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

Abstract

This paper deals with a two‐competition‐species chemotaxis‐Navier‐Stokes system with two different consumed signals false(n1false)t+boldu·∇n1=d1normalΔn1−χ1∇·false(n1∇cfalse)+μ1n1false(1−n1−a1n2false),in3.0235ptnormalΩ×false(0,∞false),ct+boldu·∇c=d2normalΔc−α1cn2,in3.0235ptnormalΩ×false(0,∞false),false(n2false)t+boldu·∇n2=d3normalΔn2−χ2∇·false(n2∇vfalse)+μ2n2false(1−a2n1−n2false),in3.0235ptnormalΩ×false(0,∞false),vt+boldu·∇v=d4normalΔv−α2vn1,in3.0235ptnormalΩ×false(0,∞false),boldut+false(boldu·∇false)boldu=normalΔboldu+∇P+false(β1n1+β2n2false)∇ϕ,in3.0235ptnormalΩ×false(0,∞false),∇·boldu=0,in3.0235ptnormalΩ×false(0,∞false), in a smooth bounded domain normalΩ⊂double-struckR3 under zero Neumann boundary conditions for n1,n2,c,v, and homogeneous Dirichlet boundary condition for boldu, where the parameters di ( i=1,2,3,4) and χj,μj,aj,αj,βj ( j=1,2) are positive. This system describes the evolution of two‐competing species which react on two different chemical signals in a liquid surrounding environment. Recently, the boundedness and stabilization of classical solutions to the above system under two‐dimensional case have been derived in the previous works. However, to the best of our knowledge, the well‐posedness problem of solutions for the above system is still open in the three dimensional setting, because of the difficulties in the Navier‐Stokes system. The aim of this paper is to construct global weak solutions and show that after some waiting time, these weak solutions become eventually smooth.

Theoretical and numerical analysis for an interspecies competition model with chemoattraction-consumption in fluids

“…In this paper we deal with the existence of global weak solutions for (1.1), as well as regularity properties. Our notion of weak solution is weaker than the one in [41] in the sense that our definition establishes that the equation for the chemoattractant is satisfied a.e. in Ω × (0, T ), and the class of initial data (and φ) is larger than the one in [41] (see Definition 2.2 below).…”

“…Our notion of weak solution is weaker than the one in [41] in the sense that our definition establishes that the equation for the chemoattractant is satisfied a.e. in Ω × (0, T ), and the class of initial data (and φ) is larger than the one in [41] (see Definition 2.2 below). In order to prove the existence of weak solutions, we consider a family of regular solutions for a suitable regularized problem, which differs from that considered in [41] (see system (2.7)-(2.8) below).…”

“…From a theoretical point of view, model (1.1) has been studied in two and three dimensional settings (e.g. [3,22,24,41]). Global existence, boundedness and stabilization of solutions in the twodimensional case were studied in [22]; while, the three-dimensional case with k = 0 (that is, considering the Stokes system instead Navier-Stokes model), was studied in [3].…”

“…In [24], the authors analyzed the convergence and derive explicit rates of convergence for any supposedly given global bounded classical solution in d-dimensional domains (d = 2, 3) for different choices of parameters a i (i = 1, 2). The global existence and eventual smoothness of weak solutions in the three-dimensional case (with k = 0) were studied in [41]. More exactly, for initial data satisfying…”

“…in Ω × (0, T ), and the class of initial data (and φ) is larger than the one in [41] (see Definition 2.2 below). In order to prove the existence of weak solutions, we consider a family of regular solutions for a suitable regularized problem, which differs from that considered in [41] (see system (2.7)-(2.8) below). Indeed, we introduce a decoupling through an auxiliary elliptic problem which allows to gain regularity for the chemical equation as well as to obtain an energy-type inequality after testing and combining conveniently the densities and the concentration equations (see details in Section 2).…”