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<p>In this work, we study the spatially inhomogeneous Kac equation with a non-cutoff cross section in a setting close to equilibrium. We prove that the solution to the Cauchy problem exhibits a sharp Gevrey-Gelfand-Shilov smoothing effect with an optimal radius. We employ a well-chosen exponential-type Fourier multiplier to establish the smoothing effect for position and velocity variables.</p>

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Sharp regularizing effect of the Cauchy problem for inhomogeneous non-cutoff Kac equation

  • 期刊论文

  • 发表日期：

    2026年3月12日

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  • Xinzhi Cai,
  • Hongmei Cao,
  • Chao-Jiang Xu

摘要

In this work, we study the spatially inhomogeneous Kac equation with a non-cutoff cross section in a setting close to equilibrium. We prove that the solution to the Cauchy problem exhibits a sharp Gevrey-Gelfand-Shilov smoothing effect with an optimal radius. We employ a well-chosen exponential-type Fourier multiplier to establish the smoothing effect for position and velocity variables.