Rosenzweig Matthew, Staffilani Gigliola
Massachusetts Institute of Technology, Cambridge, USA.
Probab Theory Relat Fields. 2023;185(3-4):1219-1262. doi: 10.1007/s00440-022-01171-8. Epub 2022 Oct 31.
Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence versus finite-time blow-up. 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 Gevrey-type Fourier-Lebesgue spaces with quantifiable high probability.
诸如抛物 - 椭圆型Patlak - Keller - Segel模型之类的聚集方程，已知对于全局存在与有限时间爆破存在一个最优阈值。特别地，如果没有扩散，那么所有具有有限二阶矩的光滑解只能在时间上局部存在。然而，可以问是否通过给方程添加合适的噪声能够恢复全局存在性，使得动力学现在是随机的。受Buckmaster等人（《国际数学研究通讯》2020年第23期：9370 - 9385）工作的启发，他们表明具有随机扩散的无粘SQG方程以高概率具有全局经典解，我们研究合适的随机扩散是否能恢复任意维数下一大类具有可能奇异速度场的有源标量方程的全局存在性。这类方程包括哈密顿流，如SQG方程及其推广形式，以及梯度流，如聚集模型中出现的那些。对于这类方程，我们证明在具有可量化高概率的Gevrey型傅里叶 - 勒贝格空间中解的全局存在性。
