PUB - Publikationen an der Universität Bielefeld

We consider a family of singular surface quasi-geostrophic equations partial derivative(t)theta + u center dot del theta = -nu(-Delta)(gamma/2)theta + (-Delta)(alpha/2) xi, u = del(perpendicular to)(-Delta)(-1/2) theta, on [0, infinity) x T-2, where nu >= 0, gamma is an element of[0,3/2), alpha is an element of[0,1/4) and xi is a space-time white noise. For the first time, we establish the existence of infinitely many non-Gaussian probabilistically strong solutions for every initial condition in C-eta, eta > 1/2; ergodic stationary solutions. The result presents a single approach applicable in the subcritical, critical as well as supercritical regime in the sense of Hairer (Invent Math 198(2):269-504, 2014). It also applies in the particular setting alpha = gamma/2 which formally possesses a Gaussian invariant measure. In our proof, we first introduce a modified Da Prato-Debussche trick which, on the one hand, permits to convert irregularity in time into irregularity in space and, on the other hand, increases the regularity of the linear solution. Second, we develop a convex integration iteration for the corresponding nonlinear equation which yields non-unique non-Gaussian solutions satisfying powerful global-in-time estimates and generating stationary as well as ergodic stationary solutions.