No-flux Boundary Conditions Research Articles

Asymptotic stability of spiky steady states for a singular chemotaxis model with signal-suppressed motility.

We study the nonlinear stability of spiky solutions to a chemotaxis model of consumption type with singular signal-suppressed motility in the half space. We show that, when the no-flux boundary condition for the bacteria density and the nonhomogeneous Dirichlet boundary condition for the nutrient are prescribed, this chemotaxis model admits a unique smooth spiky steady state, and it is nonlinearly stable under appropriate perturbations. The challenge of the problem is that there are two types of singularities involved in the model: one is the logarithmic singularity of the sensitive function; and the other is the inverse square singularity of the motility. We employ a Cole-Hopf transformation to relegate the former singularity to a nonlocality that can be resolved by the method of anti-derivative. To deal with the latter singularity, we construct an approximate system that retains a key structure of the original singular system in the local theory, and develop a new strategy, which combines a weighted elliptic estimate and the weighted energy estimate, to establish a priori estimate in the global theory.

Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan–Majda model and Taylor–Aris Dispersion

We study the long time behavior of an advection–diffusion equation with a random shear flow which depends on a stationary Ornstein–Uhlenbeck (OU) process in parallel-plate channels enforcing the no-flux boundary conditions. We derive a closed form formula for the long time asymptotics of the arbitrary N-point correlator using the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). In turn, appealing to the conclusion of the Hausdorff moment problem (Shohat and Tamarkin, 1943), we discover a diffusion equation with a random drift and deterministic enhanced diffusion possessing the exact same probability density function at long times. The strategy we presented is not only restricted to the parallel-plate channel domain. The same methods can derive effective equations for a straight channel with uniform arbitrary cross-section. Such equations enjoy many ergodic properties which immediately translate to ergodicity results for the original problem. In particular, we establish that the first two Aris moments using a single realization of the random field can be used to explicitly construct all ensemble averaged moments. Also, the first two ensemble averaged moments explicitly predict any long time centered Aris moment. Our formulae quantitatively depict the dependence of the deterministic effective diffusion on the interaction between spatial structure of flow and random temporal fluctuation. Further, this approximation provides many identities regarding the stationary OU process dependent time integral. We derive explicit formulae for the decaying passive scalar’s long time limiting probability density function (PDF) for different types of initial conditions (e.g. deterministic and random).

Spatiotemporal patterns in a general networked activator–substrate model

To understand the spatiotemporal pattern formation in the random networked system, a general activator–substrate model with network structure is introduced. Firstly, we investigate the boundedness of the non-constant steady state of the elliptic system of the continuous medium system. It is found that the non-constant steady state admits their upper and lower bounds under certain conditions. Then, we establish some properties and nonexistence of the non-constant steady state with the no-flux boundary conditions. The main results show that the diffusion rate of the activator should be less than that of the substrate. Otherwise, there might be no pattern formation of the system. Afterward, a general random networked activator–substrate model is presented. Conditions of the stability, the Hopf bifurcation, the Turing instability and the co-dimensional-two Turing–Hopf bifurcation are yielded by the method of stability analysis and bifurcation theory. Finally, a suitable sub-system of the general activator–substrate model is chosen to verify the theoretical results, and full numerical simulations have well verified these results. Especially, an interesting finding is that the stability of the positive equilibrium will switch from unstable to stable one with the change of the connection probability of the nodes, which is different from the pattern formation in the continuous medium systems.

The model reduction of the Vlasov–Poisson–Fokker–Planck system to the Poisson–Nernst–Planck system via the Deep Neural Network Approach

The model reduction of a mesoscopic kinetic dynamics to a macroscopic continuum dynamics has been one of the fundamental questions in mathematical physics since Hilbert’s time. In this paper, we consider a diagram of the diffusion limit from the Vlasov–Poisson–Fokker–Planck (VPFP) system on a bounded interval with the specular reflection boundary condition to the Poisson–Nernst–Planck (PNP) system with the no-flux boundary condition. We provide a Deep Learning algorithm to simulate the VPFP system and the PNP system by computing the time-asymptotic behaviors of the solution and the physical quantities. We analyze the convergence of the neural network solution of the VPFP system to that of the PNP system via the Asymptotic-Preserving (AP) scheme. Also, we provide several theoretical evidence that the Deep Neural Network (DNN) solutions to the VPFP and the PNP systems converge to the a priori classical solutions of each system if the total loss function vanishes.

