# Well-posedness of Weak and Strong Solutions to the Kinetic Cucker-Smale Model

## TL;DR

This paper establishes the well-posedness of weak and strong solutions for the kinetic Cucker-Smale model, including cases with noise, and rigorously justifies the vanishing noise limit using weighted energy estimates.

## Contribution

It proves well-posedness in Sobolev spaces for both deterministic and noisy kinetic Cucker-Smale models, introducing weighted Hilbert spaces and energy estimates for the analysis.

## Key findings

- Well-posedness of solutions in Sobolev spaces.
- Extension to models with noise using weighted Hilbert spaces.
- Rigorous justification of the vanishing noise limit.

## Abstract

We first prove the well-posedness of weak and strong solutions to the kinetic Cucker-Smale model in the Sobolev space with the initial data having compact velocity support. Then we study the kinetic Cucker-Smale model with noise. By introducing two weighted Hilbert spaces, we successfully establish the well-posedness of weak and strong solutions, respectively. Our proof is based on weighted energy estimates. Finally, we rigorously justify the vanishing noise limit.