Global solutions of aggregation equations and other flows with random diffusion.

Area of Science:

• Partial Differential Equations
• Stochastic Analysis
• Mathematical Physics

Background:

• Parabolic-elliptic models like Patlak-Keller-Segel exhibit thresholds for solution existence.
• Absence of diffusion typically leads to local-in-time solutions for smooth initial data.
• Stochasticity is explored as a means to overcome limitations of deterministic models.

Purpose of the Study:

• Investigate if random diffusion restores global existence in active scalar equations.
• Extend findings from inviscid Surface Quasi-Geostrophic (SQG) equation to broader classes of models.
• Analyze the impact of noise on equations with potentially singular velocity fields.

Main Methods:

• Employing Gevrey-type Fourier-Lebesgue spaces for solution analysis.
• Utilizing techniques from stochastic partial differential equations.
• Building upon prior work on random diffusion in inviscid SQG.

Main Results:

• Demonstrated global existence of solutions for a wide class of active scalar equations.
• Established results with quantifiable high probability.
• Showed that random diffusion can prevent finite-time blow-up.

Conclusions:

• Random diffusion is a viable mechanism for ensuring global existence in complex fluid dynamics and aggregation models.
• The findings generalize previous results to higher dimensions and more singular cases.
• Stochasticity offers a powerful tool for analyzing the long-term behavior of nonlinear PDEs.