arXiv subjects
Publications and source records attributed to Rowan Killip.
We prove almost sure global existence and scattering for the energy-critical nonlinear Schrödinger equation with randomized spherically symmetric initial data in $H^s(\mathbb{R}^4)$ with $\frac56<s<1$. We were inspired to consider this problem by the recent work of Dodson--Lührmann--Mendelson, which treated the analogous problem for the energy-critical wave equation.
We prove inverse Strichartz theorems at $L^2$ regularity for a family of Schrödinger evolutions in one space dimension. Prior results rely on spacetime Fourier analysis and are limited to the translation-invariant equation $i\partial_t u = -\tfrac{1}{2} Δu$. Motivated by applications to the mass-critical Schrödinger equation with external potentials (such as the harmonic oscillator) we use a physical space approach.
We prove symplectic non-squeezing for the cubic nonlinear Schrödinger equation on the line via finite-dimensional approximation.
We prove symplectic non-squeezing (in the sense of Gromov) for the cubic nonlinear Schrödinger equation on $\R^2$. This is the first symplectic non-squeezing result for a Hamiltonian PDE in infinite volume. As the underlying symplectic Hilbert space is $L^2(\R^2)$, this requires working with initial data in this space. This space also happens to be scaling-critical for this equation. Thus, we also obtain the first unconditional symplectic non-squeezing result in such a critical setting. More generally, we show that solutions of this PDE can be approximated by a finite-dimensional Hamiltonian system, despite the wealth of non-compact symmetries: scaling, translation, and Galilei boosts. This approximation result holds uniformly on bounded sets of initial data. Complementing this approximation result, we show that all solutions of the finite-dimensional Hamiltonian system can be approximated by the full PDE. A key ingredient in these proofs is the development of a general methodology for obtaining uniform global space-time bounds for suitable Fourier truncations of dispersive PDE models.
We consider random non-normal matrices constructed by removing one row and column from samples from Dyson's circular ensembles or samples from the classical compact groups. We develop sparse matrix models whose spectral measures match these ensembles. This allows us to compute the joint law of the eigenvalues, which have a natural interpretation as resonances for open quantum systems or as electrostatic charges located in a dielectric medium. Our methods allow us to consider all values of $β>0$, not merely $β=1,2,4$.
We consider the mass-subcritical nonlinear Schrödinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity $s_c\in(\max\{-1,-\frac{d}{2}\},0)$, we prove that any solution satisfying $\|\, |x|^{|s_c|}e^{-itΔ} u\|_{L_t^\infty L_x^2} <\infty$ on its maximal interval of existence must be global and scatter.
We give a simplified proof of the nonexistence of large nuclei in the liquid drop model and provide an explicit bound. Our bound is within a factor of 2.3 of the conjectured value and seems to be the first quantitative result.
We consider the focusing cubic nonlinear Schrödinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS. We also prove failure of uniform space-time bounds at the threshold.
We construct solutions with prescribed scattering state to the cubic-quintic NLS $$ (i\partial_t+Δ)ψ=α_1 ψ-α_{3}\vert ψ\vert^2 ψ+α_5\vert ψ\vert^4 ψ$$ in three spatial dimensions in the class of solutions with $|ψ(x)|\to c >0$ as $|x|\to\infty$. This models disturbances in an infinite expanse of (quantum) fluid in its quiescent state --- the limiting modulus $c$ corresponds to a local minimum in the energy density. Our arguments build on work of Gustafson, Nakanishi, and Tsai on the (defocusing) Gross--Pitaevskii equation. The presence of an energy-critical nonlinearity and changes in the geometry of the energy functional add several new complexities. One new ingredient in our argument is a demonstration that solutions of such (perturbed) energy-critical equations exhibit continuous dependence on the initial data with respect to the \emph{weak} topology on $H^1_x$.
We consider the focusing cubic NLS in the exterior $Ω$ of a smooth, compact, strictly convex obstacle in three dimensions. We prove that the threshold for global existence and scattering is the same as for the problem posed on Euclidean space. Specifically, we prove that if $E(u_0)M(u_0)<E(Q)M(Q)$ and $\|\nabla u_0\|_2\|u_0\|_2<\|\nabla Q\|_2\|Q\|_2$, the corresponding solution to the initial-value problem with Dirichlet boundary conditions exists globally and scatters to linear evolutions asymptotically in the future and in the past. Here, $Q(x)$ denotes the ground state for the focusing cubic NLS in $\mathbb{R}^3$.
We show that discrete one-dimensional Schrödinger operators on the half-line with ergodic potentials generated by the doubling map on the circle, $V_θ(n) = f(2^n θ)$, may be realized as the half-line restrictions of a non-deterministic family of whole-line operators. As a consequence, the Lyapunov exponent is almost everywhere positive and the absolutely continuous spectrum is almost surely empty.
The goal of this paper is to develop some basic harmonic analysis tools for the Dirichlet Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions $d\geq 3$. Specifically, we will discuss analogues of the Mikhlin Multiplier Theorem, Littlewood-Paley Theory, and Hardy inequalities, culminating in a proof that homogeneous Sobolev norms defined with respect to the Dirichlet and whole-space Laplacians are equivalent for the sharp ranges of integrability exponent $p$ and regularity $s$. Counterexamples are included to show that these results are indeed sharp. In particular, we precisely settle the question of boundedness of Riesz transforms on $L^p$, including the endpoint. The utility of such results in the study of nonlinear PDE is that they allow us to deduce important results, such as the fractional product and chain rules for the Dirichlet Laplacian, directly from the classical Euclidean setting. As an application, we discuss the local well-posedness and stability problems for energy-critical NLS. All the results of this paper play an essential role in the authors' proof of large-data global well-posedness and scattering for the energy-critical NLS in three dimensional exterior domains; see arXiv:1208:4904.
We consider the cubic-quintic nonlinear Schrödinger equation: \[ i\partial_t u = -Δu - |u|^2u + |u|^4u. \] In the first part of the paper, we analyze the one-parameter family of ground-state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo--Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region $\mathcal{R}$ of the mass/energy plane where the virial is positive. We show this region is partially bounded by solitons but also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial-based methodology.
We prove scale-invariant Strichartz inequalities for the Schrodinger equation on rectangular tori (rational or irrational) in all dimensions. We use these estimates to give a unified and simpler treatment of local well-posedness of the energy-critical nonlinear Schrodinger equation in dimensions three and four.
We consider the defocusing energy-critical nonlinear Schrödinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions. For the initial-value problem with Dirichlet boundary condition we prove global well-posedness and scattering for all initial data in the energy space.
We analyze the blowup behaviour of solutions to the focusing nonlinear Klein--Gordon equation in spatial dimensions $d\geq 2$. We obtain upper bounds on the blowup rate, both globally in space and in light cones. The results are sharp in the conformal and sub-conformal cases. The argument relies on Lyapunov functionals derived from the dilation identity. We also prove that the critical Sobolev norm diverges near the blowup time.
We consider the Gross--Pitaevskii equation on $\R^4$ and the cubic-quintic nonlinear Schrödinger equation (NLS) on $\R^3$ with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations.
We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments. This result was obtained previously by Colliander, Keel, Staffilani, Takaoka, and Tao.