We consider the focusing energycritical quintic nonlinear wave equation in 3D Euclidean space. It is known that this equation admits a oneparameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s1}_x({{\mathbb{R}}}^3)$, for any $s> 1/2$. By randomizing radial initial data in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s1}_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 forwardintime 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 longtime random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.
Aggregation equations, such as the parabolicelliptic Patlak–Keller–Segel model, are known to have an optimal threshold for global existence versus finitetime blowup. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. (Int Math Res Not IMRN 23:9370–9385, 2020) showing that, with high probability, the inviscid SQG equation with random diffusion has global classical solutions, we investigate whether suitable random diffusion can restore global existence for a large class of active scalar equations in arbitrary dimension with possibly singular velocity fields. This class includes Hamiltonian flows, such as the SQG equation and its generalizations, and gradient flows, such as those arising in aggregation models. For this class, we show global existence of solutions in Gevreytype Fourier–Lebesgue spaces with quantifiable high probability.
more » « less NSFPAR ID:
 10377991
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Probability Theory and Related Fields
 Volume:
 185
 Issue:
 34
 ISSN:
 01788051
 Page Range / eLocation ID:
 p. 12191262
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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