Spring 2013 Seminars
- About Existence of Solutions of Differential Equations
Catalin Georgescu (USD)
April 3, Arts & Sciences Room 105, 4 p.m. - 5 p.m.
A more than a hundred years old result (the celebrated Picard-Lindelöf theorem) asserts that if an autonomous vector field f(x) is Lipschitz, then the Cauchy problem X(0)=x of the differential equation X'(t)=f(X) has a unique solution. People wondered if the Lipschitz condition can be weakened. This proved to be a very resistant problem, but with serious theoretical and practical implications. There are two milestone steps in this development: the 1989 DiPerna –Lions paper and the 2004 Ambrosio's article, both in Inventiones Mathematicae. I will briefly present the main innovations brought by these two papers, with a focus on the Ambrosio's concept of Lagrangian flow. Then I will show why if the vector field is only bounded measurable, one can find a vector field with no solutions on a set of positive Lebesgue measure. On the other hand, if the vector field is only bounded measurable and with components staying away from zero, it is believed that the existence theorem still holds almost everywhere, but this seems to be a difficult problem in dimension higher than one. This talk wants to be a discussion revolving around the concept of solution of an ODE/PDE.
- Convergence analysis of the method of lines - A family of second to third order accuracy schemes
Jiang Nan (USD)
February 20, Arts & Sciences Room 104B, 4 p.m. - 5 p.m.