Student Harmonic Analysis and PDE Seminar (HADES): Almost conservation laws and low regularity well-posedness for nonlinear Schrodinger equations

Seminar | October 23 | 3:40-5 p.m. | 740 Evans Hall

 James Rowan, UC Berkeley

 Department of Mathematics

For energy-subcritical nonlinear Schrodinger equations, the law of conservation of energy can be used to extend the local well-posedness theory for solutions with initial data in $H_x^1$ to a global well-posedness theory in $H_x^1$. If we want a global well-posedness theory in a Sobolev space $H_x^s$ for $0 < s < 1$, the energy may be infinite, and thus conservation of energy is unavailable. Colliander, Keel, Staffilani, Takaoka, and Tao, building off of earlier ideas of Bourgain, developed a method to prove global well-posedness at regularities below $H_x^1$ via "almost conserved" quantities. After applying a Fourier multiplier $I$ which is the identity at low frequencies and is decaying at a rate like $\xi ^{s-1}$ at high frequencies, the energy of this modified function $Iu$ can be made to be finite, and the time derivative of $E[Iu]$ can be estimated. The fact that the modified energy $E[Iu]$ is "almost-conserved" is enough to extend local well-posedness to global well-posedness. I will present the proof of an almost-conservation law for the three-dimensional cubic NLS due to Colliander et al. and use it to show global well-posedness in $H_x^s$ for $s > 5/6$.