Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier–Stokes system with competitive kinetics

Abstract

This paper is concerned with the two-species chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kineticsunder homogeneous Neumann boundary conditions and initial conditions, where Ω is a bounded domain in R 3 with smooth boundary. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier-Stokes system, we can not expect existence of classical solutions to the above system. The purpose of this paper is to obtain global existence of weak solutions to the above system, and their eventual smoothness and stabilization.2010Mathematics Subject Classification. Primary: 35B40; Secondary: 35K55, 35Q30, 92C17.

Global existence and asymptotic behavior of classical solutions for a 3D two‐species chemotaxis‐Stokes system with competitive kinetics

“…However, the problem , which is the combination of chemotaxis‐fluid systems and chemotaxis‐competition systems, had not been studied. Recently, global existence, boundedness of classical solutions, and their asymptotic behavior were shown only in the 2‐dimensional setting . On the other hand, the 3‐dimensional setting is a more realistic problem; however, in the 3‐dimensional “chemotaxis‐Navier‐Stokes” setting, difficulties of the Navier‐Stokes equation strongly affect; all works that dealt with the 3‐dimensional chemotaxis‐Navier‐Stokes system could establish only “weak solutions.”…”

“…Thus, we only show convergence in the cases that a 1 , a 2 <1, and a 2 <1≤ a 1 . The same arguments as in the proofs of Hirata et al, Theorem 1.2 yield the following 2 lemmas; therefore, we write brief proofs.…”

Singular sensitivity in a Keller–Segel-fluid system

“…The most advanced developments dealing with the full chemotaxis-Navier-Stokes system in three-dimensional domains are constituted by the construction of global weak solutions in [43] and the proof of their eventual regularization and convergence in [44]. Also model variants involving logistic source terms ( [5,20]), several species ( [13,6]), rotational sensitivity functions ( [42,7,24]) and/or nonlinear diffusion ( [47,15]) have been studied and the interested reader can find additional information and references there or in [1, Section 4.1]. Chemotaxis-fluid systems describing a signal being produced by the cells themselves, as with the Keller-Segel type choice g(n, c) = +n − c, up to now have been studied in much fewer works.…”

Global weak solutions and eventual smoothness in a 3D two‐competing‐species chemotaxis‐Navier‐Stokes system with two consumed signals

“…When , system is called as two‐species chemotaxis‐Navier‐Stokes model. In the two‐dimensional setting, Hirata et al derived global existence, boundedness and stabilization of classical solutions for . Moreover, global existence of weak solutions, eventual smoothness, and stabilization are studied under the three‐dimensional case in Hirata et al When , system is called as two‐species chemotaxis‐Stokes model.…”

“…In the two‐dimensional setting, Hirata et al derived global existence, boundedness and stabilization of classical solutions for . Moreover, global existence of weak solutions, eventual smoothness, and stabilization are studied under the three‐dimensional case in Hirata et al When , system is called as two‐species chemotaxis‐Stokes model. In the three‐dimensional setting, Cao et al studied the global existence and asymptotic behavior of classical solutions for system provided that ( ) is sufficiently large.…”