arXiv subjects

Peter Y. H. Pang

Publications and source records attributed to Peter Y. H. Pang.

6 recordsLinked to original sources

arXiv2025

Global classical solutions to a two-dimensional chemotaxis-fluid system involving signal-dependent degenerate diffusion

This paper is concerned with the two-dimensional chemotaxis-fluid model \begin{equation*} \begin{cases} n_t+u\cdot\nabla n=Δ(nϕ(v))+μn(1-n),\\ v_t+u\cdot\nabla v=Δv-nv,\\ u_t+ κ(u\cdot\nabla) u=Δu+n\nablaΦ-\nabla P, \quad\nabla\cdot u=0, \end{cases} \end{equation*} accounting for signal-dependent motilities of microbial populations interacting with an incompressible liquid through transport and buoyancy, where the suitably smooth function $ϕ$ satisfies $ϕ>0$ on $(0,\infty)$ with $ϕ(0)=0$ and $ϕ'(0)>0$, and the parameter $μ\geq 0$. For all reasonably regular initial data, if $μ=0$, the corresponding initial boundary value problem possesses global classical solutions with a smallness condition on $\int_Ωn_0$; whereas if $μ>0$, this problem possesses global bounded classical solutions, which can converge toward (1,0,0) as time tends to infinity when a certain small mass is imposed on the initial data $v_0$. These results extend recent results for the fluid-free system to one in a Navier-Stokes fluid environment.

math.AP↗

arXiv2025

Small-mass solutions in a two-dimensional logarithmic chemotaxis-Navier-Stokes system with indirect nutrient consumption

This paper is concerned with the singular chemotaxis-fluid system with indirect nutrient consumption: $ n_{t}+u\cdot\nabla n=Δn-\nabla\cdot(n S(x,n,v)\cdot \nabla v);\ v_{t}+u\cdot\nabla v=Δv-vw;\ w_{t}+u\cdot\nabla w=Δw-w+n;\ u_t+(u\cdot\nabla) u=Δu-\nabla P+n\nablaΦ;\ \nabla\cdot u=0\ $ in a smooth bounded domain $Ω\subset\mathbb{R}^2$ under no-flux/Neumann/Neumann/Dirichlet boundary conditions, where $Φ\in W^{2,\infty}(Ω)$, and $S: \overlineΩ\times [0,\infty) \times (0,\infty)\rightarrow\mathbb{R}^{2\times 2}$ is a suitably smooth function that satisfies $|S(x,n,v)|\leq S_0(v) /v $ for all $(x,n,v) \in Ω\times (0,\infty)^2$ with some nondecreasing $S_0: (0,\infty)\rightarrow(0,\infty)$. For all reasonably regular initial data with a smallness assumption merely involving the quantity $\int_Ωn_0$, it is shown that the problem possesses a globally bounded classical solution, which, inter alia, exponentially stabilizes toward the spatially homogeneous state $( \frac{1}{|Ω|}\int_Ωn_0,0,\frac{1}{|Ω|}\int_Ωn_0,0)$ with respect to the norm in $L^\infty(Ω)$. This rigorously confirms that, at least in the two-dimensional setting, in comparison to the direct mechanism of nutrient consumption, an indirect mechanism can induce much more regularity of solutions to the chemotaxis--fluid system even with a singular tensor-valued sensitivity.

math.AP↗

arXiv2020

Asymptotic profile of a two-dimensional chemotaxis--Navier--Stokes system with singular sensitivity and logistic source

The chemotaxis--Navier--Stokes system \begin{equation*}\label{0.1} \left\{\begin{array}{ll} n_t+u\cdot \nabla n=\triangle n-χ\nabla\cdotp \left(\displaystyle\frac n {c}\nabla c\right)+n(r-μn), c_t+u\cdot \nabla c=\triangle c-nc, u_t+ (u\cdot \nabla) u=Δu+\nabla P+n\nablaϕ, \nabla\cdot u=0, \end{array}\right. \end{equation*} is considered in a bounded smooth domain $Ω\subset \mathbb{R}^2$, where $ϕ\in W^{1,\infty}(Ω)$, $χ>0$, $r\in \mathbb{R}$ and $μ> 0$ are given parameters. It is shown that there exists a value $μ_*(Ω,χ, r)\geq 0$ such that whenever $ μ>μ_*(Ω,χ, r)$, the global-in-time classical solution to the system is uniformly bounded with respect to $x\in Ω$. Moreover, for the case $r>0$, $(n,c,\frac {|\nabla c|}c,u)$ converges to $(\frac r μ,0,0,0)$ in $L^\infty(Ω)\times L^\infty(Ω)\times L^p(Ω)\times L^\infty(Ω)$ for any $p>1$ exponentially as $t\rightarrow \infty$, while in the case $r=0$, $(n,c,\frac {|\nabla c|}c,u)$ converges to $(0,0,0,0)$ in $(L^\infty(Ω))^4$ algebraically. To the best of our knowledge, these results provide the first precise information on the asymptotic profile of solutions in two dimensions.

