Multi-D Fast Diffusion Equation via Diffusive Scaling of Generalized Carleman Kinetic Equation

TL;DR

This paper establishes the diffusive limit of a generalized Carleman kinetic equation to the fast diffusion equation, including cases with growing initial data, using a novel barrier method and asymptotic expansion.

Contribution

It introduces a new barrier argument and explicit subsolutions for hyperbolic systems, enabling the analysis of diffusive limits with non-conserved mass and growing initial data.

Findings

Proved convergence to the fast diffusion equation in the diffusive limit.

Extended analysis to cases with growing initial data, including one-dimensional cases.

Developed a new method combining comparison principles and asymptotic expansions.

Abstract

In this paper, we investigate generalized Carleman kinetic equation for n$\ge$2 and prove convergence towards the solution of equation with fast diffusion or porous medium type, $u_t=\Delta u^m$ ($0\le m\le2$), in its diffusive hydrodynamic limit. Using comparison principle of system combined with fixed speed propagation property of transport equation, we create a new barrier argument for this hyperbolic system. It is crucial to construct explicit local sub and solution of system and this is done by employing an ansatz from second order asymptotic expansion. This allow us prove diffusive limit toward subcritical FDE, which is thought to be difficult with previous method due to the lack of mass conservation. Moreover, we can also prove convergence with growing initial data in slow diffusion range (including $n=1$), which was also unknown before.