Well-Posedness for the Dispersive Hunter–Saxton Equation

Abstract

This article represents a 1st step toward understanding the well-posedness of the dispersive Hunter–Saxton equation, which arises in the study of nematic liquid crystals. Although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character. Further, the lack of spatial decay obstructs access to dispersive tools, including local smoothing estimates. Here, we give the 1st proof of local and global well-posedness for the Cauchy problem. Secondly, we improve our well-posedness results with respect to the low regularity of the initial data. The key techniques we use include constructing modified energies to realize a normal form analysis in our quasilinear setting, and frequency envelopes to prove continuous dependence with respect to the initial data.

Global solutions for a family of GSQG front equations

“…A similar reduction in s could be made to our previous result for SQG fronts [27], where we took s = 1200. These results are still not optimal, and Ai and Avadanei have recently proved low-regularity results for the SQG [2] and GSQG [3] front equations.…”