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<p>We investigate the smoothing effect of the spatially inhomogeneous Boltzmann equation without an angular cut-off, under the Navier-Stokes scaling. For Maxwellian molecules or hard potentials with singular angular kernels, we demonstrate that the solutions become analytic at positive times when the angular singularities are sufficiently strong and lie within the optimal Gevrey class when the singularities are mild. The analysis is based on carefully selected vector fields with time-dependent coefficients and quantitative estimates of directional derivatives, which reveal the behavior of the kinetic-fluid transition.</p>

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Regularization effect of the Boltzmann equation under Navier-Stokes type scaling

  • 期刊论文

  • 发表日期：

    2025年10月29日

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  • Qi An,
  • Xin Hu,
  • Weixi Li

摘要

We investigate the smoothing effect of the spatially inhomogeneous Boltzmann equation without an angular cut-off, under the Navier-Stokes scaling. For Maxwellian molecules or hard potentials with singular angular kernels, we demonstrate that the solutions become analytic at positive times when the angular singularities are sufficiently strong and lie within the optimal Gevrey class when the singularities are mild. The analysis is based on carefully selected vector fields with time-dependent coefficients and quantitative estimates of directional derivatives, which reveal the behavior of the kinetic-fluid transition.