Hopf bifurcation of a delayed reaction–diffusion model with advection term

This article focus on the Hopf bifurcation of a delayed reaction–diffusion equation with advection term subject to Dirichlet boundary and no-flux boundary conditions in a bounded domain, respectively. It is shown that the existence of spatially non-homogeneous steady-state solutions will be obtained when the parameter λ of the model (9) closes to the principle eigenvalue λ1 of the elliptic operator Lλ. Moreover, a supercritical Hopf bifurcation occurs near the non-homogeneous positive steady-state at a series critical time delay values. Finally, we elucidate the effect of advection on Hopf bifurcation values. It is worth noting that the advective effect has accelerated the generation of Hopf bifurcation to a certain extent.

Distribution and pressure of active Lévy swimmers under confinement

Many active matter systems are known to perform Lévy walks during migration or foraging. Such superdiffusive transport indicates long-range correlated dynamics. These behavior patterns have been observed for microswimmers such as bacteria in microfluidic experiments, where Gaussian noise assumptions are insufficient to explain the data. We introduce active Lévy swimmers to model such behavior. The focus is on ideal swimmers that only interact with the walls but not with each other, which reduces to the classical Lévy walk model but now under confinement. We study the density distribution in the channel and force exerted on the walls by the Lévy swimmers, where the boundaries require proper explicit treatment. We analyze stronger confinement via a set of coupled kinetics equations and the swimmers’ stochastic trajectories. Previous literature demonstrated that power-law scaling in a multiscale analysis in free space results in a fractional diffusion equation. We show that in a channel, in the weak confinement limit active Lévy swimmers are governed by a modified Riesz fractional derivative. Leveraging recent results on fractional fluxes, we derive steady state solutions for the bulk density distribution of active Lévy swimmers in a channel, and demonstrate that these solutions agree well with particle simulations. The profiles are non-uniform over the entire domain, in contrast to constant-in-the-bulk profiles of active Brownian and run-and-tumble particles. Our theory provides a mathematical framework for Lévy walks under confinement with sliding no-flux boundary conditions and provides a foundation for studies of interacting active Lévy swimmers.

Global well-posedness to a chemotaxis-Stokes system

This paper concerns the chemotaxis-Stokes system nt+u⋅∇n=Δnm−∇⋅(n∇c)+μn(1−n),ct+u⋅∇c=Δc−cnα,ut=Δu−∇π+n∇φ,∇⋅u=0in a three dimensional bounded domain under no-flux boundary conditions for n,c and no-slip boundary conditions for u. The purpose of this paper is to study the global solvability and large time asymptotic behavior of solutions. Here, it is worth mentioning that the nonlinear consumption term cnα (when α>1) will lead to some higher order nonlinear terms in the proof of some uniformly bounded prior estimates of the approximation solutions, which brings great difficulties to the study of the problem. To overcome these difficulties, we make some very precise analysis, combined with some iterative techniques, and finally establish the uniform boundedness of weak solutions for m>1, 0<α<2m−1. Then, the global solvability of weak solutions is derived for any large initial data. Furthermore, we focus on the convergence of weak solutions, and prove that the solutions will converge to the constant steady state (1,0,0) in the large time limit.

