On the $8\pi$-Critical-Mass Threshold of a Patlak--Keller--Segel--Navier--Stokes System

Partial regularity of suitable weak solution to a three-dimensional fractional parabolic-elliptic chemotaxis-Navier–Stokes system

“…Kozono et al [24] deduced a global mild solution to the two-dimensional Keller-Segel-Navier-Stokes system for small initial data and proved a blow-up criterion for an appropriately large initial mass. Recently, Gong and He [14] proposed a parabolic-elliptic chemotaxis-Navier-Stokes system driven by friction force n∇c:…”

“…Since the global existence of the solution to the chemotaxis-Navier-Stokes system (1.2) depends on the size of the initial mass (see, e.g. [14]), to eliminate the limitation of the smallness condition on the initial data, we consider a repulsive chemotactic signal +∇ • (n∇c) to obtain the global existence of weak solution with large initial data. Furthermore, we investigate a Navier-Stokes equation driven by friction force to discuss the regularity of the solution.…”

An interpolation inequality involving LlogL$L\log L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes system

“…Since the experimental and modeling-oriented works by Goldstein et al [8,35], problems of this and some closely related forms have received considerable interest in the mathematical literature [4,10,11,14,21,25,39,42]. Especially in settings involving chemoattractant production through cells, as underlying (1.13), a predominant focus has been on the question how far fluid interaction can influence the explosion-supporting properties known as a core characteristic of classical two-component Keller-Segel systems ( [18,38]; see [23] for a sufficient criterion for the occurrence of blow-up in (1.13) under additional assumptions on fluid regularity).…”

Well‐posedness and time decay of fractional Keller–Segel–Navier‐Stokes equations in homogeneous Besov spaces

“…where 𝑛, 𝑣 denote the density of cells and the chemical density, the vector function 𝑢 stands for the fluid velocity field, 𝑃 is a scalar function that represents the pressure of the fluid. If 𝛼 = 2, the system (1.1) corresponds to the classical Keller-Segel-Navier-Stokes equations, which is initially proposed by Gong and He in [15]. Fractional Keller-Segel-Navier-Stokes equations (1.1) describe the time evolution of cells transported by ambient fluid flow 𝑢 under Lévy diffusion and chemotactic attraction and the fluid motion subject to forcing induced by the cells.…”

“…The second equation of Equation (1.1) models the fact that the cells generate chemical substance. Gong and He [15] explained the meaning of the coupling force 𝑛∇𝑣 in the Navier-Stokes equation, that is, in order to make the cells move without acceleration, the fluid exerts frictional force on the moving cells, so reaction forces act on the fluid.…”