| Abstract: |
| In this talk, we first consider the Korteweg-de Vries equation (KdV) with a modulated dispersion.
We observe the regularization-by-noise effects resulting from this modulation: we establish
well-posedness of the modulated KdV on the circle in the regime where the unmodulated KdV
is ill-posed. In particular, we show that the modulated KdV on the circle is locally well-posed
in Sobolev spaces of arbitrarily low regularity, provided that the modulation is sufficiently
irregular.
Then I will present more recent results on the stochastic modulated KdV on the circle
with multiplicative fractional-in-time noise, where we establish a new regularization-by-noise
phenomenon on the stochastic convolution in a pathwise manner, where a gain of spatial
regularity becomes (arbitrarily) larger for more irregular modulations. We then prove that
the stochastic modulated KdV is pathwise locally well-posed in Sobolev spaces of arbitrarily
low regularity, provided that the modulation is sufficiently irregular. If time permits, I will
mention the results when the multiplicative noise is white-in-time.
Based on joint works with Khalil Chouk (formerly Edinburgh), Massimiliano Gubinelli (Oxford), Jiawei Li & Tadahiro Oh (Edinburgh), and Andreia Chapouto (CNRS & Monash). |
|