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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

3 votes
1 answer
119 views

energy trapping for NLS

I am trying to understand the proof of Theorem 3.9, p.9. We consider NLS $$i\partial_t u + \Delta + |u|^{4/(d-2)}u=0, u(x,0)=u_0 \in H^1(\mathbb R^d)$$ where $u:\mathbb R^{d+1} \to \mathbb C, u_0:\ …
abcd's user avatar
  • 233
1 vote
0 answers
66 views

How to define spectral multiplier for −Δ?

Put $e_{n}(t)=e^{int}=\prod_{j=1}^{d}e^{in_{j}t_{j}}$, $t\in \mathbb T^d, n\in \mathbb Z^d.$ ($t=(t_{1},..., t_d), n=(n_1,..., n_d)$) We note that $\{e_{n}\}_{n\in \mathbb Z^d}$ forms an orthonormal …
abcd's user avatar
  • 233
3 votes
2 answers
400 views

finite time blow-up criterion in nonlinear Schrodinger

I am trying to understand the philosophy for finite time blow-up criterion and global existence for ODE/PDEs, any suggestion and comments will be useful to me. I hope this question is OK for MO. Cons …
abcd's user avatar
  • 233
2 votes
1 answer
156 views

Defect of Compactness for the Strichartz Estimates

I am trying to understand the motivation behind the main theorem of Keraani: See the top of page 356 and which is the motivation behind his Theorem 1.6 We consider $$i\partial_t u + \Delta u =0, u …
abcd's user avatar
  • 233