Qualitative Properties of Dispersive PDEs
Qualitative Properties of Dispersive PDEs
复制标题
色散偏微分方程的定性性质
DOI：
10.1007/978-981-19-6434-3_2
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Bellazzini J
中科院分区：
文献类型：
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作者：
Bellazzini J
我们回顾了一些关于控制非捕获偶极量子气体的Gross-Pitaevskii方程（GPE）解的长时间动力学的最新结果。我们描述了能量空间中初始基准的不同初始构型解的渐近行为，特别是对于质能阈值以下、上方和处的数据。利用抛物型双调和方程的积分核的衰减性质，重新讨论了Riesz变换幂的一些性质。这些衰减性质在建立所研究的GPE解的动力学特征方面起着重要作用。
We review some recent results on the long-time dynamics of solutions to the Gross–Pitaevskii equation (GPE) governing non-trapped dipolar quantum gases. We describe the asymptotic behaviors of solutions for different initial configurations of the initial datum in the energy space, specifically for data below, above, and at the mass–energy threshold. We revisit some properties of powers of the Riesz transforms by means of the decay properties of the integral kernel associated to the parabolic biharmonic equation. These decay properties play a fundamental role in establishing the dynamical features of the solutions to the studied GPE.