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Periodic stochastic Korteweg-de Vries equation with additive space-time white noise. (English) Zbl 1190.35202

Summary: 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 \(\widehat{b}^s_{p,\infty}(\mathbb T)\) for \(s=-\frac12+\), \(p=2+\) such that \(sp<-1\). In establishing local well-posedness, we use a variant of the Bourgain space adapted to \(\widehat{b}^s_{p,\infty}(\mathbb T)\) 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 \(M(\mathbb T)\), the space of finite Borel measures on \(T\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35B35 Stability in context of PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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