Higher Order Coercive Inequalities
We study the higher order q- Poincaré and other coercive inequalities for a class probability measures satisfying Adam's regularity condition.
math.FA↗
arXiv subjects
Yifu Wang
Publications and source records attributed to Yifu Wang.
Asymptotic behavior of a quasilinear Keller--Segel system with signal-suppressed motility
This paper is concerned with the density-suppressed motility model: $u_{t}=Δ(\displaystyle\frac{u^m}{v^α}) +βuf(w), v_{t}=DΔv-v+u, w_{t}=Δw-uf(w)$ in a smoothly bounded convex domain $Ω\subset {\mathbb{R}}^2$, where $m>1$, $α>0, β>0$ and $D>0$ are parameters, the response function $f$ satisfies $f\in C^1([0,\infty)), f(0)=0, f(w)>0$ in $(0,\infty)$. This system describes the density-suppressed motility of Eeshcrichia coli cells in process of spatio-temporal pattern formation via so-called self-trapping mechanisms. Based on the duality argument, it is shown that for suitable large $D$ the problem admits at least one global weak solution $(u,v,w)$ which will asymptotically converge to the spatially uniform equilibrium $(\overline{u_0}+β\overline{w_0},\overline{u_0}+β\overline{w_0},0)$ with $\overline{u_0}=\frac1{|Ω|}\int_Ωu(x,0)dx $ and $\overline{w_0}=\frac1{|Ω|}\int_Ωw(x,0)dx $ in $L^\infty(Ω)$.
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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.
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Multiple solutions for a class of semilinear elliptic equations
By using truncation technique, minimization method and Morse theory, we obtain three nontrivial solutions for a class of semilinear elliptic equations.
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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.
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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.
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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.
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Global solvability and stability to a nutrient-taxis model with porous medium slow diffusion
In this paper, we study a nutrient-taxis model with porous medium slow diffusion \begin{align*} \left\{ \begin{aligned} &u_t=Δu^m-χ\nabla\cdot(u\nabla v)+ξuv-ρu, \\ &v_t-Δv=-vu+μv(1-v), \end{aligned}\right. \end{align*} in a bounded domain $Ω\subset \mathbb R^3$ with zero-flux boundary condition. It is shown that for any $m>\frac{11}4-\sqrt 3$,the problem admits a global weak solution for any large initial datum. We divide the study into three cases,(i) $ξμ=0, ρ\ge 0$; (ii) $ξμρ>0$; (iii) $ξμ>0$, $ρ=0$. In particular, for Case (i) and Case (ii), the global solutions are uniformly bounded. Subsequently, the large time behavior of these global bounded solutions are also discussed. At last, we also extend the results to the coupled chemotaxis-Stokes system. Important progresses for chemotaxis-Stokes system with $m>\frac 76$, $m>\frac 87$ and $m>\frac 98$ have been carried out respectively by \cite{W2, TW2, W3}, but leave a gap for $1<m\le \frac98$. Our result for chemotaxis-Stokes system supplements part of the gap $(\frac{11}4-\sqrt 3, \frac 98)$. Here $\frac{11}4-\sqrt 3\approx 1.018$.
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