arXiv subjects

Bao Quoc Tang

Publications and source records attributed to Bao Quoc Tang.

At least 19 recordsLinked to original sources

arXiv2026

Global classical solutions by transport noise for reaction-diffusion systems with entropy dissipation

The existence of global classical solutions for reaction-diffusion systems arising from chemical reaction networks remains a major open problem in the deterministic setting, especially for reactions with high polynomial growth. We prove that a suitably chosen, physically motivated transport noise yields unique strong solutions for complex balanced chemical reaction networks that are global in time with arbitrarily high probability. These solutions possess paths in $C^θ_t C^{\infty}_x$ for all $θ<1/2$ and are, in particular, classical in space. Furthermore, we show that a suitable transport noise can enhance the dissipation of spatial fluctuations at an arbitrarily prescribed exponential rate. Our proofs rely on a combination of scaling-limit arguments, maximal $L^p(L^q)$-regularity, and entropy-entropy dissipation estimates.

math.AP↗

arXiv2026

Two-sided estimates of Lyapunov exponents for Milstein schemes of non-autonomous stochastic differential equations

The stabilising effect of multiplicative noise for stochastic differential equations, though counterintuitive, has been observed and investigated extensively in last decades. In practice, it is desirable to know if such stabilisation holds also for the discretised setting. In this paper, we address this problem by means of sharp upper and lower estimates for Lyapunov exponents of Milstein schemes for non-autonomous stochastic differential equations. These estimates provide precise large time behaviour in both almost sure and $p$-moment sense. In particular, our results show the preservation of stabilisation from the continuum setting to the discretised setting. One main idea of our analysis is to exploit the second order term concerning the stochastic noise from the Milstein scheme to obtain precise estimates for Taylor expansions of logarithmic and power functions.

math.NA↗

arXiv2026

Existence, asymptotic behaviour and convergence of a generalised 3D Muskat problem in stable regime

We address a generalised three-dimensional $α$-Muskat model that comes from the fluid interface problem given by two incompressible fluids with different densities in the stable regime. We establish local-in-time wellposedness when $α\in[0,1)$ and also prove global-in-time existence for strong solutions when $α\in[0,\frac{1}{2})$ with initial data controlled by explicit constants. We obtain maximum principles for the $L^{\infty}$-norms of both the solutions and their gradients, and we further acquire the corresponding decay rates of these $L^{\infty}$-norms. Finally, some convergence results for strong solutions as $α\to0^+$ are also proved.

math.AP↗

arXiv2026

Fast reaction limits and convergence rate for nonlinear bulk-surface reaction-diffusion systems modeling reversible chemical reactions

The fast reaction limit for a nonlinear bulk-surface reaction-diffusion system is investigated. This system describes a reversible reaction with arbitrary stoichiometric coefficients, where one chemical is present in a bounded vessel $Ω$ and the other chemical lies only on the boundary $\partialΩ$ where the reaction takes place. In the limit as the reaction rate constant tends to infinity, we prove that the solution converges in $L^p(0,T;L^p(Ω))$ to the solution of a heat equation with nonlinear dynamical boundary condition. This is obtained by showing a-priori estimates of solutions which are uniform in the reaction rate constants. In order to overcome the difficulty caused by the bulk-surface coupling, we consider the limit in suitable product spaces where the Aubin-Lions lemma is applicable. Moreover, in the case of equal stoichiometric coefficients, we obtain the convergence rate of the fast reaction limit by exploiting suitable estimates of the limiting system.

math.AP↗

arXiv2026

Global well-posedness for nonlinear generalized Camassa-Holm equation

We establish local and global well-posedness for the Cauchy problem of a generalized Camassa-Holm equation where orders of the momentum and the nonlinearity can be arbitrarily high. More precisely, we consider the equation \begin{equation*} m_t + m_x u^p + b m u^{p-1}u_x = -(g(u))_x + (b+1)u^p u_x, \quad m = (1-\partial_x^2)^k u, \end{equation*} where $p \geq 1$, $k \geq 1$ are arbitrary, $b$ is a real parameter, and $g(u)$ is a smooth function. %The standard Camassa-Holm equation corresponds to $k=1$, $p=1$, $b=2$, and $g(u)=0$. The local well-posedness is shown by using Kato's semigroup approach, where we treat the nonlinearity directly using commutator estimates and the fractional Leibniz rule without having to transform it in any specific differential form. This well-posedness is obtained in the phase space $H^s$ for $s > 2(k-1) + 3/2$, which is consistent with the results for the classical Camassa-Holm equation. We also prove the global existence of solutions by obtaining conserved quantity and applying the same idea from our local theory.

