The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data
| Autor: | Carlos E. Kenig, Dana Mendelson |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Euclidean space
General Mathematics Operator (physics) 010102 general mathematics 01 natural sciences Quintic function 010101 applied mathematics Sobolev space Dispersive partial differential equation symbols.namesake Fourier transform symbols Soliton 0101 mathematics Energy (signal processing) Mathematical physics Mathematics |
| Zdroj: | International Mathematics Research Notices. 2021:14508-14615 |
| ISSN: | 1687-0247 1073-7928 |
| Popis: | We consider the focusing energy-critical quintic nonlinear wave equation in 3D Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$, for any $s> 1/2$. By randomizing radial initial data in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$ for $s> 5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton that give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the 1st long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime. |
| Databáze: | OpenAIRE |
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