Global existence, blow-up and stability for a stochastic transport equation with non-local velocity. (English) Zbl 1494.60070
Summary: In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish a local-in-time theory, i.e., existence, uniqueness and blow-up criterion for pathwise solutions in Sobolev spaces \(H^s\) with \(s > 3\). Thereafter, we give a picture of the long time behavior of the solutions based on the type of noise we consider. On one hand, we identify a family of noises such that blow-up can be prevented with probability 1, guaranteeing the existence and uniqueness of global solutions almost surely. On the other hand, in the particular linear noise case, we show that singularities occur in finite time with positive probability, and we derive lower bounds of these probabilities. To conclude, we introduce the notion of stability of exiting times and show that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35Q51 | Soliton equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B44 | Blow-up in context of PDEs |
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
Keywords:
stochastic evolution equations; pathwise solution; blow-up criterion; noise prevents blow-up; weak instabilityReferences:
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