Graduate Student Analysis/Applied/PDE Seminar:Derek Smith (UCSB), 'Local Well-Posedness of Nonlinear Schrödinger Equation with L^2 Data'

Event Date: 

Tuesday, April 16, 2013 -
10:00am to 11:00am

Event Location: 

  • 6617 South Hall

Event Contact: 

Jon Lo Kim Lin

Email: jlokimlin@math.ucsb.edu

Abstract: We will review properties of the linear Schrödinger equation, in particular stating the Strichartz estimates. I will introduce a nonlinearity of the form $|u|^{alpha-1}u$ into the problem. Formulating this new PDE as an integral equation using Duhamel's principle, we can prove existence of a solution by a contraction mapping argument. The hard part is showing the mapping we introduce is really a contraction mapping and Strichartz's inequalities provide the necessary technical glue. The result is local, with length of existence depending on the size of the initial data.