We plan to start by introducing dispersive partial differential equations and recalling definition of well-posedness in our settings. We shall discuss failure of well-posedness which will cover ill-posedness, norm inflation and infinite loss of regularity. The data spaces involved are mainly Fourier amalgam spaces, Fourier Lebesgue spaces and modulation spaces.
Next we shall discuss a resent work (jointly with Rowan Killip, Monica Vișan and Yungfeng Zhang at UCLA) dealing with global well-posedness (GWP) of modified Korteweg-de Vries equation in modulation space \( M^{s,2}_p (\mathbb{R}) \). Our result extends the range of the regularity index \( s \) for which GWP holds in \( M^{s,2}_p(\mathbb{R}) \) to a new range where usual method of contraction mapping principle fails. To present a sketch of proof, we shall discuss few tools from Harmonic analysis e.g. trace, Hilbert-Schmidt norm of an operator in \( L^2(\mathbb{R}) \). It turns out that our method also works to establish GWP for a cubic nonlinear Schrödinger (NLS) equation.
Lastly, I shall briefly discuss our published works on dispersive PDEs and conclude with mentioning some future research directions.
