# 2D Stochastic Chemotaxis-Navier-Stokes System

## TL;DR

This paper proves the existence and uniqueness of solutions for a complex 2D stochastic chemotaxis and fluid dynamics system, advancing mathematical understanding of such coupled stochastic PDEs.

## Contribution

It introduces a novel approach to establish both mild and weak solutions for the 2D stochastic chemotaxis-Navier-Stokes system, including fixed point and martingale methods.

## Key findings

- Existence of mild/variational solutions via fixed point theorem.
- Existence of martingale weak solutions.
- Pathwise uniqueness of solutions.

## Abstract

In this paper, we establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through a fixed point argument in a purposely constructed Banach space. To get the weak solution we first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution.