Bilinear estimates in the presence of a large potential and a critical NLS in 3D (Q6605405)

The general framework of this manuscript is the study of a nonlinear evolution equation in the presence of an external potential \(V\),\N\[\Ni\partial_t u +L(-i\nabla_x)u + V(x) u=\mathcal N (u).\N\]\NThe situation analyzed in detail corresponds to the Schrödinger equation, \(L(-i\nabla) =\Delta\), but the authors also have in mind the Klein-Gordon or the wave equation. Such an equation appears typically when studying the stability of solitary waves \(e^{i\omega t}Q(x)\) for the nonlinear Schrödinger equation (without potential): after linearizing about the solitary wave, the potential \(V\) appears, as a function of \(Q\), and the perturbation of the solitary wave satisfies an equation of the form considered here.\N\NThe core of the analysis consists of the analysis of the distorted Fourier transform, adapted to the Schrödinger operator \(H=-\Delta+V\), in dimension \(d=3\), under the assumption that \(V\) is smooth and rapidly decaying,\N\[\N\int_{\mathbb R^3}(1+|x|)^M |\nabla^\alpha V(x)|dx<\infty,\quad 0\le |\alpha|\le M,\quad M\gg 1,\N\]\Nand that \(H\) has no eigenvalues or resonances. Generalized eigenfunctions \(\psi(x,k)\) solve\N\[\NH\psi = |k|^2\psi,\quad \forall k\in \mathbb R^3\setminus\{0\},\N\]\Nand can be viewed as generalization, in the presence of a potential, of the plane waves \(e^{ik\cdot x}\). The distorted Fourier transform \(\widetilde {\mathcal F}\) is defined by\N\[\N\widetilde{\mathcal F} g(k) =\widetilde g(k)= \frac{1}{(2\pi)^{3/2}}\int_{\mathbb R^3} \overline{\psi(x,k)}g(x)dx.\N\]\NLike the Fourier transform for the Laplacian, it diagonalizes the Schrödinger operator \(H\), \(\widetilde{\mathcal F}H = |k|^2\widetilde {\mathcal F}\), and the inverse formula is given by\N\[\N\widetilde{\mathcal F}^{-1} g(x) = \frac{1}{(2\pi)^{3/2}}\int_{\mathbb R^3} \psi(x,k)g(k)dk.\N\]\NWhen considering the nonlinear Schrödinger equation\N\[\Ni\partial_t u +Hu= u^2,\quad u_{\mid t=0}=u_0,\quad x\in \mathbb R^3,\N\]\Nwhich is the application the manuscript focuses on (proving global-in-time bounds and scattering, for small data \(u_0\)), and after the change of unknown function\N\[\Nf(t,x) = \left(e^{-itH} u(t,\cdot)\right)(x),\N\]\NDuhamel's formula reads, on the distorted Fourier side,\N\[\N\widetilde {\mathcal F} f(t,k) = \widetilde u_0(k) -i\mathcal D(t)(f,f),\N\]\Nwhere\N\[\N\mathcal D(t)(f,f) =\int_0^t \iint_{\mathbb R^3\times \mathbb R^3}e^{is(-|k|^2+|\ell|^2+|m|^2)} \widetilde f(s,\ell)\widetilde f(s,m)\mu(k,\ell,m)d\ell dm ds,\N\]\Nwith\N\[\N\mu(k,\ell,m) =\frac{1}{(2\pi)^{9/2}}\int_{\mathbb R^3}\overline{\psi(x,k)}\psi(x,\ell)\psi(x,m)dx.\N\]\NThe major part of this work consists of estimates of this interaction coefficient \(\mu\), called Nonlinear Spectral Distribution here (NSD), involving the analysis of the generalized eigenfunctions \(\psi(x,k)\), viewed as perturbations of the plane waves \(e^{ik\cdot x}\),\N\begin{align*}\N\psi(x,k)&= e^{ik\cdot x} -e^{i\lvert k\rvert \lvert x\rvert}\frac{1}{4\pi|x|}\psi_1(x,k),\\\N\psi_1(x,k)&:= \int_{\mathbb R^3} e^{i\lvert k\rvert \left( |x-y|-|x|\right)}\frac{|x|}{|x-y|}V(y)\psi(y,k)dy.\N\end{align*}\NThe manuscript consists of nine chapters:\N\begin{itemize}\N\item[1.] Introduction\N\item[2.] Main ideas and strategy\N\item[3.] Linear spectral theory\N\item[4.] Preliminary bounds: linear estimates and high frequencies\N\item[5.] Analysis of the NSD I: structure of the leading order\N\item[6.] Bilinear estimates for the leading order of the NSD\N\item[7.] Weighted estimates for leading order terms\N\item[8.] Analysis of the NSD II: lower order terms\N\item[9.] Weighted estimates for lower order terms\N\end{itemize}\NThe analysis is technically involved, addressed to readers already familiar with such aspects, as some details or quantifiers remain implicit. Several results regarding the estimates of generalized eigenfunctions have potential applications to various questions and equations.