[Paper Review] Scattering for non-radial 3D NLS with combined nonlinearities

Jacopo Bellazzini, Van Duong Dinh+1 more|arXiv (Cornell University)|Sep 4, 2022

Advanced Mathematical Physics Problems1 citations

TL;DR

This paper establishes energy scattering for non-radial solutions to the 3D nonlinear Schrödinger equation with competing focusing-defocusing nonlinearities (|u|^{q-1}u - |u|^{p-1}u) below the ground state energy, using a novel combination of interaction Morawetz estimates and a new bound for the Pohozaev functional to overcome the lack of monotonicity. The method avoids concentration-compactness and rigidity, providing a new scattering criterion for non-radial, intercritical NLS with combined nonlinearities in 3D.

ABSTRACT

We give a new proof of the scattering below the ground state energy level for a class of nonlinear Schr\"odinger equations (NLS) with mass-energy intercritical competing nonlinearities. Specifically, the NLS has a focusing leading order nonlinearity with a defocusing perturbation. Our strategy combines interaction Morawetz estimates \`a la Dodson-Murphy and a new crucial bound for the Pohozaev functional of localized functions, which is essential to overcome the lack of a monotonicity condition. Furthermore, we give the rate of blow-up for symmetric solutions.

Motivation & Objective

• Address the open problem of non-radial scattering for 3D NLS with combined focusing-defocusing nonlinearities below the ground state energy.
• Develop a scattering criterion that avoids the concentration-compactness and rigidity method, which typically requires radial symmetry.
• Overcome the lack of monotonicity in the Pohozaev functional for non-radial solutions by establishing a new localized bound.
• Provide a rate of blow-up for symmetric solutions in the energy-subcritical regime.
• Extend the applicability of interaction Morawetz estimates to intercritical NLS with competing nonlinearities.

Proposed method

• Adapt interaction Morawetz estimates in the style of Dodson and Murphy to control the growth of solutions in the non-radial setting.
• Introduce a new bound for the Pohozaev functional applied to localized functions, which replaces the need for monotonicity in the energy space.
• Use Strichartz estimates with mixed-norm spaces (L^{m_i}_t(L^{b_i}_x) and L^{a_j}_t(L^{b_j}_x)) to control the nonlinear terms.
• Apply a small data scattering argument via a contraction mapping in a complete metric space of functions with bounded Strichartz and Sobolev norms.
• Establish a scattering criterion based on the decay of the linear propagator in mixed-norm spaces, ensuring convergence of the nonlinear flow.
• Employ persistence of regularity and Duhamel's formula to propagate regularity and control the nonlinear evolution.

Experimental results

Research questions

• RQ1 Can energy scattering be established for non-radial 3D NLS with competing nonlinearities (focusing q-power and defocusing p-power) below the ground state energy without using the concentration-compactness and rigidity method?
• RQ2 How can the lack of monotonicity in the Pohozaev functional for non-radial solutions be overcome in the context of scattering theory?
• RQ3 What is the rate of blow-up for symmetric solutions in the 3D NLS with combined nonlinearities when the energy is above the ground state threshold?
• RQ4 Can interaction Morawetz estimates be effectively extended to intercritical NLS with two competing nonlinearities?
• RQ5 Is there a viable small data scattering criterion in mixed-norm Strichartz spaces for this class of NLS equations?

Key findings

• The paper establishes scattering for non-radial solutions to the 3D NLS with 7/3 < q < p < 5 and combined focusing-defocusing nonlinearities below the ground state energy, using a new method that avoids the concentration-compactness and rigidity framework.
• A new bound for the Pohozaev functional of localized functions is derived, which is crucial for overcoming the absence of monotonicity in the non-radial case.
• Interaction Morawetz estimates are successfully applied in the non-radial setting via a refined analysis of the nonlinear interaction term.
• Small data scattering is proven in mixed-norm Strichartz spaces, with the solution norm controlled by the linear propagator norm.
• The rate of blow-up for symmetric solutions is explicitly given, providing a quantitative description of finite-time blow-up in the energy-supercritical regime.
• Global existence and scattering are established under the condition that the linear propagator norm in mixed-norm spaces is sufficiently small, providing a new scattering criterion.