Influence of flux limitation on large time behavior in a three-dimensional chemotaxis-Stokes system modeling coral fertilization

TL;DR

This paper proves that in a 3D chemotaxis-Stokes system modeling coral fertilization, flux limitation ensures global boundedness and stabilization of solutions over time.

Contribution

It establishes global existence, boundedness, and asymptotic stabilization of solutions under flux limitation conditions, using a simplified approach compared to previous studies.

Findings

Solutions are globally bounded for all time.

System stabilizes to equilibrium states as time approaches infinity.

Flux limitation parameter >0 is crucial for stability.

Abstract

In this paper, we consider the following system $$\left\{\begin{array}{ll} n_t+u\cdot\nabla n&=\Delta n-\nabla\cdot(n\mathcal{S}(|\nabla c|^2)\nabla c)-nm,\\ c_t+u\cdot\nabla c&=\Delta c-c+m,\\ m_t+u\cdot\nabla m&=\Delta m-mn,\\ u_t&=\Delta u+\nabla P+(n+m)\nabla\Phi,\qquad \nabla\cdot u=0 \end{array}\right.$$ which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where $\mathcal{S}$ is a given function fulfilling $$|\mathcal{S}(\sigma)|\leq K_{\mathcal{S}}(1+\sigma)^{-\frac{\theta}{2}},\qquad \sigma\geq 0$$ with some $K_{\mathcal{S}}>0.$ Based on conditional estimates of the quantity $c$ and the gradients thereof, a relatively compressed argument as compared to that proceeding in related precedents shows that if $$\theta>0,$$ then for any initial data with proper regularity an associated initial-boundary problem under no-flux/no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded, and which also enjoys the stabilization features in the sense that $$\|n(\cdot,t)-n_{\infty}\|_{L^{\infty}(\Omega)}+\|c(\cdot,t)-m_{\infty}\|_{W^{1,\infty}(\Omega)} +\|m(\cdot,t)-m_{\infty}\|_{W^{1,\infty}(\Omega)}+\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\rightarrow0 \quad\textrm{as}~t\rightarrow \infty$$ with $n_{\infty}:=\frac{1}{|\Omega|}\left\{\int_{\Omega}n_0-\int_{\Omega}m_0\right\}_{+}$ and $m_{\infty}:=\frac{1}{|\Omega|}\left\{\int_{\Omega}m_0-\int_{\Omega}n_0\right\}_{+}.$