A class of supercritical/critical singular stochastic PDEs: Existence, non-uniqueness, non-Gaussianity, non-unique ergodicity *
HofmanovĂˇ, Martina
HofmanovĂˇ
Martina
Zhu, Rongchan
Zhu
Rongchan
Zhu, Xiangchan
Zhu
Xiangchan
We study the surface quasi-geostrophic equation with an irregular spatial perturbation partial differential te + u center dot ve = -v(-Delta)gamma/2e + sigma, u = v perpendicular to(-Delta)-1e, on [0, oo) x T2, with v 0,-y E [0, 3/2) and sigma E B-2+kappa infinity,infinity (T2) for some iota c > 0. This covers the case of sigma = (-Delta)alpha/2 xi for alpha < 1 and xi a spatial white noise on T2. Depending on the relation between-y and alpha, our setting is subcritical, critical or supercritical in the language of Hairer's regularity structures [38]. Based on purely analytical tools from convex integration and without the need of any probabilistic arguments including renormalization, we prove existence of infinitely many analytically weak solutions in Lp loc(0, oo; B infinity,1-1/2)nCb([0, oo); B-1/2-delta infinity,1 ) nCb 1 ([0, oo); B-3/2-delta infinity,1 ) for all p E [1, oo) and delta > 0. We are able to prescribe an initial as well as a terminal condition at a finite time T > 0, and to construct steady state, i.e. time independent, solutions. In all cases, the solutions are non-Gaussian, but we may as well prescribe Gaussianity at some given times. Moreover, a coming down from infinity with respect to the perturbation and the initial condition holds. Finally, we show that our solutions generate statistically stationary solutions as limits of ergodic averages, and we obtain existence of infinitely many non-Gaussian time dependent ergodic stationary solutions. We also extend our results to a more general class of singular SPDEs.(c) 2023 Elsevier Inc. All rights reserved.
285
5
Elsevier
2023