Solving the Kuramoto-Sivashinsky-Burgers equation until the $6p$-th dimension: the Brownian-time paradigm

Abstract

We use our earlier Brownian-time framework to formulate and establish global uniqueness and local-in-time existence of the Burgers incarnation of the Kuramoto-Sivashinsky PDE on ${\mathbb{R}}_{+}\times {\mathbb{R}}^d$, in the class of time-continuous $L^{2p}$-valued solutions, $p\ge 1$, for every $d<6p$. We assume neither space compactness, nor spatial coordinates dependence, nor smallness of initial data. The surprising discovery of the $6p$-th dimension bound, even for local solutions, is revealed by our approach and the Brownian-time kernel -- the Brownian average of an angled $d$-dimensional Schrödinger propagator -- at its heart. We use this kernel to give a systematic approach, for all dimensions simultaneously, including a novel formulation -- even in the well-known $d=1$ case -- of the KS equation. This yields the estimates leading to this article's conclusions. We achieve the stated results by fusing some of our earlier Brownian-time stochastic processes constructions and ideas -- encoded in the aforementioned kernel -- with analytic ones, including complex and harmonic analysis; by employing suitable $N$-ball approximations together with fixed point theory; and by an adaptation of the stochastic analytic stopping-time technique to our deterministic setting. Using a separate strategy, that is also built on our Brownian-time paradigm, we treat the global wellposedness of the multidimensional KS equation in a followup upcoming article. This work also serves as a template for another forthcoming article in which we prove similar results for the time-fractional Burgers equation in multidimensional space.

27 pages, 3 figures. This is the first installment in my multipart multidimensional KS program. I finished v1 of the math for this part in 2016. The finalized writeup was delayed due to many program reorganizations/expansions. This article was also template for a joint upcoming time-fractional Burgers paper. In the next part, in final prep, I treat time-global solutions to the multidimensional KS