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

Oh, Tadahiro

doi:10.2140/apde.2009.2.281

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

Tadahiro Oh

Vol. 2 (2009), No. 3, 281–304

DOI: 10.2140/apde.2009.2.281

Abstract

We prove the local well-posedness of the periodic stochastic Korteweg–de Vries equation with the additive space-time white noise. To treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type space ${\hat{b}}_{p, \infty}^{s} (T)$ for $s = - \frac{1}{2} +$ , $p = 2 +$ such that $s p < - 1$ . In establishing local well-posedness, we use a variant of the Bourgain space adapted to ${\hat{b}}_{p, \infty}^{s} (T)$ 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 $T$ .

Keywords

stochastic KdV, white noise, local well-posedness

Mathematical Subject Classification 2000

Primary: 35Q53, 60H15

Milestones

Received: 2 January 2009

Revised: 28 August 2009

Accepted: 19 October 2009

Published: 9 February 2010

Authors

Tadahiro Oh
Department of Mathematics University of Toronto 40 St. George Street, Rm 6290 Toronto, ON M5S 2E4 Canada

Vol. 2, No. 3, 2009