## Probability

•  Hassan Allouba, Kent State University
•  Well-posedness for the Kuramoto-Sivashinsky equation until the 6p-th dimension
•  12/07/2022
•  3:00 PM - 3:50 PM
•  C405 Wells Hall
•  Dapeng Zhan (zhan@msu.edu)

Lately, I’ve been finalizing a couple of papers on the local and global analysis of the KS PDE in multidimensional space. This includes the well known open problem of the global well-posedness of KS for d ≥ 1. In this talk, I will discuss my Brownian-time (or L-KS kernel) approach to the Burgers incarnation of the KS equation for all dimensions. This yields a new formulation for the KS equation (deterministic or stochastic), even in the better known one dimensional case. I will then focus on my latest results on the uniqueness and local well-posedness of the KS PDE for d ≥ 1. In particular, I’ll show—using careful estimates on the fundamental L-KS kernel—the existence of $L^{2p}$ solution, locally in time, till the 6p-th dimension, as well as the global uniqueness of KS solutions. Since this Brownian-time setting includes time fractional equations, I will briefly mention the corresponding results for time-fractional Burgers equations in multidimensional space—which I’m also finalizing in separate papers. The rest of the global well-posedness results (existence and more) use additional new tools, some of which are inspired in part by my earlier work on Brownian-time processes and L-KS equations; and I plan to discuss this in the near future. This work—both local and global—is at the heart of other work on the delicate path properties of the SPDEs version of the KS and time-fractional Burgers equations, currently under investigation with Yimin Xiao.

## Contact

Department of Mathematics
Michigan State University