple spatial dimensions, including 1D and 2D.

Abstract

We prove existence of solutions to the Kuramoto-Sivashinsky equation with low-regularity data, in function spaces based on the Wiener algebra and in pseudomeasure spaces. In any spatial dimension, we allow the data to have its antiderivative in the Wiener algebra. In one spatial dimension, we also allow data which is in a pseudomeasure space of negative order. In two spatial dimensions, we also allow data which is in a pseudomeasure space one derivative more regular than in the one-dimensional case. In the course of carrying out the existence arguments, we show a parabolic gain of regularity of the solutions as compared to the data. Subsequently, we show that the solutions are in fact analytic at any positive time in the interval of existence.