No-flux Boundary Conditions Research Articles

Bifurcation and stability of a two-species reaction–diffusion–advection competition model

This paper is concerned with the dynamics of a two-species reaction–diffusion–advection competition model subject to the no-flux boundary condition in a bounded domain. By the signs of the associated principal eigenvalues, we derive the existence and local stability of the trivial and semi-trivial steady-state solutions. Moreover, the nonexistence and existence of the coexistence steady-state solutions stemming from the two boundary steady states are obtained as well. In particular, we describe the feature of the coincidence of bifurcating coexistence steady-state solution branches. At the same time, the effect of advection on the stability of the bifurcating solution is also investigated, and our results suggest that the advection term may change the stability. Finally, we point out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov–Schmidt reduction, and bifurcation theory.

Can simultaneous density-determined enhancement of diffusion and cross-diffusion foster boundedness in Keller–Segel type systems involving signal-dependent motilities?

The reaction-(cross-)diffusion system is considered under no-flux boundary conditions in smoothly bounded convex domains , where m ⩾ 1 and n ⩾ 2, and where ϕ generalizes the prototype obtained on letting with a ⩾ 0, b > 0, d ⩾ 0 and α ⩾ 0. In this framework, it is firstly seen that if then finite-time blow-up is excluded in the sense that for all suitably regular initial data an associated initial-boundary value problem admits a globally defined weak solution (u, v) with u being locally bounded in. Under the assumption that additionally these solutions are moreover shown to be bounded throughout Ω × (0, ∞) in both components. In view of results known for the case m = 1, this particularly indicates that increasing m in () goes along with a certain regularizing effect despite the circumstance that thereby both the diffusion and the cross-diffusion mechanisms implicitly contained in () are simultaneously enhanced.

Steady states of thin film droplets on chemically heterogeneous substrates

Abstract We study steady-state thin films on chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1D steady-state solutions that exist on such substrates into six different branches and develop asymptotic estimates for the steady states on each branch. Using perturbation expansions, we show that leading-order solutions provide good predictions of the steady-state thin films on stepwise-patterned substrates. We show how the analysis in one dimension can be extended to axisymmetric solutions. We also examine the influence of the wettability contrast of the substrate pattern on the linear stability of droplets and the time evolution for dewetting on small domains. Results are also applied to describe 2D droplets on hydrophilic square patches and striped regions used in microfluidic applications.

Long-term behaviour in a parabolic–elliptic chemotaxis–consumption model

Global existence and boundedness of classical solutions of the chemotaxis–consumption systemnt=Δn−∇⋅(n∇c),0=Δc−nc, under no-flux boundary conditions for n and Robin-type boundary conditions∂νc=(γ−c)g for c (with γ>0 and C1+β(∂Ω)∋g>0 for some β∈(0,1)) are established in bounded domains Ω⊂RN, N≥1. Under a smallness condition on γ, moreover, we show convergence to the stationary solution.

A structure-preserving discontinuous Galerkin scheme for the Fisher\u2013KPP equation

An implicit Euler discontinuous Galerkin scheme for the Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation for population densities with no-flux boundary conditions is suggested and analyzed. Using an exponential variable transformation, the numerical scheme automatically preserves the positivity of the discrete solution. A discrete entropy inequality is derived, and the exponential time decay of the discrete density to the stable steady state in the L^1 norm is proved if the initial entropy is smaller than the measure of the domain. The discrete solution is proved to converge in the L^2 norm to the unique strong solution to the time-discrete Fisher–KPP equation as the mesh size tends to zero. Numerical experiments in one space dimension illustrate the theoretical results.

Synchronization features of target wave structures with an incoherent center

Abstract In the present paper we study numerically the interaction of two 2-dimensional networks each representing a lattice of coupled van der Pol oscillators. The isolated lattices can demonstrate spiral and target wave structures for the cases of no-flux boundary conditions and random initial conditions. We explore the effect of mutual synchronization of target waves and target wave chimeras for dissipative and inertial inter-layer coupling between the lattices and for local and nonlocal intra-layer coupling between the lattice oscillators. We show for the first time the possibility of complete synchronization of target wave chimera states, including synchronization of incoherent cores. Our findings demonstrate that there is a difference with synchronization of spiral wave chimera states in which incoherent cores are not completely synchronized. The obtained results can have an important application and can be particularly used to study the dynamics of cardiac muscle in biophysics.

