## Analysis & PDE

• Anal. PDE
• Volume 2, Number 3 (2009), 281-304.

### 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. To treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type space $b̂p,∞s(T)$ for $s=−12+$, $p=2+$ such that $sp<−1$. In establishing local well-posedness, we use a variant of the Bourgain space adapted to $b̂p,∞s(T)$ 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 $T$.

#### Article information

Source
Anal. PDE, Volume 2, Number 3 (2009), 281-304.

Dates
Revised: 28 August 2009
Accepted: 19 October 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513798038

Digital Object Identifier
doi:10.2140/apde.2009.2.281

Mathematical Reviews number (MathSciNet)
MR2603800

Zentralblatt MATH identifier
1190.35202

#### Citation

Oh, Tadahiro. Periodic stochastic Korteweg–de Vries equation with additive space-time white noise. Anal. PDE 2 (2009), no. 3, 281--304. doi:10.2140/apde.2009.2.281. https://projecteuclid.org/euclid.apde/1513798038

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