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

TL;DR

This paper demonstrates that solutions to the inhomogeneous non-cutoff Kac equation near equilibrium experience a sharp smoothing effect in Gevrey-Gelfand-Shilov spaces, achieved through a specialized Fourier multiplier.

Contribution

It establishes the optimal smoothing effect for the inhomogeneous non-cutoff Kac equation using a novel exponential Fourier multiplier approach.

Findings

Proves sharp Gevrey-Gelfand-Shilov smoothing for the solution.

Employs exponential Fourier multiplier to analyze regularization.

Achieves optimal radius of smoothing effect.

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.