Tadahiro Oh — Page 7

arXiv2009

Invariance of the white noise for KdV

We prove the invariance of the mean 0 white noise for the periodic KdV. First, we show that the Besov-type space \hat{b}^s_{p, \infty}, sp <-1, contains the support of the white noise. Then, we prove local well-posedness in \hat{b}^s_{p, \infty} for p= 2+, s = -{1/2}+ such that sp <-1. In establishing the local well-posedness, we use a variant of the Bourgain spaces with a weight. This provides an analytical proof of the invariance of the white noise under the flow of KdV obtained in Quastel-Valko.

math.AP↗

arXiv2009

Diophantine Conditions in Well-Posedness Theory of Coupled KdV-Type Systems: Local Theory

We consider the local well-posedness problem of a one-parameter family of coupled KdV-type systems both in the periodic and non-periodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter αwhen α\ne 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish sharp local well-posedness of the system in H^s(\mathbb{T}_λ), s \geq s^\ast, where s^\ast = s^\ast(α) \in ({1/2}, 1] is determined by the Diophantine characterization of certain constants derived from the coupling parameter α. We also present a sharp local (and global) result in L^2(\mathbb{R}). In the appendix, we briefly discuss the local well-posedness result in H^{-{1/2}}(\mathbb{T}_λ) for α= 1 without the mean 0 assumption, by introducing the vector-valued X^{s, b} spaces.

math.AP↗