Norm inflation for a non-linear heat equation with Gaussian initial conditions

TL;DR

This paper demonstrates that a non-linear heat equation with Gaussian initial conditions exhibits norm inflation, leading to local ill-posedness in certain Besov spaces and highlighting limitations in extending well-posedness results.

Contribution

It proves norm inflation for a non-linear heat equation with Gaussian initial data, showing ill-posedness at the critical Besov space and limitations on extending well-posedness for related flows.

Findings

Norm inflation occurs with high probability for Gaussian initial conditions.

No Banach space can carry the Gaussian free field and extend the flow continuously.

The equation is locally ill-posed at the critical Besov space B^{-1/2}_{ abla, abla}.

Abstract

We consider a non-linear heat equation $\partial_t u = \Delta u + B(u,Du)+P(u)$ posed on the $d$-dimensional torus, where $P$ is a polynomial of degree at most $3$ and $B$ is a bilinear map that is not a total derivative. We show that, if the initial condition $u_0$ is taken from a sequence of smooth Gaussian fields with a specified covariance, then $u$ exhibits norm inflation with high probability. A consequence of this result is that there exists no Banach space of distributions which carries the Gaussian free field on the 3D torus and to which the DeTurck-Yang-Mills heat flow extends continuously, which complements recent well-posedness results in arXiv:2111.10652 and arXiv:2201.03487. Another consequence is that the (deterministic) non-linear heat equation exhibits norm inflation, and is thus locally ill-posed, at every point in the Besov space $B^{-1/2}_{\infty,\infty}$; the space $B^{-1/2}_{\infty,\infty}$ is an endpoint since the equation is locally well-posed for $B^{\eta}_{\infty,\infty}$ for every $\eta>-\frac12$.