Isolating Patterns in Open Reaction\u2013Diffusion Systems

Realistic examples of reaction–diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of ‘open’ reaction–diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction–diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction–diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

Existence analysis of a degenerate diffusion system for heat-conducting gases

The existence of global weak solutions to a parabolic energy-transport system in a bounded domain with no-flux boundary conditions is proved. The model can be derived in the diffusion limit from a kinetic equation with a linear collision operator involving a non-isothermal Maxwellian. The evolution of the local temperature is governed by a heat equation with a source term that depends on the energy of the distribution function. The limiting model consists of cross-diffusion equations with an entropy structure. The main difficulty is the nonstandard degeneracy, i.e., ellipticity is lost when the gas density or temperature vanishes. The existence proof is based on a priori estimates coming from the entropy inequality and the H^{-1} method and on techniques from mathematical fluid dynamics (renormalized formulation, div-curl lemma).

Comparison methods for a Keller–Segel-type model of pattern formations with density-suppressed motilities

This paper is concerned with global existence of classical solutions as well as occurrence of infinite-time blowups to the following fully parabolic system 1 $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\varDelta (\gamma (v)u)\\ v_t-\varDelta v+v=u \end{array}\right. } \end{aligned}$$ in a smooth bounded domain $$\varOmega \subset {\mathbb {R}}^n$$ , $$n\ge 1$$ with no-flux boundary conditions. This model was recently proposed in Fu et al. (Phys Rev Lett 108:198102, 2012) and Liu et al. (Science 334:238, 2011) to describe the process of stripe pattern formations via the so-called self-trapping mechanism. The system features a signal-dependent motility function $$\gamma (\cdot )$$ , which is decreasing in v and will vanish as v tends to infinity. An essential difficulty in analysis comes from the possible degeneracy as $$v\nearrow \infty .$$ In this work we develop a novel comparison method to tackle the degeneracy issue, which greatly differs from the conventional energy method in literature. An explicit point-wise upper-bound estimate for v is obtained for the first time, which shows that v(x, t) grows point-wisely at most exponentially in time. An intrinsic mechanism is then unveiled that the finite-time degeneracy is prohibited in any spatial dimension with a generic decreasing $$\gamma $$ . With such new findings, we further study global existence of classical solutions when $$n\le 3$$ and discuss uniform-in-time boundedness when $$\gamma (\cdot )$$ decreases algebraically at large signal concentrations. Besides, a new critical-mass phenomenon in dimension two is observed if $$\gamma (v)=e^{-v}$$ . Indeed, we prove that the classical solution always exists globally and remains uniformly-in-time bounded in the sub-critical case, while in the super-critical case a blowup may take place in infinite time rather than finite time.

Global existence of weak solutions to a signal-dependent Keller–Segel model for local sensing chemotaxis

This paper is devoted to the global existence of weak solutions to the following degenerate kinetic model of chemotaxis (0.1)ut=Δ(γ(v)u)τvt=Δv−v+uin a smooth bounded domain with no-flux boundary conditions under the assumption 0≤τ<(sup(0,∞)γ)−1. The problem features a positive signal-dependent motility function γ(⋅) which may vanish as v becomes unbounded. In this paper, we first modify the comparison approach developed recently in Fujie and Jiang (2020); Fujie and Jiang (2021) to derive the upper bounds of v under the slightly weakened above assumptions on γ(⋅). Then by introducing a suitable approximation scheme which is compatible with the comparison method, we establish the global existence of weak solutions in any spatial dimension via compactness argument. Our weak solution has higher regularity than those obtained in previous literature (Burger et al., 2020; Desvillettes et al., 2019; Tao and Winkler, 2017).

