On the $8π$-critical mass threshold of a Patlak-Keller-Segel-Navier-Stokes system

Abstract

In this paper, we proposed a coupled Patlak-Keller-Segel-Navier-Stokes system, which has dissipative free energy. On the plane R 2 , we proved that if the total mass of the cells is strictly less than 8π, then classical solutions exist for any finite time and their H s -Sobolev norms are almost uniformly bounded in time. On the torus T 2 , we proved that under the 8π subcritical mass constraint, the solutions are uniformly bounded in time.

Global existence of free-energy solutions to the 2D Patlak--Keller--Segel--Navier--Stokes system with critical and subcritical mass

“…In this paper, we consider the Cauchy problem of the following 2D Patlak-Keller-Segel-Navier-Stokes system (PKS-NS system) [43]…”

“…where the first and the second term are the entropy and a potential energy of the density n, respectively, and the third term is the kinetic energy of the velocity field u. As shown in [43,Lemma 1.1], the free energy is dissipative along the dynamics (1.1).…”

“…In [43], the local and global existence of solutions to the Patlak-Keller-Segel-Navier-Stokes system (1.1) are established in the Sobolev space H s , s ≥ 2, when the initial mass is strictly less than 8π. The H s bound of their solutions grows exponentially in time.…”

“…The sign of the last term in (1.7) is unknown. It is unclear how to control the integral unless the solution is radially symmetric [43,Corollary 1], in which case the integral vanishes (the system gets decoupled). The lack of control over the second moment is the main obstacle to bound solutions uniform in time.…”

“…The regularity of u allows us to control all additional terms arising from the coupling when deriving the a priori entropy estimate of the advection-Patlak-Keller-Segel system (1.1) 1 -(1.1) 2 in Section 4. For the subcritical regime M < 8π, we improve the result of [43] by reducing the regularity assumption on the initial data (n 0 , u 0 ). In fact, we show that solutions with the initial conditions (1.3) and (1.4) become H s in short time, thus the global-in-time existence follows directly from [43].…”