arXiv subjects

Anudeep K. Arora

Publications and source records attributed to Anudeep K. Arora.

4 recordsLinked to original sources

arXiv2022

Threshold solutions for the Hartree equation

We consider the focusing $5$d Hartree equation, which is $L^2$-supercritical, with finite energy initial data, and investigate the solutions at the mass-energy threshold. We establish the existence of special solutions following the work of Duyckaerts-Roudenko [11] for the $3$d focusing cubic nonlinear Schrödinger equation (NLS). In particular, apart from the ground state solution $Q$, which is global but non-scattering, there exist special solutions $Q^+$ and $Q^-$, which in one time direction approach $Q$ exponentially, and in the other time direction $Q^+$ blows up in finite time and $Q^-$ exists for all time, exhibiting scattering behavior. We then characterize all radial threshold solutions either as scattering and blow up solutions in both time directions (similar to the solutions under the mass-energy threshold, see Arora-Roudenko [3]), or as the special solutions described above. To obtain the existence and classification result, in this paper we perform a thorough and meticulous investigation of the spectral properties of the linearized operator associated to the Hartree equation.

math.AP↗

arXiv2021

Self-Bound vortex states in nonlinear Schrödinger equations with LHY correction

We study the cubic-quartic nonlinear Schrödinger equation (NLS) in two and three spatial dimension. This equation arises in the mean-field description of Bose-Einstein condensates with Lee-Huang-Yang correction. We first prove global existence of solutions in natural energy spaces which allow for the description of self-bound quantum droplets with vorticity. Existence of such droplets, described as central vortex states in 2D and 3D, is then proved using an approach via constrained energy minimizers. A natural connection to the NLS with repulsive inverse-square potential in 2D arises, leading to an orbital stability result under the corresponding flow.

math.AP↗

arXiv2021

Well-posedness in weighted spaces for the generalized Hartree equation with $p<2$

We investigate the well-posedness in the generalized Hartree equation $iu_t + Δu + (|x|^{-(N-γ)} \ast |u|^p)|u|^{p-2}u=0$, $x \in \mathbb{R}^N$, $0<γ<N$, for low powers of nonlinearity, $p<2$. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin [6]. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the $L^2$-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.

math.AP↗

arXiv2019

On well-posedness and blow-up in the generalized Hartree equation

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential $iu_t + Δu + (|x|^{-b} \ast |u|^p)|u|^{p-2}u=0, x \in \mathbb{R}^N$.We establish the local well-posedness at the non-conserved critical regularity $\dot{H}^{s_c}$ for $s_c \geq 0$, which also includes the energy-supercritical regime $s_c>1$ (thus, complementing the work in [3], where the authors obtained the $H^1$ well-posedness in the intercritical regime together with classification of solutions under the mass-energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). Both of these results hold regardless of the criticality of the equation. In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass-energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.

math.AP↗