arXiv subjects

Qingyou He

Publications and source records attributed to Qingyou He.

9 recordsLinked to original sources

arXiv2025

Sharp power concavity of two relevant free boundary problems of reaction-diffusion type

The porous medium type reaction-diffusion equation and the Hele-Shaw problem are two free boundary problems linked through the incompressible (Hele-Shaw) limit. We investigate and compare the sharp power concavities of the pressures on their respective supports for the two free boundary problems. For the pressure of the porous medium type reaction-diffusion equation, the $\frac{1}{2}$-concavity preserves all the time, while $\alpha$-concavity for $\alpha\in[0,\frac{1}{2})\cup(\frac{1}{2},1]$ does not persist in time. In contrast, in the case of the pressure for the Hele-Shaw problem, $\alpha$-concavity with $\alpha\in[0,\frac{1}{2}]$ is maintained all the while and $\frac{1}{2}$ acts as the largest index. The intuitive explanation for the difference between the two free boundary problems is that, although the Hele-Shaw problem is the incompressible limit of the porous medium-type reaction-diffusion equation, it is no longer a degenerate parabolic equation. Furthermore, for the pressure of the porous medium type reaction-diffusion equation, the non-degenerate estimate is established by means of the derived concave properties, indicating that the spatial Lipschitz regularity in the whole space is sharp.

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arXiv2025

Hele-Shaw limit of chemotaxis-Navier-Stokes flows

This paper investigates the connection between the chemotaxis--Navier--Stokes system with porous medium type nonlinear diffusion and the Hele--Shaw problem in $\mathbb{R}^d$ ($d\geq2$). First, we prove the global-in-time existence of weak solutions for the Cauchy problem of the chemotaxis-Navier-Stokes system with the general initial data, uniformly in the diffusion range $m\in [3,\infty)$. Then, we rigorously justify the Hele--Shaw limit for this system as $m\rightarrow\infty$, showing the convergence to a free boundary problem of Hele--Shaw type, where the bacterium (cell) diffusion is governed by the stiff pressure law. Moreover, the complementarity relation characterizing the limiting bacterium (cell) pressure via a degenerate elliptic equation is verified by a novel application of the Hele--Shaw framework.

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arXiv2024

On the incompressible limit of Keller-Segel system with volume-filling effects

We consider the Keller-Segel system with a volume-filling effect and study its incompressible limit. Due to the presence of logistic-type sensitivity, $K=1$ is the critical threshold. When $K>1$, as the diffusion exponent tends to infinity, by supposing the weak limit of $u^2_m$, we prove that the limiting system becomes a Hele-Shaw type free boundary problem. For $K\le 1$, we justify that the stiff pressure effect ($\Delta P_\infty$) vanishes, resulting in the limiting system being a hyperbolic Keller-Segel system. Compared to previous studies, the new challenge arises from the stronger nonlinearity induced by the logistic chemotactic sensitivity. To address this, our first novel finding is the proof of strong convergence of the density on the support of the limiting pressure, thus confirming the validity of the \emph{complementarity relation} for all $K>0$. Furthermore, specifically for the case $K\le1$, by introducing the \emph{kinetic formulation}, we verify the strong limit of the density required to reach the incompressible limit.

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arXiv2024

Porous medium type reaction-diffusion equation: large time behaviors and regularity of free boundary

We consider the Cauchy problem of the porous medium type reaction-diffusion equation \begin{equation*} \partial_t\rho=\Delta\rho^m+\rho g(\rho),\quad (x,t)\in \mathbb{R}^n\times \mathbb{R}_+,\quad n\geq2,\quad m>1, \end{equation*} where $g$ is the given monotonic decreasing function with the density critical threshold $\rho_M>0$ satisfying $g(\rho_M)=0$. We prove that the pressure $P:=\frac{m}{m-1}\rho^{m-1}$ in $L_{loc}^{\infty}(\mathbb{R}^n)$ tends to the pressure critical threshold $P_M:=\frac{m}{m-1}(\rho_M)^{m-1}$ at the time decay rate $(1+t)^{-1}$. If the initial density $\rho(x,0)$ is compactly supported, we justify that the support $\{x: \rho(x,t)>0\}$ of the density $\rho$ expands exponentially in time. Furthermore, we show that there exists a time $T_0>0$ such that the pressure $P$ is Lipschitz continuous for $t>T_0$, which is the optimal (sharp) regularity of the pressure, and the free surface $\partial \{(x,t): \rho(x,t)>0\}\cap \{t>T_0\}$ is locally Lipschitz continuous. In addition, under the same initial assumptions of compact support, we verify that the free boundary $\partial \{(x,t): \rho(x,t)>0\}\cap \{t>T_0\}$ is a local $C^{1,\alpha}$ surface.

