arXiv subjects

Bao-Ngoc Tran

Publications and source records attributed to Bao-Ngoc Tran.

6 recordsLinked to original sources

arXiv2026

Global weak solvability for doubly degenerate nutrient taxis system in physical dimension

Motivated by the study of bacteria's response to environmental conditions, we consider the doubly degenerate nutrient taxis system \begin{align*} \begin{cases} u_t=\nabla\cdot(uv\nabla u)-χ\nabla\cdot(u^αv\nabla v)+\ell uv,\\ v_t=Δv-uv, \end{cases} \end{align*} subjected to no-flux boundary conditions and smooth initial data, where $α\ge0$ is the bacterial response parameter. Global solvability of weak solutions to this taxis system appears to be highly challenging due to the difficulty of quantifying the dissipation of the doubly degenerate diffusive flux, the strong chemotactic effect, particularly as $α$ is close to $2$, and the dimensionally dependent limitation, which is the most difficult one. Recent findings on the global weak solvability for the considered system are summarised as follows \begin{itemize} \item In [M. Winkler, \textit{Trans. Amer. Math. Soc.}, 2021] for $α=2$, $N=1$; \item In [M. Winkler, \textit{J. Differ. Equ.}, 2024] for $1\leα\le 2$, $N=2$with initial data of small size if $α=2$; \item In [Z. Zhang, Y. Li, \textit{Math. Models Methods Appl. Sci.}, 2026] for $α=2$, $N=2$; \item In [G. Li, \textit{J. Differ. Equ.}, 2022] for $\frac{7}{6}<α<\frac{13}{9}$, $N=3$; and \item In [X.M. De-Ji, A. Huang, Y. Wang, \textit{Eur. J. Appl. Math.}] for $\frac{3}{2}<α<\frac{19}{12}$, $N=3$. \end{itemize} Our work aims to provide a picture of global weak solvability for $0\le α<2$ in the physically dimensional setting $N=3$. As suggested by the analysis, it is divided into three separable cases, including (i) $0\leα\le1$: Weak chemotaxis effect; (ii) $1<α\le3/2$: Moderate chemotaxis effect; and (iii) $3/2<α<2$: Strong chemotaxis effect.

math.AP↗

arXiv2025

Parabolic-elliptic and indirect-direct simplifications in chemotaxis systems driven by indirect signalling

Singular limits for the following indirect signalling chemotaxis system \begin{align*} \left\{ \begin{array}{lllllll} \partial_t n = Δn - \nabla \cdot (n \nabla c ) & \text{in } Ω\times(0,\infty) , \varepsilon \partial_t c = Δc - c + w & \text{in } Ω\times(0,\infty), \varepsilon \partial_t w = τΔw - w + n & \text{in } Ω\times (0,\infty), \partial_νn = \partial_νc = \partial_νw = 0, &\text{on } \partialΩ\times (0,\infty) %(n,c,w)_{t=0} = (n_0,c_0,w_0) & \text{on } Ω, \end{array} \right. \end{align*} are investigated. More precisely, we study parabolic-elliptic simplification, or PES, $\varepsilon\to 0^+$ with fixed $τ>0$ up to the critical dimension $N=4$, and indirect-direct simplification, or IDS, $(\varepsilon,τ)\to (0^+,0^+)$ up to the critical dimension $N=2$. These are relevant in biological situations where the signalling process is on a much faster time scale compared to the species diffusion and all interactions. Showing singular limits in critical dimensions is challenging. To deal with the PES, we carefully combine the entropy function, an Adam-type inequality, the regularisation of slow evolution, and an energy equation method to obtain strong convergence in representative spaces. For the IDS, a bootstrap argument concerning the $L^p$-energy function is devised, which allows us to obtain suitable uniform bounds for the singular limits. Moreover, in both scenarios, we also present the convergence rates, where the effect of the initial layer and the convergence to the critical manifold are also revealed.

