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Results tagged with ap.analysis-of-pdes
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user 149364
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
vote
1
answer
71
views
Scaling invariance for Hartree equation
Consider generalized Hartree Equation
$ i \partial_t u + \Delta u + \left( \frac{1}{|x|^{\gamma}} \ast |u|^p \right)|u|^{p-2}u, \quad u(x, 0)=u_0(x)$
where $0<\gamma<d, p\geq 2, u(x,t)\in \mathbb C …
- 161
2
votes
1
answer
104
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WKB expansion for NLS
We consider the equation (NLS)
\begin{eqnarray}\label{gnls}
i \epsilon\partial_t u^{\epsilon} + \frac{\epsilon^2}{2}\Delta_{\eta}u^{\epsilon} = \epsilon |u^{\epsilon}|^{2}u^{\epsilon}, \quad x \in \m …
- 161
2
votes
0
answers
300
views
Marcinkiewicz-Mihlin-Hormander Fourier multiplier theorem
I'm trying to understand the hypothesis of the Marcinkiewicz-Mihlin-Hörmander multiplier theorem. See for instance Theorem A in this paper of Elias Stein.
Theorem A: Assume that $m: (0, \infty)\to \m …
- 161