Connectivity of fractures and groundwater flows analyses into the Western Andean Front by means of a topological approach (Aconcagua Basin, Central Chile)

The misunderstanding of hydrogeological processes together with the oversimplification of aquifer conceptual models result in numerous inaccuracies in the management of groundwater resources. In Central Chile (32–36°S), hydrogeological studies have exclusively focused to alluvial aquifers in valleys (~15% of total area) and mountain-front zones remain considered as no-flux boundary conditions. By a topological approach and an analysis of fractures, the hydrogeological potential of the Western Andean Front along the N–S-oriented Pocuro Fault Zone (PFZ) in the Aconcagua Basin were determined. Perennial springs (23) show evidence of groundwater flows into the fractured Principal Cordillera. Topology allows for quantification of the density of connected fractures within the fault zone and its relationship with groundwater circulation. The study results highlight two areas where the density of fractures and connected nodes (Nc) is high (>2.4 km/km2, 2.5 Nc/km2). Both areas are topologically related to the main springs of the PFZ: Termas de Jahuel (discharge ~14.0 m3/h at 22 °C) and Termas El Corazon (discharge ~7.2 m3/h at 20 °C). Outcrop-scale mapping reveals that groundwater outflows from NW–SE fractures, which is consistent with the preferential orientation of the fracture network (N30–60 W) within the PFZ. The results indicate that oblique basement faults are discrete high-permeability structures conducting groundwater across the Western Andean Front from the Principal Cordillera up to adjacent alluvial aquifers (focused recharge). Therefore, the simplistic hydrogeological view of the Western Andean Front (i.e. impervious limit) is partially erroneous.

Global solvability and eventual smoothness in a chemotaxis-fluid system with weak logistic-type degradation

We consider the coupled chemotaxis–Navier–Stokes system with logistic source term [Formula: see text] in a bounded, smooth domain [Formula: see text], where [Formula: see text] and where [Formula: see text], [Formula: see text] and [Formula: see text] are given parameters. Although the degradation here is weaker than the usual quadratic case, it is proved that for any sufficiently regular initial data, the initial-value problem for this system under no-flux boundary conditions for [Formula: see text] and [Formula: see text] and homogeneous Dirichlet boundary condition for [Formula: see text] possesses at least one globally defined weak solution. And this weak solution becomes smooth after some waiting time provided [Formula: see text].

Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction

<p style='text-indent:20px;'>We consider the haptotaxis system <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{lcl} u_t &amp; = &amp; \Delta u - \nabla \cdot (u\nabla v), \\ v_t &amp; = &amp; - (u+w)v, \\ w_t &amp; = &amp; D_w \Delta w - w + uz, \\ z_t &amp; = &amp; D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>which arises as a simplified version of a recently proposed model for oncolytic virotherapy. When posed under no-flux boundary conditions in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document}</tex-math></inline-formula>, with positive parameters <inline-formula><tex-math id="M2">\begin{document}$ D_w $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ D_z $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>, and along with initial conditions involving suitably regular data, this system is known to admit global classical solutions. <p style='text-indent:20px;'>It is shown that with respect to infinite-time blow-up, this system exhibits a critical mass phenomenon related to the quantity <inline-formula><tex-math id="M5">\begin{document}$ m_c: = \frac{1}{(\beta-1)_+} $\end{document}</tex-math></inline-formula>: In fact, it is seen that each solution fulfilling <inline-formula><tex-math id="M6">\begin{document}$ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) &gt; m_c $\end{document}</tex-math></inline-formula> must be unbounded, and this is complemented by a boundedness result which inter alia asserts that for any choice of <inline-formula><tex-math id="M7">\begin{document}$ m&lt;m_c $\end{document}</tex-math></inline-formula> one can find a nontrivial set of solutions, particularly containing spatially heterogeneous solutions, each of which is bounded though satisfying <inline-formula><tex-math id="M8">\begin{document}$ \frac{1}{|\Omega|} \int_\Omega u(\cdot,0) = m $\end{document}</tex-math></inline-formula>.

Global existence for a kinetic model of pattern formation with density-suppressed motilities

In this paper, we consider global existence of classical solutions to the following kinetic model of pattern formation(0.1){ut=Δ(γ(v)u)+μu(1−u)−Δv+v=u in a smooth bounded domain Ω⊂Rn, n≥1 with no-flux boundary conditions. Here, μ≥0 is any given constant. The function γ(⋅) represents a signal-dependent diffusion motility and is decreasing in v which models a density-suppressed motility in the process of stripe pattern formation through the self-trapping mechanism [9,20].The major difficulty in analysis lies in the possible degeneracy of diffusion as v↗+∞. In the present contribution, based on a subtle observation of the nonlinear structure, we develop a new method to rule out finite-time degeneracy in any spatial dimension for all smooth motility functions satisfying γ(v)>0 and γ′(v)≤0 for v≥0. Then we prove global existence of classical solution for (0.1) in the two-dimensional setting with any μ≥0. Moreover, the global solution is proven to be uniform-in-time bounded if either 1/γ satisfies certain polynomial growth condition or μ>0.Besides, we pay particular attention to the specific case γ(v)=e−v with μ=0. Under the circumstances, system (0.1) becomes of great interest because it shares the same set of equilibria as well as the Lyapunov functional with the classical Keller–Segel model. A novel critical phenomenon in the two-dimensional setting is observed that with any initial datum of sub-critical mass, the global solution is proved to be uniform-in-time bounded, while with certain initial datum of super-critical mass, the global solution will become unbounded as time goes to infinity. Namely, blowup takes place in infinite time rather than finite time in our model which is distinct from the well-known fact that certain initial data of super-critical mass will enforce a finite-time blowup for the classical Keller–Segel system.