math.AP↗

arXiv2026

Renormalised solutions to reaction-diffusion systems with interface conditions: Global existence and weak-strong uniqueness

We introduce an extension of the concept of renormalised solutions for entropy-dissipating reaction-diffusion systems due to J. Fischer (Arch. Ration. Mech. Anal. 218, 2015) to systems coupled by nonlinear interface conditions. For this notion of solution, we establish global existence as well as a weak-strong stability estimate. Our framework allows to handle entropy-dissipating interfacial transmission rates without growth restrictions, including power-law nonlinearities as arising in the thermodynamic modelling of dissipative bulk-interface systems via generalised gradient structures. Our analysis relies on suitable extensions of the species' densities across the interface as well as on a non-local truncated variant of the relative entropy.

math.AP↗

arXiv2026

How Recommendation Algorithms Shape Social Networks: An Adaptive Voter Model Approach

The rise of social media and recommendation algorithms has sparked concerns about their role in fostering opinion polarization and echo chambers. We study these phenomena using an adaptive voter model to compare two connection mechanisms: "free" global rewiring, where individuals connect with anyone sharing their opinion, and "friend-of-a-friend" local rewiring, which mimics algorithmic link recommendations on platforms like Facebook or LinkedIn. Simulations across different network topologies reveal that local rewiring increases final-state polarization of the system and fragments social networks into many disconnected components. The usual phase transition into two disconnected components turns into a fragmentation of smaller components, leading to an increase in echo chambers as well as many isolated nodes. This effect is most pronounced in clustered networks with high homophily in rewiring, illustrating how recommendation algorithms can intensify social fragmentation by changing the very structure of the network.

physics.soc-ph↗

arXiv2025

Explicit spectral gap estimates for the linearized Boltzmann operator modeling reactive gaseous mixtures

We consider hard-potential cutoff multi-species Boltzmann operators modeling microscopic binary elastic collisions and bimolecular reversible chemical reactions inside a gaseous mixture. We prove that the spectral gap estimate derived for the linearized elastic collision operator can be exploited to deduce an explicit negative upper bound for the Dirichlet form of the linearized chemical Boltzmann operator. Such estimate may be used to quantify explicitly the rate of convergence of close-to-equilibrium solutions to the reactive Boltzmann equation toward the global chemical equilibrium of the mixture.

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.

math.AP↗

arXiv2025

Turing instability for nonlocal heterogeneous reaction-diffusion systems: A computer-assisted proof approach

This paper provides a computer-assisted proof for the Turing instability induced by heterogeneous nonlocality in reaction-diffusion systems. Due to the heterogeneity and nonlocality, the linear Fourier analysis gives rise to \textit{strongly coupled} infinite differential systems. By introducing suitable changes of basis as well as the Gershgorin disks theorem for infinite matrices, we first show that all $N$-th Gershgorin disks lie completely on the left half-plane for sufficiently large $N$. For the remaining finitely many disks, a computer-assisted proof shows that if the intensity $δ$ of the nonlocal term is large enough, there is precisely one eigenvalue with positive real part, which proves the Turing instability. Moreover, by detailed study of this eigenvalue as a function of $δ$, we obtain a sharp threshold $δ^*$ which is the bifurcation point for Turing instability.

math.AP↗

arXiv2025

Turing Instability Suppressed and Induced by Multiplicative Noise in Brusselator System

The effect of multiplicative noise to the Turing instability of the Brusselator system is investigated. We show that when the noise acts on both of the concentrations with the same intensities, then the Turing instability is suppressed provided that the intensities are sufficiently large. This aligns with the stabilizing effect of multiplicative noise in partial differential equations. Utilizing the linearized system, we can quantify the magnitude of noise which stabilizes the system. On the other hand, when the noise is involving only one concentration, then the Turing instability can be triggered with suitable intensities. These are confirmed by numerical simulations.

math.AP↗

arXiv2025

Well-Posedness for Fractional Reaction-Diffusion Systems with Mass Dissipation in $\mathbb R^N$