Lie group solutions of advection-diffusion equations

Transport phenomena in homogeneous and inhomogeneous media are commonly encountered in many practical and industrial applications, which are modeled by advection-diffusion equations (ADEs) with constant or variable diffusivities, respectively. This paper provides a new perspective on how to solve advection-diffusion equations that model different transport phenomena in low Reynolds number flows. A mathematical description of the Lie group method is conducted first and then its potential in solving advection-diffusion equations for passive scalars transport with no-slip and no-flux boundary conditions is explored. The key step is to recast advection-diffusion equations as homogeneous diffusion processes on unimodular matrix Lie groups. Consequently, an approximate solution can be obtained from mean and covariance propagation techniques developed for diffusion equations on these Lie groups. The motivation to transform the advection-diffusion equation from Euclidean space to Lie groups is to exploit the available solutions of diffusion equation on these Lie groups so that the original equation can be solved in a simple way. In this paper, methodological details have been illustrated in solving ADEs modeling three kinds of transport phenomena. Two of them govern homogeneous transport and the solutions from mean and covariance propagation on the Lie group agree well with available results in published papers. We also use this method to solve more complicated ADEs governing inhomogeneous transport in one-dimensional compressible flows with spatially varying diffusivity, which is beyond the capabilities of existing approaches. The three real problems solved by the Lie group method illustrate the potential of this method. Instead of numerical calculations, the proposed closed-form method provides a simple alternative to study mass transfer encountered in various complex physical and industrial processes.

Stability and bifurcation in a two-species reaction–diffusion–advection competition model with time delay

In this paper, we are concerned with the dynamics of a class of two-species reaction–diffusion–advection competition models with time delay subject to the homogeneous Dirichlet boundary condition or no-flux boundary condition in a bounded domain. The existence of steady state solution is investigated by means of the Lyapunov–Schmidt reduction method. The stability and Hopf bifurcation at the spatially nonhomogeneous steady-state are obtained by analyzing the distribution of the associated eigenvalues. Finally, the effect of advection on Hopf bifurcation is explored, which shows that with the increase of convection rate, the Hopf bifurcation phenomenon is more likely to emerge.

Global boundedness and large time behavior of solutions to a chemotaxis–consumption system with signal-dependent motility

This paper deals with the following chemotaxis–consumption system with signal-dependent motility $$\begin{aligned} \left\{ \begin{array}{llll} u_{t}=\Delta (\gamma (v)u),\,\,\, &{}x\in \Omega ,\,\,\, t>0,\\ v_{t}=\Delta v-uv, \,\,\,&{}x\in \Omega ,\,\,\, t>0,\\ \end{array} \right. \end{aligned}$$ under no-flux boundary conditions in a smoothly bounded domain $$\Omega \subset {\mathbb {R}}^{n}$$ . $$\gamma (s)$$ is the motility function. For the case of positive motility function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, it is asserted that the solution to the system exponentially converges to constant equilibria in the large time. Finally, if the motility function is zero at some point, we obtain the existence of the weak solution.

Control of a reaction-diffusion system: application on a SEIR epidemic mode

In this work, we are interested in the study of a spatiotemporal SEIR epidemiological model, with no-flux boundary conditions. This model includes a constant inflow of new susceptible, exposed, infectious and recovered. In addition, it also incorporates a contact rate depending on the size of the population and another death related to the disease. Our objective is to characterize the optimal control pair, which minimizes exposed, infected individuals and the corresponding effort and treatment costs. We have demonstrated the existence of the state system solution and optimal control. The characterization of the optimal control pair is determined in terms of state functions and adjoint functions. The numerical resolution of the optimal system, has shown the effectiveness of our adopted strategy.

Fourier Spectral High-Order Time-Stepping Method for Numerical Simulation of the Multi-Dimensional Allen–Cahn Equations

This paper introduces the Fourier spectral method combined with the strongly stable exponential time difference method as an attractive and easy-to-implement alternative for the integration of the multi-dimensional Allen–Cahn equation with no-flux boundary conditions. The main advantages of the proposed method are that it utilizes the discrete fast Fourier transform, which ensures efficiency, allows an extension to two and three spatial dimensions in a similar fashion as one-dimensional problems, and deals with various boundary conditions. Several numerical experiments are carried out on multi-dimensional Allen–Cahn equations including a two-dimensional Allen–Cahn equation with a radially symmetric circular interface initial condition to demonstrate the fourth-order temporal accuracy and stability of the method. The numerical results show that the proposed method is fourth-order accurate in the time direction and is able to satisfy the discrete energy law.