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arXiv2024

Global dynamics for the generalized chemotaxis-Navier-Stokes system in $\mathbb{R}^3$

We consider the chemotaxis-Navier-Stokes system with generalized fluid dissipation in $\mathbb{R}^3$: \begin{eqnarray*} \begin{cases} \partial_t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi(c)n \nabla c),\\ \partial_t c+u \cdot \nabla c=\Delta c-nf(c),\\ \partial_t u +u \cdot \nabla u+\nabla P=-(-\Delta)^\alpha u-n\nabla \phi,\\ \nabla \cdot u=0, \end{cases} \end{eqnarray*} which describes the motion of swimming bacteria or bacillus subtilis suspended to water flows. First, we prove some blow-up criteria of strong solutions to the Cauchy problem, including the Prodi-Serrin type criterion ($\alpha>\frac{3}{4}$) and the Beir${\rm\tilde{a}}$o da Veiga type criterion $(\alpha>\frac{1}{2})$. Then, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and bacteria density for $\alpha\geq \frac{5}{4}$. Furthermore, in the scenario of $\frac{3}{4}<\alpha<\frac{5}{4}$, we establish uniform regularity estimates and optimal time-decay rates of global solutions if the $L^2$-norm of initial data is small. To our knowledge, this is the first result concerning the global existence and large-time behavior of strong solutions for the chemotaxis-Navier-Stokes equations with possibly large oscillations.

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arXiv2024

Nonsymmetric traveling wave solution to a Hele-Shaw type tumor growth model

We consider a Hele-Shaw model that describes tumor growth subject to nutrient supply. The model is derived by taking the incompressible limit of porous medium type equations, and the boundary instability of this model was recently studied in \cite{feng2022tumor} using asymptotic analysis. In this paper, we further prove the existence of nonsymmetric traveling wave solutions to the model in a two dimensional tube-like domain, which reflect intrinsic boundary instability in tumor growth dynamics.

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arXiv2023

Incompressible limit of porous media equation with chemotaxis and growth

We revisit the problem of proving the incompressible limit for the compressible porous media equation with Newtonian drift and growth. The question is motivated by models of living tissues development including chemotaxis. We extend the problem, already treated by the authors and several other contributions, in using a simplified approach, in treating dimensions two or higher, and in incorporating the pressure driven growth term. We also complete the analysis with stronger $L^4$ estimates on the pressure gradient. The major difficulty is to prove the strong convergence of the pressure gradient which is obtained here by a new observation on an algebraic relation involving the pressure gradient for weak limits.

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arXiv2022

Incompressible limits of Patlak-Keller-Segel model and its stationary state

We complete previous results about the incompressible limit of both the $n$-dimensional $(n\geq3)$ compressible Patlak-Keller-Segel (PKS) model and its stationary state. As in previous works, in this limit, we derive the weak form of a geometric free boundary problem of Hele-Shaw type, also called congested flow. In particular, we are able to take into account the unsaturated zone, and establish the complementarity relation which describes the limit pressure by a degenerate elliptic equation. Not only our analysis uses a completely different framework than previous approaches, but we also establish a novel uniform $L^3$ estimate of the pressure gradient, regularity à la Aronson-Bénilan, and a uniform $L^1$ estimate for the time derivative of the pressure. Furthermore, for the Hele-Shaw problem, we prove the uniqueness of solutions, meaning that the incompressible limit of the PKS model is unique. In addition, we establish the corresponding incompressible limit of the stationary state for the PKS model with a given mass, where, different from the case of PKS model, we obtain the uniform bound of pressure and the uniformly bounded support of density.

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arXiv2022

The hyperbolic-parabolic chemotaxis system for vasculogenesis: global dynamics and relaxation limit toward a Keller-Segel model

An Euler-type hyperbolic-parabolic system of chemotactic aggregation describing the vascular network formation is investigated in the critical regularity setting. For small initial data around a constant equilibrium state, the well-posedness of the global classical solution to the Cauchy problem with general pressure laws is established in homogeneous hybrid Besov spaces. Then, the optimal time-decay rates of the global solution are analyzed under an additional regularity assumption on the initial data. Furthermore, the relaxation limit (large friction limit) of the hyperbolic-parabolic system is justified rigorously. It is shown that as the friction coefficient tends to zero, the global solution of the hyperbolic-parabolic chemotaxis system converges to the global solution of the Keller-Segel equations with an explicit convergence rate. To capture the dissipative properties of the nonlinear system, our approach relies on the introduction of new effective unknowns in low frequencies and the construction of a Lyapunov functional in the spirit of Beauchard and Zuazua's in [5] to treat the high frequencies.

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