Searcharxiv⌕ Search

arXiv subjects

Rowan Killip

Publications and source records attributed to Rowan Killip.

At least 73 records · Page 4Linked to original sources
arXiv2011

Smooth solutions to the nonlinear wave equation can blow up on Cantor sets

We construct $C^\infty$ solutions to the one-dimensional nonlinear wave equation $$ u_{tt} - u_{xx} - \tfrac{2(p+2)}{p^2} |u|^p u=0 \quad \text{with} \quad p>0 $$ that blow up on any prescribed uniformly space-like $C^\infty$ hypersurface. As a corollary, we show that smooth solutions can blow up (at the first instant) on an arbitrary compact set. We also construct solutions that blow up on general space-like $C^k$ hypersurfaces, but only when $4/p$ is not an integer and $k > (3p+4)/p$.

math.AP↗
arXiv2010

Energy-critical NLS with quadratic potentials

We consider the defocusing $\dot H^1$-critical nonlinear Schrödinger equation in all dimensions ($n\geq 3$) with a quadratic potential $V(x)=\pm \tfrac12 |x|^2$. We show global well-posedness for radial initial data obeying $\nabla u_0(x), xu_0(x) \in L^2$. In view of the potential $V$, this is the natural energy space. In the repulsive case, we also prove scattering. We follow the approach pioneered by Bourgain and Tao in the case of no potential; indeed, we include a proof of their results that incorporates a couple of simplifications discovered while treating the problem with quadratic potential.

math.AP↗
arXiv2010

Scattering for the cubic Klein--Gordon equation in two space dimensions

We consider both the defocusing and focusing cubic nonlinear Klein--Gordon equations $$ u_{tt} - Δu + u \pm u^3 =0 $$ in two space dimensions for real-valued initial data $u(0)\in H^1_x$ and $u_t(0)\in L^2_x$. We show that in the defocusing case, solutions are global and have finite global $L^4_{t,x}$ spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state. These results rely on analogous statements for the two-dimensional cubic nonlinear Schrödinger equation, which are known in the defocusing case and for spherically-symmetric initial data in the focusing case. Thus, our results are mostly unconditional. It was previously shown by Nakanishi that spacetime bounds for Klein--Gordon equations imply the same for nonlinear Schrödinger equations.

math.AP↗
arXiv2010

The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions

We consider the defocusing nonlinear wave equation $u_{tt}-Δu + |u|^p u=0$ with spherically-symmetric initial data in the regime $\frac4{d-2} \frac4{d-2}$. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximal-lifespan solutions with bounded critical Sobolev norm are global and scatter.

math.AP↗
arXiv2010

The defocusing energy-supercritical nonlinear wave equation in three space dimensions

We consider the defocusing nonlinear wave equation $u_{tt}-Δu + |u|^p u=0$ in the energy-supercritical regime p>4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.

math.AP↗
arXiv2009

On the mass-critical generalized KdV equation

We consider the mass-critical generalized Korteweg--de Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global well-posedness and scattering conjecture for the mass-critical nonlinear Schrödinger equation $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, there exists a minimal-mass blowup solution to the mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-low frequency cascade solution.

math.AP↗
arXiv2008

Energy-supercritical NLS: critical $\dot H^s$-bounds imply scattering

We consider two classes of defocusing energy-supercritical nonlinear Schrödinger equations in dimensions $d\geq 5$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot H^{s_c}_x$, then $u$ is global and scatters.

math.AP↗
arXiv2008

Perturbations of Orthogonal Polynomials With Periodic Recursion Coefficients

We extend the results of Denisov-Rakhmanov, Szego-Shohat-Nevai, and Killip-Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.

math.SP↗
arXiv2008

Characterization of minimal-mass blowup solutions to the focusing mass-critical NLS

Let $d\geq 4$ and let $u$ be a global solution to the focusing mass-critical nonlinear Schrödinger equation $iu_t+Δu=-|u|^{\frac 4d}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state $Q$. We prove that if $u$ does not scatter then, up to phase rotation and scaling, $u$ is the solitary wave $e^{it}Q$. Combining this result with that of Merle \cite{merle2}, we obtain that in dimensions $d\geq 4$, the only spherically symmetric minimal-mass blowup solutions are, up to phase rotation and scaling, the pseudo-conformal ground state and the solitary wave.

math.AP↗
arXiv2008

The cubic nonlinear Schrödinger equation in two dimensions with radial data

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation $iu_t + Δu = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.

math.AP↗
arXiv2007

The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation $iu_t + Δu = \pm |u|^{4/d} u$ for large spherically symmetric L^2_x(R^d) initial data in dimensions $d\geq 3$. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.

math.AP↗
arXiv2007

Gaussian fluctuations for βEnsembles

We study the Circular and Jacobi $β$-Ensembles and prove Gaussian fluctuations for the number of points in one or more intervals in the macroscopic scaling limit.

math.PR↗
arXiv2006

Eigenvalue Statistics for CMV Matrices: From Poisson to Clock via Circular Beta Ensembles

We study CMV matrices (a discrete one-dimensional Dirac-type operator) with random decaying coefficients. Under mild assumptions we identify the local eigenvalue statistics in the natural scaling limit. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process. For a certain critical rate of decay we obtain the circular beta ensembles of random matrix theory. The temperature β^{-1} appears as the square of the coupling constant.

math-ph↗
arXiv2006

Absence of reflection as a function of the coupling constant

We consider solutions of the one-dimensional equation $-u'' +(Q+ λV) u = 0$ where $Q: \mathbb{R} \to \mathbb{R}$ is locally integrable, $V : \mathbb{R} \to \mathbb{R}$ is integrable with supp$(V) \subset [0,1]$, and $λ\in \mathbb{R}$ is a coupling constant. Given a family of solutions $\{u_λ \}_{λ\in \mathbb{R}}$ which satisfy $u_λ(x) = u_0(x)$ for all $x<0$, we prove that the zeros of $b(λ) := W[u_0, u_λ]$, the Wronskian of $u_0$ and $u_λ$, form a discrete set unless $V \equiv 0$. Setting $Q(x) := -E$, one sees that a particular consequence of this result may be stated as: if the fixed energy scattering experiment $-u'' + λV u = Eu$ gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then $V \equiv 0$.

math-ph↗
arXiv2005

Schroedinger Operators With Few Bound States

We show that whole-line Schrödinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential spectrum exponentially fast. We also prove the following result for one- and two-dimensional Schrödinger operators, $H$, with bounded positive ground states: Given a potential $V$, if both $H\pm V$ are bounded from below by the ground-state energy of $H$, then $V\equiv 0$.

math-ph↗