Stabilization in a two‐dimensional fractional chemotaxis‐Navier–Stokes system with logistic source.

This paper investigates a chemotaxis‐Navier–Stokes system with logistic source and a fractional diffusion of order α∈(0,1)$$ \alpha \in \left(0,1\right) $$nt+u·∇n=−(−Δ)αn−χ∇·(n∇c)+an−bn2,ct+u·∇c=Δc+n−c,ut+(u·∇)u=Δu−∇P+n∇ϕ+f,∇·u=0,$$ \left\{\begin{array}{l}{n}_t+u\cdotp \nabla n=-{\left(-\Delta \right)}^{\alpha }n-\chi \nabla \cdotp \left(n\nabla c\right)+ an-b{n}^2,\\ {}{c}_t+u\cdotp \nabla c=\Delta c+n-c,\\ {}{u}_t+\left(u\cdotp \nabla \right)u=\Delta u-\nabla P+n\nabla \phi +f,\nabla \cdotp u=0,\end{array}\right. $$on the two‐dimensional periodic torus 핋2. It is proved that if α∈12,1,a,χ≥0,b>0,f∈C1핋2×[0,∞)∩L∞핋2×(0,∞)∩L2(0,∞);L2핋2 and the initial data (n0,c0,u0)$$ \left({n}_0,{c}_0,{u}_0\right) $$ satisfy some regular conditions, the considered problem admits a global classical solution. Through constructing a new Lyapunov functional different from the previous literature, we can obtain the following asymptotic stabilization: If a=0$$ a=0 $$, the solution components n$$ n $$ and c$$ c $$ satisfy n→0andc→0inL∞핋2ast→∞.If a>0$$ a>0 $$ and b>B04$$ b>\frac{B_0}{4} $$, where B0$$ {B}_0 $$ is a constant determined in (6.24), the solution components n$$ n $$ and c$$ c $$ satisfy n→abandc→abinL∞핋2ast→∞.In either case, the solution component u$$ u $$ satisfies u→0inL∞핋2ast→∞. [ABSTRACT FROM AUTHOR]