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We use our earlier Brownian-time framework to formulate and establish globaluniqueness and local-in-time existence of the Burgers incarnation of theKuramoto-Sivashinsky PDE on $\mathbb{R}_+\times\mathbb{R}^d$, in the class oftime-continuous $L^{2p}$-valued solutions, $p\ge1$, for every $d<6p$. We assumeneither space compactness, nor spatial coordinates dependence, nor smallness ofinitial data. The surprising discovery of the $6p$-th dimension bound, even forlocal solutions, is revealed by our approach and the Brownian-time kernel --the Brownian average of an angled $d$-dimensional Schr\"odinger propagator --at its heart. We use this kernel to give a systematic approach, for alldimensions simultaneously, including a novel formulation -- even in thewell-known $d=1$ case -- of the KS equation. This yields the estimates leadingto this article's conclusions. We achieve the stated results by fusing some ofour earlier Brownian-time stochastic processes constructions and ideas --encoded in the aforementioned kernel -- with analytic ones, including complexand harmonic analysis; by employing suitable $N$-ball approximations togetherwith fixed point theory; and by an adaptation of the stochastic analyticstopping-time technique to our deterministic setting. Using a separatestrategy, that is also built on our Brownian-time paradigm, we treat the globalwellposedness of the multidimensional KS equation in a followup upcomingarticle. This work also serves as a template for another forthcoming article inwhich we prove similar results for the time-fractional Burgers equation inmultidimensional space.

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