Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator
supercritical nonlinear Schrödinger equation
This is a new study on the randomized Cauchy problem for nonlinear dispersive equations. Here, the NLS equation in \(\mathbb{R}^d,\;d\geq 2\), is considered with a power nonlinearity with odd exponent \(p\geq 3\). The solutions are studied in Sobolev spaces whose exponent \(s\) is for most results such that \(s_c<s<d/2\) with \(s_c=d/2-2/(p-1)\).NEWLINENEWLINESeveral randomization techniques have emerged. The authors use the one developed by \textit{N. Burq} and \textit{G. Lebeau} [Ann. Sci. Éc. Norm. Supér. (4) 46, No. 6, 917--962 (2013; Zbl 1296.46031)] in a semi-classical framework on compact domains. Here, the domain is not compact but thanks to the so-called ``lens transform'', one gets a new NLS equation with harmonic potential whose linear operator has discrete spectrum which makes the mentioned method work.NEWLINENEWLINEFirstly, the authors give a local existence result for the NLS with harmonic potential. As expected in this random context, a gain of \(d/2\) derivatives is obtained. The result also shows that for any time \(T\) there exists a non-zero measure set of initial data for which the solutions time existence is greater than \(T\). The result is based on a stochastic Strichartz-like inequality whose proof relies on Burq-Lebeau's quoted paper.NEWLINENEWLINESecondly, upon assuming that the random variables can take arbitrarily small values, the authors give a global existence result and a scattering result for the NLS when \(d=2\) and \(2<p<3\). An analogous result is given in an \(L^2\)-subcritical context.NEWLINENEWLINEFinally, a global existence result is given for the NLS with harmonic potential for \(d=p=3\) and \(1/6<s<1\) (supercritical) and also for \(d=2\), \(p=3\) and \(0<s<1\). These results rely on the Strichartz estimate.
scientific article; zbMATH DE number 6340686