Rough Axisymmetric Euler Dynamics: Global Energy Compactness and Kinetic Passage Across Regularity Breakdown

Abstract: We establish global energy compactness for rough three-dimensional axisymmetric Euler flow without swirl and use it to construct hydrodynamic limits of collisional kinetic equations beyond finite-time Euler regularity breakdown. For relative vorticity in $L^1\cap L<sup>p$, regular approximate Euler flows are globally compact in $C_tL_x<sup>2$ for $p\ge4/3$, without a sign condition or any spatial-moment assumption. Below $4/3$, the same conclusion holds under uniform control of the absolute impulse. We identify the sharp radial dynamics governing this confinement: the absolute impulse is propagated without a sign condition for $p\ge5/3$, while for $1&lt;p\&lt;5/3$ it is controlled under one-sided $L^{p^\sharp}$ integrability, where $p^\sharp= {4p} / (3p-1)$, and this exponent is sharp. In particular, for one-sign finite-impulse vorticity the resulting global energy compactness holds throughout the full range $$p\&gt;1$$. For the Coulomb Landau equation and a class of non-cutoff soft-potential Boltzmann equations, we construct strong solutions on lifespans $T<sup>\varepsilon\to\infty$ realizing prescribed rough finite-energy Euler initial data and obtain energy-conserving axisymmetric Euler limits on every fixed time interval. If the corresponding regular Euler evolution loses regularity in finite time, the hydrodynamic fields converge to the regular Euler solution up to the singular time and have a common strong trace there, while subsequences yield global energy-conserving Euler continuations past that time.