Random perturbations for the chemotaxis-fluid model with fractional dissipation: Global pathwise weak solutions

TL;DR

This paper proves the existence and uniqueness of global solutions for a stochastically perturbed chemotaxis-fluid model with fractional dissipation, introducing a new approximation method and deriving novel inequalities.

Contribution

It introduces a new three-layer approximation system for the chemotaxis-fluid model with fractional dissipation and establishes global solutions with novel entropy-energy inequalities.

Findings

Existence of global probabilistically strong solutions.

Uniqueness of pathwise solutions using Littlewood-Paley theory.

Development of new interpolation inequalities in Besov spaces.

Abstract

This paper considers a stochastically perturbed Keller-Segel-Navier-Stokes (KS-SNS) system arising from the biomathematics in two dimensions, where the diffusion of fluid is expressed by a fractional Laplacian with an exponent in $[1/2,1]$. Our main result demonstrates that, under appropriate assumptions, the Cauchy problem of the KS-SNS system has a unique global probabilistically strong and analytically weak solution, which also confirms that the quadratic logistic source authentically contributes to the global existence of solutions. First, a three-layer approximate system is introduced, this system seems to be new in the studying of chemotaxis-fluid model and it enables one to construct approximate solutions in regular Hilbert spaces $H^s(\mathbb{R}^2)$. Second, to accomplish the convergence progressively, a series of crucial entropy-energy inequalities for the approximations are derived, whose derivation strongly depends on the fine structure of the system and requires a novel strategy to adapt to the appearance of fractional dissipation and the unboundedness of domain. Third, based on these uniform bounds, we establish the existence of global martingale weak solutions by virtue of a stochastic compactness method. Finally, by applying the Littlewood-Paley theory, we show that the pathwise uniqueness holds for these martingale solution. As a by-product, a few interpolation inequalities in Besov spaces are obtained, which seems to be new and may have their own interest.