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

Abstract

Abstract We consider a non-linear heat equation $$\partial _t u = \Delta u + B(u,Du)+P(u)$$ ∂ t u = Δ u + B ( u , D u ) + 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 $$u_0$$ 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 of Cao–Chatterjee and the author with Chandra–Hairer–Shen. 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 $$B^{-1/2}_{\infty ,\infty }$$ B ∞ , ∞ - 1 / 2 ; the space $$B^{-1/2}_{\infty ,\infty }$$ B ∞ , ∞ - 1 / 2 is an endpoint since the equation is locally well-posed for $$B^{\eta }_{\infty ,\infty }$$ B ∞ , ∞ η for every $$\eta >-\frac{1}{2}$$ η > - 1 2 .