Research Projects

This project focuses on the Keller–Segel–Navier–Stokes system, specifically the classical Keller–Segel model coupled with the Navier–Stokes equations and a logistic source term of the form (\rho n - \mu n^{\alpha}) in the first equation. Existing literature establishes the existence of global generalized solutions when \alpha =2. However, our interest lies in investigating whether weaker damping (i.e., \alpha \in (1,2) ) is still sufficient to guarantee the existence of such solutions. In addition, we are also interested in analyzing the blow-up behavior of solutions in the absence of the logistic source term. Studying the blow-up of solutions in chemotaxis–fluid systems is crucial for understanding the formation of singularities, which correspond to extreme aggregation of cells in some region in the biological and physical systems. Studying these models has several applications, such as white blood cells (leukocytes) migrate toward infection sites in the bloodstream by detecting chemical signals from pathogens. KS-NS helps in simulating and understanding this behaviour in the presence of blood flow. Tumour cells migrate via chemotaxis through body fluids toward favourable environments. Modelling this can help in predicting metastasis patterns and developing targeted therapies. Microswimmers or bio-inspired robots that mimic bacteria and navigate the human body to deliver drugs. KS-NS models help simulate their movement in complex fluids like mucus, blood, or tissue fluids, improving design and control strategies. In general, the Keller-Segel-Navier-Stokes system has wide-ranging practical applications in fields such as biomedicine, environmental science, marine ecology, and engineering. It provides a powerful framework for simulating and understanding the complex interplay between chemical signalling and fluid motion in real-world systems.