Ph.D. Chulkwang Kwak
Department of Mathematics
Ewha Womans University, Republic of Korea

In this talk, we discuss the well-posedness issues for the one-dimensional generalized Benjamin--Bona--Mahony (gBBM) equations. We first establish unconditional local well-posedness in $H^s$ for $ s > \frac{k-2}{2k}, $ a threshold proven to be sharp by demonstrating the failure of $C^p$ regularity for the flow map below this level. Furthermore, we present the global well-posedness results below the $H^1$ energy space for $p=3$ and $p=5$, achieved by implementing a novel Hamiltonian conservation law approach. These results extend the classical Bona--Tzvetkov framework and provide a complete picture of the regularity requirements for the gBBM model. If time allows, we will also discuss ongoing work regarding the improvement of global well-posedness results for more general cases.