| Abstract: |
| We investigate $C^1$ blow-up phenomena for a class of transport-type equations. The classical method of characteristics provides only limited information on singularity formation, typically capturing the temporal blow-up rate along characteristic curves. While this approach has been successfully applied to the Burgers equation and various models of wave breaking in water waves, it does not fully describe the structure of singularities. It is well known that singularities in the Burgers equation exhibit $C^{1/3}$ regularity, a feature also observed in certain fluid models such as the Euler equations. This has led to the common belief that all $C^1$ blow-up solutions arising in transport equations share the same regularity profile. In this work, we show that this paradigm is incomplete. We construct self-similar blow-up profiles for a general class of transport-type equations-often arising as leading-order models in water wave dynamics-that exhibit singular behaviors distinct from those of the Burgers equation. In particular, these profiles may possess different H\older regularities. Our framework applies to several important models, including the Camassa--Holm equation, the Hunter--Saxton equation, and the $b$-family of fluid transport equations. We present the construction of explicit stable blow-up profiles and outline the proof of their stability in a suitable functional setting. This yields sharp regularity results and a refined description of blow-up dynamics.If time permits, we will also discuss more exotic self-similar blow-up profiles arising within a broader scaling framework. |
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