Boundedness in a two‐dimensional self‐consistent chemotaxis–Navier–Stokes system with nonlinear diffusion

Abstract

This paper deals with the self‐consistent chemotaxis–Navier–Stokes system with porous medium diffusion {left left leftarrayarraynt+u·∇n=Δnm−∇·(n∇c)+∇·(n∇ϕ),arrayx∈Ω,t>0,arrayarrayct+u·∇c=Δc−nc,arrayx∈Ω,t>0,arrayarrayut+(u·∇)u+∇P=Δu−n∇ϕ+n∇c,arrayx∈Ω,t>0,arrayarray∇·u=0,arrayx∈Ω,t>0$$ \left\{\begin{array}{lll}& {n}_t+u\cdotp \nabla n=\Delta {n}^m-\nabla \cdotp \left(n\nabla c\right)+\nabla \cdotp \left(n\nabla \phi \right),\kern2em & x\in \Omega, t>0,\\ {}& {c}_t+u\cdotp \nabla c=\Delta c- nc,\kern2em & x\in \Omega, t>0,\\ {}& {u}_t+\left(u\cdotp \nabla \right)u+\nabla P=\Delta u-n\nabla \phi +n\nabla c,\kern2em & x\in \Omega, t>0,\\ {}& \nabla \cdotp u=0,\kern2em & x\in \Omega, t>0\end{array}\right. $$ in the two‐dimensional smoothly bounded domain, where the gravitational potential ϕ∈W1,∞false(normalΩfalse)$$ \phi \in {W}^{1,\infty}\left(\Omega \right) $$ is given function. Systems of this type involve the effect of gravity on cells ( ∇·false(n∇ϕfalse)$$ \nabla \cdotp \left(n\nabla \phi \right) $$) and the effect of the chemotactic force on fluid ( n∇c$$ n\nabla c $$), which leads to a stronger coupling than usual chemotaxis‐fluid system that proposed by Tuval et al. (Proc. Nat. Acad. Sci. 102 (2005), 2277‐2282) and studied widely in the most existing literatures. Under no‐flux boundary conditions for n$$ n $$ and c$$ c $$ and no‐slip boundary condition for u$$ u $$, it is proved that the system admits at least one global weak solution which is uniformly bounded if m>1$$ m>1 $$. In particular, our result improves that established by Wang (Discr. Cont. Dyn. Syst. S 13 (2020), 329‐349), which asserts the global existence of weak solutions under the constraint m>1$$ m>1 $$.

A novel growing wavelet neural network algorithm for solving chemotaxis systems with blow‐up

“…Theorem 4.2. Assume that the solution v to (10) belongs to C 1 . Then for any 𝛿 > 0, there exists Θ and a sequence of the GWNN solutions, denoted by u = u(t, x; Θ), such that when the number of parameters is sufficiently large,…”

“…In [7], the authors review the available variations of chemotaxis models and the numerical methods for solving such problems during the last few decades. Although there are some interesting theoretical studies, such as the existence and boundedness of solutions for chemotaxis models [8][9][10], finding an analytical solution of chemotaxis problems is difficult or impossible in most cases. So, one should obtain some approximate solutions.…”

“…Next, we prove the stability of GWNN which, along with its consistency proved in Theorem 4.2, implies convergence. Let u be the approximate solution of the proposed scheme and denote their difference with the scheme's exact solution to (10), v, by 𝜺 = u − v. Clearly, we have…”