The drift-diffusion system in two-dimensional critical Hardy space (Q957966)

The authors establish the well-posedness of a time local solution to two-dimensional drift-diffusion system for large initial data in the critical Hardy space \(\mathcal H^1(\mathbb R^2)\). This solution is obtained in the Hardy class, for which the natural free energy is well defined from the initial time. The time global well-posedness result for small initial data is also derived in a similar class. The crucial parts of the proof are to employ the end-point type of maximal regularity for the homogeneous heat equation and some new bilinear estimates in the Hardy space.