arXiv 1502.009372015-02-03math.AP

变系数NLS的能量空间散射

Scattering in the energy space for the NLS with variable coefficients

Biagio Cassano, Piero D'Ancona

摘要：

我们考虑在$\mathbb{R}^{n}$或更一般地在具有Dirichlet边界条件的外部区域上的$n\ge3$ \begin{equation*} i \partial_t u - Lu +f(u)=0, \qquad Lv=\nabla^{b}\cdot(a(x)\nabla^{b}v)-c(x)v, \qquad \nabla^{b}=\nabla+ib(x), \end{equation*}中具有变系数的NLS，对于规范不变的幂型散焦非线性$f(u)\simeq|u|^{γ-1}u$。我们假设$L$是$Δ$的一个小的、长范围的扰动，加上一个具有大的正部分的势。本文的第一个主要结果是双线性平滑（相互作用Morawetz）估计的解决方案。作为应用，在线性流$e^{itL}$的Ehartz估计成立的条件下，证明了次临界幂$γ<1+\frac{4}{n-2}$和散射提供的$γ>1+\frac4n$在能量空间的整体适定性.当区域为$\mathbb{R}^{n}$时，通过推广Tataru [Tataru08]的Escherichartz估计，我们证明了条件假设是满足的，并推导出了能量空间中的适定性和散射性.

英文摘要：

We consider the NLS with variable coefficients in dimension $n\ge3$ \begin{equation*} i \partial_t u - Lu +f(u)=0, \qquad Lv=\nabla^{b}\cdot(a(x)\nabla^{b}v)-c(x)v, \qquad \nabla^{b}=\nabla+ib(x), \end{equation*} on $\mathbb{R}^{n}$ or more generally on an exterior domain with Dirichlet boundary conditions, for a gauge invariant, defocusing nonlinearity of power type $f(u)\simeq|u|^{γ-1}u$. We assume that $L$ is a small, long range perturbation of $Δ$, plus a potential with a large positive part. The first main result of the paper is a bilinear smoothing (interaction Morawetz) estimate for the solution. As an application, under the conditional assumption that Strichartz estimates are valid for the linear flow $e^{itL}$, we prove global well posedness in the energy space for subcritical powers $γ<1+\frac{4}{n-2}$, and scattering provided $γ>1+\frac4n$. When the domain is $\mathbb{R}^{n}$, by extending the Strichartz estimates due to Tataru [Tataru08], we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space.

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