Abstract:We study self-similar singularity formation for the one-dimensional weak-advection Hou--Li model, a reduced model motivated by the axisymmetric Euler equations. In the periodic setting, we construct exact finite-time self-similar blowup solutions for $2/3<a<1$, with profiles that are neither focusing nor expanding. In the whole-space setting with a Neumann condition, we construct exact finite-time self-similar blowup solutions for the full range $0<a\leq1$, with profiles of focusing, non-expanding/non-focusing, or expanding form depending on the sign of the self-similar scaling parameter. The construction is based on a fixed-point formulation near the origin, followed by an ODE extension argument. We also establish regularity, asymptotic behavior, monotonicity properties of the profiles, and uniqueness up to the natural scaling invariance.
From: Xiang Qin [view email]
[v1]
Thu, 25 Jun 2026 06:45:21 UTC (290 KB)
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