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Nonlinear Schrödinger equations and sharp interpolation estimates

yhhuang 添加于 2012-6-3 11:41 | 2036 次阅读 | 0 个评论

作者
Weinstein MI
详细资料
- 文献种类:期刊
- 期刊名称: Communications in Mathematical Physics
- 期卷页: 1983年第87卷第4期 567-576页
- 出版社: Springer
标签

inequality
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Sharp GN inequality, NLS and sharp estimates

The one family of GS inequalities
$\|f\|_{2\sigma+2}^{2\sigma+2} \leq C_{\sigma,N}^{2\sigma+2}\|\nabla f\|_2^{\sigma N}\|f\|_2^{2+\sigma(2-N)}$
and the solution to the equation
$\frac{\sigma N}{2}\Delta\psi - \left(1+\frac{\sigma}{2}(2-N)\right)\psi+ \psi^{2\sigma+1}=0$
General existence for NLS
$2i\frac{\partial \phi}{\partial t} + \Delta \phi + |\phi|^{2\sigma}\phi=0$
The most important thing is conserved quantities of the energy and Hamiltonian
$\mathcal{N}(\phi)\equiv \|\phi(\cdot,t)\|_2$
and
$\mathcal{H}(\phi) \equiv \|\nabla\phi(\cdot,t)\|_2^2 - \frac{1}{\sigma+1}\|\phi(\cdot,t)\|_{2+2\sigma}^{2+2\sigma}$
Sharp condition for some special cases