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arXiv2025

Rigorous fast signal diffusion limit and convergence rates with the initial layer effect in a competitive chemotaxis system

We study a chemotaxis system that includes two competitive prey and one predator species in a two-dimensional domain, where the movement of prey (resp. predators) is driven by chemicals secreted by predators (resp. prey), called mutually repulsive (resp. mutually attractive) chemotactic effect. The kinetics for all species are chosen according to the competitive Lotka--Volterra equations for prey and to a Holling type functional response for the predator. Under the biologically relevant scenario that the chemicals diffuse much faster than the individual diffusion of all species and a suitable re-scaling, equations for chemical concentrations are parabolic with slow evolution depending on the relaxation time $0<\varepsilon\ll 1$. The first main result shows the global existence of a unique classical solution to the system for each $\varepsilon$. Second, we study rigorously the so-called fast signal diffusion limit, passing from the system including parabolic equations with the slow evolution of the chemical concentrations to elliptic equations for the chemical concentrations, i.e. the limit as $\varepsilon \to 0$. This explains why elliptic equations can be proposed for chemical concentration instead of parabolic ones with slow evolution. Third, the $L^\infty$-in-time convergence rates for the fast signal diffusion limit are estimated, where the effect of the initial layer is carefully treated. Finally, the differences between the systems with and without the slow evolution, and between the systems with one or two prey, as well as their dynamics, are discussed numerically.

math.AP↗

arXiv2025

Slow Manifolds for PDE with Fast Reactions and Small Cross Diffusion

Multiple time scales problems are investigated by combining geometrical and analytical approaches. More precisely, for fast-slow reaction-diffusion systems, we first prove the existence of slow manifolds for the abstract problem under the assumption that cross diffusion is small. This is done by extending the Fenichel theory to an infinite dimensional setting, where a main idea is to introduce a suitable space splitting corresponding to a small parameter, which controls additional fast contributions of the slow variable. These results require a strong convergence in $L^\infty(0,T;H^2(Ω))$, which is the subtle analytical issue in fast reaction problems in comparison to previous works. By considering a nonlinear fast reversible reaction in one dimension, we successfully prove this convergence and therefore obtain the slow manifold for a PDE with fast reactions. Moreover, the obtained convergence also shows the influence of the cross diffusion term and illuminates the role of the initial layer. Explicit approximations of the slow manifold are also carried out in the case of linear systems.

math.AP↗

arXiv2024

Singular limit and convergence rate via projection method in a model for plant-growth dynamics with autotoxicity

We investigate a fast-reaction--diffusion system modelling the effect of autotoxicity on plant-growth dynamics, in which the fast-reaction terms are based on the dichotomy between healthy and exposed roots depending on the toxicity. The model was proposed in [Giannino, Iuorio, Soresina, forthcoming] to account for stable stationary spacial patterns considering only biomass and toxicity, and its fast-reaction (cross-diffusion) limit was formally derived and numerically investigated. In this paper, the cross-diffusion limiting system is rigorously obtained as the fast-reaction limit of the reaction-diffusion system with fast-reaction terms by performing a bootstrap argument involving energies. Then, a thorough well-posedness analysis of the cross-diffusion system is presented, including a $L^\infty_{x,t}$-bound, uniqueness, stability, and regularity of weak solutions. This analysis, in turn, becomes crucial to establish the convergence rate for the fast reaction limit, thanks to the key idea of using an inverse Neumann Laplacian operator. Finally, numerical experiments illustrate the analytical findings on the convergence rate.

math.AP↗

arXiv2023

Rigorous derivation of Michaelis-Menten kinetics in the presence of diffusion

Reactions with enzymes are critical in biochemistry, where the enzymes act as catalysis in the process. One of the most used mechanisms for modeling enzyme-catalyzed reactions is the Michaelis-Menten (MM) kinetic. In the ODE level, i.e. concentrations are only on time-dependent, this kinetic can be rigorously derived from mass action law using quasi-steady-state approximation. This issue in the PDE setting, for instance when molecular diffusion is taken into account, is considerably more challenging and only formal derivations have been established. In this paper, we prove this derivation rigorously and obtain MM kinetic in the presence of spatial diffusion. In particular, we show that, in general, the reduced problem is a cross-diffusion-reaction system. Our proof is based on improved duality method, heat regularisation and a suitable modified energy function. To the best of our knowledge, this work provides the first rigorous derivation of MM kinetic from mass action kinetic in the PDE setting.

math.AP↗