The global existence of bounded solutions to reaction-diffusion systems with fractional diffusion in the whole space $\mathbb R^N$ is investigated. The systems are assumed to preserve the non-negativity of initial data and to dissipate total mass. We first show that if the nonlinearities are at most quadratic then there exists a unique global bounded solution regardless of the fractional order. This is done by combining a regularizing effect of the fractional diffusion operator and the Hölder continuity of a non-local inhomogeneous parabolic equation. When the nonlinearities might be super-quadratic, but satisfy some intermediate sum conditions, we prove the global existence of bounded solutions by adapting the well-known duality methods to the case of fractional diffusion. In this case, the order of the intermediate sum conditions depends on the fractional order. These results extend the existing theory for mass dissipated reaction-diffusion systems to the case of non-local diffusion and unbounded domains.

math.AP↗

arXiv2025

Pathwise mild solutions for superlinear stochastic evolution equations and their attractors

We investigate stochastic parabolic evolution equations with time-dependent random generators and locally Lipschitz continuous drift terms. Using pathwise mild solutions, we construct an infinite-dimensional stationary Ornstein-Uhlenbeck type process, which is shown to be tempered in suitable function spaces. This property, together with a bootstrapping argument based on the regularizing effect of parabolic evolution families, is then applied to prove the global well-posedness and the existence of a random attractor for reaction-diffusion equations with random non-autonomous generators and nonlinearities satisfying certain growth and dissipativity assumptions.

math.PR↗

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

Volume-surface systems with sub-quadratic intermediate sum on the surface: Global existence and boundedness

The global existence and boundedness of solutions to volume-surface reaction diffusion systems with a mass control condition are investigated. Such systems arise typically in e.g. cell biology, ecology or fluid mechanics, when some concentrations or densities are inside a domain and some others are on its boundary. Comparing to previous works, the difficulty of systems under consideration here is that the nonlinearities on the surface can have a sub-quadratic growth rates in all dimensions. To overcome this, we first use the Moser iteration to get some uniform bounds of the time integration of the solutions. Then by combining these bounds with an $L^p$-energy method and a duality argument, we obtain the global existence of solutions. Moreover, under mass dissipation conditions, the solution is shown to be bounded uniformly in time.

math.AP↗

arXiv2024

On the equilibriation of chemical reaction-diffusion systems with degenerate reactions

The trend to equilibrium for reaction-diffusion systems modelling chemical reaction networks is investigated, in the case when reaction processes happen on subsets of the domain. We prove the convergence to equilibrium by directly showing functional inequalities in terms of entropy method. Our approach allows us to deal with nonlinearities of arbitrary orders, for which only global renormalised solutions are known to globally exist. For bounded solutions, we also prove the convergence to equilibrium when the diffusion as well as the reaction are degenerate, that is both diffusion and reaction processes only act on specific subsets of the domain.

math.AP↗

arXiv2024

Stability analysis of irreversible chemical reaction-diffusion systems with boundary equilibria

Large time dynamics of reaction-diffusion systems modeling some irreversible reaction networks are investigated. Depending on initial masses, these networks possibly possess boundary equilibria, where some of the chemical concentrations are completely used up. In the absence of these equilibria, we show an explicit convergence to equilibrium by a modified entropy method, where it is shown that reactions in a measurable set with positive measure is sufficient to combine with diffusion and to drive the system towards equilibrium. When the boundary equilibria are present, we show that they are unstable (in Lyapunov sense) using some bootstrap instability technique from fluid mechanics, while the nonlinear stability of the positive equilibrium is proved by exploiting a spectral gap of the linearized operator and the uniform-in-time boundedness of solutions.

math.AP↗

arXiv2024

Existence, Stability and Optimal Drug Dosage for a Reaction-Diffusion System Arising in a Cancer Treatment

In this paper, a reaction-diffusion system modeling injection of a chemotherapeutic drug on the surface of a living tissue during a treatment for cancer patients is studied. The system describes the interaction of the chemotherapeutic drug and the normal, tumor and immune cells. We first establish well-posedness for the nonlinear reaction-diffusion system, then investigate the long-time behavior of solutions. Particularly, it is shown that the cancer cells will be eliminated assuming that its reproduction rate is sufficiently small in a short time period in each treatment interval. The analysis is then essentially exploited to study an optimal drug injection rate problem during a chemotherapeutic drug treatment for tumor cells, which is formulated as an optimal boundary control problem with constraints. For this, we show that the existence of an optimal drug injection rate through the boundary, and derive the first-order optimality condition.

math.AP↗