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

Abstract

We consider a non-linear heat equation ${\partial }_{t}u=\Delta u+B(u,Du)+P(u)$ 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 $B_{\infty ,\infty }^{-1/2}$; the space $B_{\infty ,\infty }^{-1/2}$ is an endpoint since the equation is locally well-posed for $B_{\infty ,\infty }^{\eta }$ for every $\eta >-\frac{1}{2}$.

21 pages. Minor corrections, added Appendix B on well-posedness in classical regime. To appear in Stoch PDE: Anal Comp

References in corpus (6)

• Random data Cauchy theory for supercritical wave equations I: Local theory
• Langevin dynamic for the 2D Yang-Mills measure
• Global dynamics for the two-dimensional stochastic nonlinear wave equations
• Stochastic quantisation of Yang-Mills
• Global well-posedness of the two-dimensional stochastic nonlinear wave equation on an unbounded domain
• Norm inflation for a non-linear heat equation with Gaussian initial conditions

Cited by in corpus (6)

• Stochastic quantisation of Yang-Mills-Higgs in 3D
• Stochastic quantisation of Yang-Mills
• Norm inflation for a non-linear heat equation with Gaussian initial conditions
• Uniqueness of gauge covariant renormalisation of stochastic 3D Yang-Mills-Higgs
• Non-Gaussianity of invariant measures to SPDEs in Da Prato-Debussche regime
• Local well-posedness of subcritical non-linear heat equations with Gaussian initial data