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

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 .

Stochastic quantisation of Yang–Mills–Higgs in 3D

“…Unfortunately, these spaces are sufficiently badly behaved so that there do exist vector-valued X ∈ C − 1 2 for which the map t → P t X ⊗ ∇P t X fails to be locally integrable at t = 0 (this does not happen when u ∈ H 1 2 ). In fact, there exists no Banach space of distributions supporting the 3D Gaussian free field to which the YM flow extends continuously [31].…”

“…5.2, we will linearise Y around PY 0 and U 0 , and then apply Lemma A.4 with appropriate "input" distributions, in a similar way as we did for the last line of (5.20). 31 Recall the conditions η ∧ α > −2 and η = η ∧ α + β in [56,Prop. 6.16].…”