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Results tagged with ap.analysis-of-pdes
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user 96950
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
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 …
- 233
2
votes
0
answers
60
views
When are solutions of the Schrödinger equation radial?
Let $S$ be a nonnegative self-adjoint operator on a complex Hilbert space $X$. (For example, $X$ consists of functions on $\mathbb R^d$; it could be $L^2(\mathbb R^d), \dot{H}^2(\mathbb R^d)$, etc. …
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2
votes
1
answer
156
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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 …
- 233
1
vote
0
answers
105
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$(i\partial_t)^{\frac{1}{2}} e^{it\Delta} f = (-\Delta )^{\frac{1}{2}} e^{it\Delta} f$?
Put $\langle x \rangle ^s = (1 + |x|)^{1/2},$ and $|\nabla |^s $ denotes the Fourier multiplier with symbol $|\xi|^s$, that is, $\widehat{|\nabla |^s f} = |\xi|^s \hat{f}.$
Put $ \langle \nabla \r …
- 233
0
votes
0
answers
47
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$u$ satisfies Schrödinger equation implies $\mathcal{F}^{-1} \left(\chi_{2}(\xi) \hat{u} \ri...
Consider the Schrödinger equation (SE):
$i \frac{\partial }{\partial t}u (x,t )+ \Delta u(x,t) =0, (x, t)\in \mathbb R^{N}\times \mathbb R.$
$u(0,x)=\phi(x).$
Then, formally,
the solution of (SE) c …
- 233