Sharp Regularizing Effect of the Cauchy Problem for the Inhomogeneous Non-Cutoff Kac Equation

2025/12/09 by Cai, Xinzhi, Cao, Hongmei, Xu, Chao-jiang
#35B65 #35Q20 #35S30 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.08380

Abstract

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.

Citations

• On the Boltzmann equation with soft potentials: Existence, uniqueness and smoothing effect of mild solutions
• Sharp regularization effect for the non-cutoff Boltzmann equation with hard potentials
• Analytic smoothing effect of the spatially inhomogeneous Landau equations for hard potentials
• Gevrey regularity of mild solutions to the non-cutoff Boltzmann equation
• Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules
• The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space
• Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation
• Gelfand-Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation
• Analytical regularizing effect for the radial and spatially homogeneous\n Boltzmann equation
• Spectral and phase space analysis of the linearized non-cutoff Kac collision operator
• Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential
• Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation
• Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules

Related

• Gelfand-Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation
• The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space
• Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation
• Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off
• Sharp regularization effect for the non-cutoff Boltzmann equation with hard potentials