We show that if the dispersion of the KdV equation is replaced by a higher order dispersion , where is an odd integer, then the critical Sobolev exponent for local well-posedness on the circle does not change. That is, the resulting equation is locally well-posed in , .
In this second talk, the results leading to the main Theorem (proved in Part 1) will be established.
Keywords: KdV equation, Local well-posedness, Sobolev spaces, Fourier transform.
Well-posedness of KdV with higher dispersion - Part 2
Feb. 04, 2015 4pm (MATH 350)
Grad Student Seminar
Robert Hines (CU-Boulder)
X
e is transcendental [and maybe pi -PIZZA PIE- not that I'd describe Blackjack pizza as transcendent, but I digress].
You know that e is transcendental, right? Have you seen a proof? Well here's your big chance! We'll warm up by showing that e is irrational and proving the existence of transcendental numbers (a la Liouville). Hermite's (1873) proof of the transcendence of e uses nothing more than basic calculus, arithmetic, and a heaping tablespoon of genius. Time permitting (or forcing), we will discuss generalizations due to Lindemann-Weierstrass, Gelfond-Schneider, and Baker.