Well-posedness and $L^1-L^p$ Smoothing Effect of the Porous Media Equation under Poincaré Inequality
Lukang Sun
TL;DR
This work establishes well-posedness and uniqueness for the Cauchy problem of a porous media equation with a potential on $\mathbb{R}^d$ under a Poincaré inequality. It develops a barrier-based framework and Cauchy-Dirichlet approximations to overcome the lack of a fundamental solution, yielding an $L^1$-$L^p$ smoothing effect: for $t>0$, the solution lies in all $L^p(\mathbb{R}^d,\pi)$ with $p\in[1,\infty)$. The authors derive an Aronson-Bénilan type estimate and establish mass-preserving, contractive solutions in $L^1$ that extend from smooth to general $L^1$ initial data, with detailed large-time asymptotics showing a two-phase decay in $L^p$ norms. These results illuminate the smoothing and long-time behavior of the porous media flow with potential, and the role of the Poincaré inequality in guaranteeing regularization and stabilization. $
Abstract
We investigate the well-posedness and uniqueness of the Cauchy problem for a class of porous media equations defined on $\mathbb{R}^d$, and demonstrate the $L^1-L^p$ smoothing effect. In particular, we establish that the logarithm of the ratio of the $L^p$ norm to the $L^1$ norm decreases super-exponentially fast during the initial phase, subsequently decaying to zero exponentially fast in the latter phase. This implies that if the initial data is solely in $L^1$, then for $t>0$, the solution will belong to $L^p$ for any $p\in [1,\infty)$. The results are obtained under the assumption of a Poincaré inequality.