math.AP↗

arXiv2019

Global classical small-data solutions for a three-dimensional Keller--Segel--Navier--Stokes system modeling coral fertilization

We are concerned with the Keller--Segel--Navier--Stokes system \begin{equation*} \left\{ \begin{array}{ll} ρ_t+u\cdot\nablaρ=Δρ-\nabla\cdot(ρ\mathcal{S}(x,ρ,c)\nabla c)-ρm, &\!\! (x,t)\in Ω\times (0,T), \\ m_t+u\cdot\nabla m=Δm-ρm, &\!\! (x,t)\in Ω\times (0,T), \\ c_t+u\cdot\nabla c=Δc-c+m, & \!\! (x,t)\in Ω\times (0,T), \\ u_t+ (u\cdot \nabla) u=Δu-\nabla P+(ρ+m)\nablaϕ,\quad \nabla\cdot u=0, &\!\! (x,t)\in Ω\times (0,T) \end{array}\right. \end{equation*} subject to the boundary condition $(\nablaρ-ρ\mathcal{S}(x,ρ,c)\nabla c)\cdot ν\!\!=\!\nabla m\cdot ν=\nabla c\cdot ν=0, u=0$ in a bounded smooth domain $Ω\subset\mathbb R^3$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $\mathcal{S}\in C^2(\overlineΩ\times [0,\infty)^2)^{3\times 3}$ satisfies $|\mathcal{S}(x,ρ,c)|\leq C_S $ for some $C_S>0$, and the initial data satisfy certain smallness conditions.

math.AP↗

arXiv2019

Asymptotic behavior of solutions to a tumor angiogenesis model with chemotaxis--haptotaxis

This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain $Ω\subset \mathbb{R}^N$ ($N=1,2$): $$\label{0} \left\{\begin{array}{ll} p_t=Δp-\nabla\cdotp p(\displaystyle\frac α{1+c}\nabla c+ρ\nabla w)+λp(1-p),\,& x\in Ω, t>0, c_t=Δc-c-μpc,\, &x\in Ω, t>0,\\ w_t= γp(1-w),\,& x\in Ω, t>0, \end{array}\right. $$ where $α, ρ, λ, μ$ and $γ$ are positive parameters. For any reasonably regular initial data $(p_0, c_0, w_0)$, we prove the global boundedness ($L^\infty$-norm) of $p$ via an iterative method. Furthermore, we investigate the long-time behavior of solutions to the above system under an additional mild condition, and improve previously known results. In particular, in the one-dimensional case, we show that the solution $(p,c,w)$ converges to $(1,0,1)$ with an explicit exponential rate as time tends to infinity.

math.AP↗

arXiv2018

Global boundedness and decay property of a three-dimensional Keller--Segel--Stokes system modeling coral fertilization

This paper is concerned with the four-component Keller--Segel--Stokes system modelling the fertilization process of corals: \begin{equation*} \left\{ \begin{array}{ll} ρ_t+u\cdot\nablaρ=Δρ-\nabla\cdot(ρ\mathcal{S}(x,ρ,c)\nabla c)-ρm, & \quad (x,t)\in Ω\times (0,T), \\ m_t+u\cdot\nabla m=Δm-ρm, & \quad (x,t)\in Ω\times (0,T), \\ c_t+u\cdot\nabla c=Δc-c+m, & \quad (x,t)\in Ω\times (0,T), \\ u_t=Δu-\nabla P+(ρ+m)\nablaϕ,\quad \nabla\cdot u=0, & \quad (x,t)\in Ω\times (0,T) \end{array}\right. \end{equation*} subject to the boundary conditions $\nabla c\cdot ν=\nabla m\cdot ν=(\nablaρ-ρ\mathcal{S}(x,ρ,c)\nabla c)\cdot ν=0$ and $u=0$, and suitably regular initial data $(ρ_0(x),m_0(x), c_0(x),u_0(x))$, where $T\in (0,\infty]$, $Ω\subset\mathbb R^3$ is a bounded domain with smooth boundary $\partialΩ$. This system describes the spatio-temporal dynamics of the population densities of sperm $ρ$ and egg $m$ under a chemotactic process facilitated by a chemical signal released by the egg with concentration $c$ in a fluid-flow environment $u$ modeled by the incompressible Stokes equation. In this model, the chemotactic sensitivity tensor $\mathcal{S}\in C^2(\overlineΩ\times [0,\infty)^2)^{3\times 3}$ satisfies $|\mathcal{S}(x,ρ,c)|\leq C_S(1+ρ)^{-α}$ with some $C_S>0$ and $α\geq 0$. We will show that for $α\geq \frac 13$, the solutions to the system are globally bounded and decay to a spatially homogeneous equilibrium exponentially as time goes to infinity. In addition, we will also show that, for any $α\geq 0$, a similar result is valid when the initial data satisfy a certain smallness condition.

math.AP↗