论文标题
在存在巨大潜力和3D中的临界NLS的情况下，双线性估计值
Bilinear estimates in the presence of a large potential and a critical NLS in 3d
论文作者
论文摘要
我们提出了一种具有较大且腐烂的外部电势的非线性演化方程的方法，该方程解决了在这种情况下全球控制局部波的非线性相互作用的问题。在研究围绕（可能是未折扣的）特殊解决方案的局部扰动时，就会出现此问题，并试图将投影控制到非线性辐射相互作用的连续光谱上。我们的主要工具之一是适合Schrödinger操作员$ H =-Δ+V $的傅立叶变换，我们在非线性级别采用。作为第一步，我们分析了$ h $的三个广义本征函数的产物的空间积分，并确定其奇异性的确切结构。这导致研究了具有某些奇异核的双线性操作员，为此，我们得出了Coifman-Meyer类型的产品估计。然后，该分析可以与多线性谐波分析工具以及对分散和波程方程的加权估计值（扭曲的傅立叶空间类似物）进行研究。作为第一个应用程序，我们将$ 3 $ d的非线性schrödinger方程式视为具有无界状态的巨大衰减潜力，并且具有$ u^2 $非线性。主要的困难是，对于Strauss指数，$ 3 $ d的二次非线性至关重要。此外，即使$ v = 0 $，这种非线性也具有非平凡的完全连贯交互。我们证明了针对小解决方案的定量全球范围和散射。
We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem arises when studying localized perturbations around (possibly non-decaying) special solutions of evolution PDEs, and trying to control the projection onto the continuous spectrum of the nonlinear radiative interactions. One of our main tools is the Fourier transform adapted to the Schrödinger operator $H=-Δ+V$, which we employ at a nonlinear level. As a first step we analyze the spatial integral of the product of three generalized eigenfunctions of $H$, and determine the precise structure of its singularities. This leads to study bilinear operators with certain singular kernels, for which we derive product estimates of Coifman-Meyer type. This analysis can then be combined with multilinear harmonic analysis tools and the study of oscillations to obtain (distorted Fourier space analogues of) weighted estimates for dispersive and wave equations. As a first application we consider the nonlinear Schrödinger equation in $3$d in the presence of large decaying potential with no bound states, and with a $u^2$ non-linearity. The main difficulty is that a quadratic nonlinearity in $3$d is critical with respect to the Strauss exponent; moreover, this nonlinearity has non-trivial fully coherent interactions even when $V=0$. We prove quantitative global-in-time bounds and scattering for small solutions.