Control of a reaction-diffusion system: application on a SEIR epidemic mode

In this work, we are interested in the study of a spatiotemporal SEIR epidemiological model, with no-flux boundary conditions. This model includes a constant inflow of new susceptible, exposed, infectious and recovered. In addition, it also incorporates a contact rate depending on the size of the population and another death related to the disease. Our objective is to characterize the optimal control pair, which minimizes exposed, infected individuals and the corresponding effort and treatment costs. We have demonstrated the existence of the state system solution and optimal control. The characterization of the optimal control pair is determined in terms of state functions and adjoint functions. The numerical resolution of the optimal system, has shown the effectiveness of our adopted strategy.

A Convergent Structure-Preserving Finite-Volume Scheme for the Shigesada--Kawasaki--Teramoto Population System

An implicit Euler finite-volume scheme for an n-species population cross-diffusion system of Shigesada-Kawasaki-Teramoto-type in a bounded domain with no-flux boundary conditions is proposed and analyzed. The scheme preserves the formal gradient-flow or entropy structure and preserves the nonnegativity of the population densities. The key idea is to consider a suitable mean of the mobilities in such a way that a discrete chain rule is fulfilled and a discrete analog of the entropy inequality holds. The existence of finite-volume solutions, the convergence of the scheme, and the large-time asymptotics to the constant steady state are proven. Furthermore, numerical experiments in one and two space dimensiona for two and three species are presented. The results are valid for a more general class of cross-diffusion systems satisfying some structural conditions.

A minimizing Movement approach to a class of scalar reaction–diffusion equations

The purpose of this paper is to introduce a Minimizing Movement approach to scalar reaction–diffusion equations of the form [see formula in PDF] with parameters Λ, Σ &gt; 0 and no-flux boundary condition [see formula in PDF] which is built on their gradient-flow-like structure in the space [see formula in PDF] of finite nonnegative Radon measures on [see formula in PDF], endowed with the recently introduced Hellinger-Kantorovich distance HKΛ,Σ. It is proved that, under natural general assumptions on [see formula in PDF] and [see formula in PDF], the Minimizing Movement scheme [see formula in PDF] for [see formula in PDF] yields weak solutions to the above equation as the discrete time step size τ ↓ 0. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance HKΛ,Σ, which will play an important role in this context, is established in the general setting of a separable Hilbert space; that result will constitute a starting point for the study of the differentiability of HKΛ,Σ along absolutely continuous curves which will be carried out in a subsequent paper.

Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation

Abstract The Keller-Segel-Stokes system (*) n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n ∇ c ) + ρ n − μ n α , c t + u ⋅ ∇ c = Δ c − c + n , u t = Δ u + ∇ P − n ∇ Λ , ∇ ⋅ u = 0 , $$\begin{eqnarray*} \left\{ \begin{array}{lcll} n_t + u\cdot\nabla n &amp;=&amp; \it\Delta n - \nabla \cdot (n\nabla c) + \rho n - \mu n^\alpha, \\[1mm] c_t + u\cdot\nabla c &amp;=&amp; \it\Delta c-c+n, \\[1mm] u_t &amp;=&amp; \it\Delta u + \nabla P - n\nabla \it\Lambda, \qquad \nabla\cdot u =0, \end{array} \right. \end{eqnarray*}$$ is considered in a bounded domain Ω ⊂ ℝ3 with smooth boundary, with parameters ρ ≥ 0, μ &gt; 0 and α &gt; 1, and with a given gravitational potential Λ ∈ W 2,∞(Ω). It is shown that in this general setting, when posed under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u, and for any suitably regular initial data, an associated initial value problem possesses at least one globally defined solution in an appropriate generalized sense. Since it is well-known that in the absence of absorption, already the corresponding fluid-free subsystem with u ≡ 0 and μ = 0 admits some solutions blowing up in finite time, this particularly indicates that any power-type superlinear degradation of the form in (*) goes along with some significant regularizing effect.