Construction of two-bubble solutions for the energy-critical NLS

Abstract

We construct pure two-bubbles for the energy-critical focusing nonlinear Schrödinger equation in space dimension $N\ge 7$. The constructed solution is global in (at least) one time direction and approaches a superposition of two stationary states both centered at the origin, with the ratio of their length scales converging to $0$. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $|t{|}^{-\frac{2}{N-6}}$. The phases of the two bubbles form the right angle.

33 pages

References in corpus (5)

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• Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions
• Nonexistence of radial two-bubbles with opposite signs for the energy-critical wave equation
• Global dynamics above the ground state for the energy-critical Schrodinger equation with radial data
• The $L^2$ weak sequential convergence of radial mass critical NLS solutions with mass above the ground state

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• On pseudoconformal blow-up solutions to the self-dual Chern-Simons-Schrödinger equation: existence, uniqueness, and instability
• Uniqueness of two-bubble wave maps
• On classification of global dynamics for energy-critical equivariant harmonic map heat flows and radial nonlinear heat equation
• Existence and stability of infinite time bubble towers in the energy critical heat equation
• On the multi-bubble blow-up solutions to rough nonlinear Schrödinger equations
• Bubble towers in the ancient solution of energy-critical heat equation