Periodic stochastic Korteweg–de Vries equation with additive space-time white noise

Abstract

We prove the local well-posedness of the periodic stochastic Korteweg-de Vries equation with the additive space-time white noise. In order to treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type space bb s∞(T) for s = 1 +, p = 2+ such that sp < 1. In establishing the local well-posedness, we use a variant of the Bourgain space adapted to b s∞(T) and establish a nonlinear estimate on the second iteration on the integral formulation. The deterministic part of the nonlinear estimate also yields the local well-posedness of the deterministic KdV in M(T), the space of finite Borel measures on T.