Morawetz Reading Notes
2011-03-02
J. Colliander
(The MultiMarkDown source file for these notes is here.)
$ \rightarrow $ Kyle Thompson: The Oberwolfach talk outlines the main advance made in this paper. (See also [Planchon-Vega] where the same key estimate was found indepdently and simultaneously.) I suggest you try to understand the vector-commutator approach to proving the 2-particle (bilinear) Morawetz estimate. This is the most recent and perhaps most flexible proof found to date. It is not, at least for me, the most intuitive.
Research Directions
- I want to understand how to “lift” these Morawetz ideas to higher complexity problems like quantum many body theory, density functional descriptions, nonlinear Dirac, ….
- What is the relationship between these Morawetz-type estimates and Mourre estimates?
- I want to understand why the interaction estimate for the nonlinear Klein-Gordon equation fails. I want to understand the relationship between NLS and NLKG in the nonrelativisity limit. What is the impact of the interaction Morawetz estimate in the limit when NLS approximates NLKG well?
- M. Keel and T. Tao have recent work which establishes interaction Morawetz type estimates in certain Dirac equations. I don’t recall the details but Magda might have some notes. (I think d’Ancona has a recent paper in this direction on arXiv.)
- M. Czubak has generalized some aspects of these estimates to the setting of magnetic Schrodinger operators. This sets the stage to consider Schrodinger systems where the EM field moves according to a different dynamic.
- What is the log-Sobolev inequality? Can you formulate an analogous log-Strichartz inequality? Can you “bilinearize” the log-Strichartz inequality? Is there a many particle Morawetz based proof of the log-Strichartz inequality? Is there a many particle style proof of the log-Sobolev inequality?
- A main idea: Suppose we know that $f’ = A + B$, that $A \geq 0$, that $B \geq 0$ and that $f \leq \Lambda.$ By integrating, we can conclude
$$
\int_0^t B ds \leq 2 \Lambda.
$$
(In our case, $B$ will be an integral over the spatial domain so we infer a time integrated spacetime upper bound.)
- Morawetz-type estimates emerge from monotonicity properties for certain virial-type identities.
- Virial-type identities emerge from integrating the pointwise local conservation laws against clever weight functions.
- The local conservation laws may be inferred through Nöther’s theorem.
A request
- Can you upload the papers discussed here into the Toronto PDE group on Zotero?