Bilinear local smoothing estimate for Airy equation. (English) Zbl 1249.35288

The authors present a refinement of bilinear local smoothing estimates for the operator associated with the initial value problem for the Airy equation \(\partial _t u + \partial _x^3 u =0\), when the frequency supports of two waves are separated. The gain comes from the fact that the interaction of two linear waves at different frequencies is weaker due to the curvature of the characteristic curve in the space-time frequency space. As a corollary, a smoothing property of bilinear form is observed, where the refined bilinear estimate helps to move some derivatives to the low frequency part.


35Q53 KdV equations (Korteweg-de Vries equations)
35E15 Initial value problems for PDEs and systems of PDEs with constant coefficients
Full Text: arXiv