9,320 reputation
12878
bio website williewong.wordpress.com
location
age 31
visits member for 5 years, 1 month
seen Mar 24 at 15:59

A mathematician specializing in nonlinear hyperbolic PDEs, geometric analysis of pseudo-Riemannian manifolds, and general relativity.


Mar
24
comment Uninteresting questions with interesting answers
I am guessing that $C$ is supposed to be closed?
Mar
24
reviewed Close The product of the power and the natural number in the short interval
Mar
24
comment Global existence of power-type nonlinear Schrodinger equations on compact manifolds
Hyperbolic space arxiv.org/abs/0801.3523
Mar
24
comment Global existence of power-type nonlinear Schrodinger equations on compact manifolds
Manifold with boundary arxiv.org/abs/1004.3976
Mar
24
comment Global existence of power-type nonlinear Schrodinger equations on compact manifolds
Have you even tried searching mathscinet and arxiv? There's a ton of work starting from Bourgain's paper ams.org/mathscinet-getitem?mr=1209299 For example, Burq has done a bit of work, such as this one and this one. Or you can look at some of Zaher Hani's results. Even google turns up a bunch of results.
Mar
24
comment Which way for reading the proofs?
I am not entirely sure that this question is on topic here (seems that Math Educators may be a better fit). But in any case, your first question was very kindly answered by Martin Hairer here (the question and answer are on the last page of the transcript of the interview).
Mar
23
comment global well posedness of cubic NLS in for initial data in $H^{s}(\mathbb R), 0<s<1$
@Inquisitive: you don't use Kato for local existence. Persistence of regularity allows you to say that for initial data in $H^s$ and for any interval (finite or infinite) including the initial time over which the right hand side of the inequality is finite, the solution is in $H^s$ on that interval. The Strichartz estimates allow you to show that for any finite interval the RHS is finite. Therefore you can exhaust $\mathbb{R}$ with a sequence of increasing finite intervals and obtain the desired result. As I mentioned, doing so you lose control of the $H^s$ norm as $t\to \pm\infty$.
Mar
23
comment global well posedness of cubic NLS in for initial data in $H^{s}(\mathbb R), 0<s<1$
What part do you not understand? The proposition above immediately implies that the solution which we guarantee to exist in $C(\mathbb{R}, L^2)$ is in fact in $C(\mathbb{R}, H^s)$. Uniqueness follows trivially since we already have the result on local wellposedness from Kato,
Mar
20
comment structure of metrics on a compact manifold
Two follow up questions: (1) is it true that, as your notations indicated, that you want those $n$ eigenvalues to be the same? (2) Can I assume that $f'_i$ should be obtained by $A_{ij} f_j$ where $A_{ij}$ is some (given, fixed) invertible matrix?
Mar
20
revised structure of metrics on a compact manifold
edited tags
Mar
19
awarded  Nice Answer
Mar
19
comment Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero
Actually, re-reading your answer I noticed that it is much more elegant than I originally thought. So +1 and I won't bother thinking about it more. // and yes, I absent-mindedly forgot to type the $m$ in my previous comment.
Mar
19
comment Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero
@FedorPetrov: oops, you are right. I did over look that fact. In fact I only have $[(1-x^2) W]' = 2 (\ell + 1) P_\ell P_{\ell + 1}$ which is not enough. This should be fixable, let me think about it and delete my previous comment in the mean while.
Mar
19
comment Famous examples of PhD advisors younger than their student
Though I wouldn't be surprised at more examples considering mathoverflow.net/questions/3591/… mathoverflow.net/questions/7120/… and mathoverflow.net/questions/43341/technical-phd-after-33
Mar
19
answered Famous examples of PhD advisors younger than their student
Mar
18
comment Which is the smallest space $X\subset L^{2}$ where the conservation law holds in the norm of $X$?
My (deleted) comment was not 100% correct. See my answer on mathoverflow.net/questions/200331/…
Mar
18
answered global well posedness of cubic NLS in for initial data in $H^{s}(\mathbb R), 0<s<1$
Mar
18
comment Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'
There seems to be a small typo in the statement in the paper I linked to. The number $m$ should be defined as $[ (\lambda - n)/p ]$ rather than $[(n-\lambda )/ p]$ as written, noting that $\lambda > n$ in the regime where you get Holder spaces.
Mar
18
answered Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'
Mar
17
comment Wedge product of Endomorphism-Valued Forms
Yes, just do it on the base. You should be able to find something in any exposition about Yang-Mills theories (on curved backgrounds) (there instead of End(E) your form takes value in a Lie algebra, but the principles are the same).