Transformation and Upwelling of Bottom Water in Fracture Zone Valleys

AbstractClosing the overturning circulation of bottom water requires abyssal transformation to lighter densities and upwelling. Where and how buoyancy is gained and water is transported upward remain topics of debate, not least because the available observations generally show downward-increasing turbulence levels in the abyss, apparently implying mean vertical turbulent buoyancy-flux divergence (densification). Here, we synthesize available observations indicating that bottom water is made less dense and upwelled in fracture zone valleys on the flanks of slow-spreading midocean ridges, which cover more than one-half of the seafloor area in some regions. The fracture zones are filled almost completely with water flowing up-valley and gaining buoyancy. Locally, valley water is transformed to lighter densities both in thin boundary layers that are in contact with the seafloor, where the buoyancy flux must vanish to match the no-flux boundary condition, and in thicker layers associated with downward-decreasing turbulence levels below interior maxima associated with hydraulic overflows and critical-layer interactions. Integrated across the valley, the turbulent buoyancy fluxes show maxima near the sidewall crests, consistent with net convergence below, with little sensitivity of this pattern to the vertical structure of the turbulence profiles, which implies that buoyancy flux convergence in the layers with downward-decreasing turbulence levels dominates over the divergence elsewhere, accounting for the net transformation to lighter densities in fracture zone valleys. We conclude that fracture zone topography likely exerts a controlling influence on the transformation and upwelling of bottom water in many areas of the global ocean.

Boundedness and global stability of a diffusive prey–predator model with prey-taxis

ABSTRACT A diffusive prey–predator model with prey-taxis and trophic interactions of three levels is proposed and analyzed, in which the predator could move in the direction of prey gradient. We prove the global existence and boundedness of solutions of the system on bounded domains with no-flux boundary condition. The result holds for arbitrary spatial dimension and any prey-taxis sensitivity coefficient. Moreover, we also prove that such prey-taxis does not qualitatively affect the existence and stability of nonnegative equilibria under some conditions.

Evolution of dispersal in advective homogeneous environments

The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been investigated. In contrast, the role of intermediate advection still remains poorly understood. This paper is devoted to studying a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates and advection rates. Zhou (P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp) considered the system under the no-flux boundary conditions. It is pointed that, in this paper, we focus on the case where the upstream end has the Neumann boundary condition and the downstream end has the hostile condition. By employing a new approach, we firstly determine necessary and sufficient conditions for the persistence of the corresponding single species model, in forms of the critical diffusion rate and critical advection rate. Furthermore, for the two-species model, we find that (i) the strategy of slower diffusion together with faster advection is always favorable; (ii) two species will also coexist when the faster advection with appropriate faster diffusion.

Some global dynamics of a Lotka-Volterra competition-diffusion-advection system

This paper studies some population dynamics of a diffusive Lotka-Volterra competition advection model under no-flux boundary condition. We establish the main results that determine the stability of semi-trivial steady states.

An immersed boundary method for simulations of flow and mixing in micro-channels with electro kinetic effects

Immersed boundary method (IBM) is a methodology in which any real stationary or moving complex bodies in flow or heat transfer or mixing field is treated by distinct forcing functions introduced in the respective governing equations, thereby its presence is dealt with much easiness in the computational domain. This work proposes an extension of an IBM based on discrete forcing approach to simulate pressure driven electro-osmotic flow and mixing enhancement in constricted micro channels. The fractional-step based temporal discretisation and finite-volume basedspatial discretisation on a staggered mesh are used to solve the governing equations. Concentration forcing function is a parameter, which is newly defined in this model to satisfy the no-flux boundary condition for species concentration on the IB and is introduced in convection-diffusion equation. The present computational model is validated with the published experimental and numerical results in the literature and finally some sample results are also exemplified.

Cross-diffusion systems and fast-reaction limits

The rigorous asymptotics from reaction-cross-diffusion systems for three species with known entropy to cross-diffusion systems for two variables is investigated. The equations are studied in a bounded domain with no-flux boundary conditions. The global existence of very weak (integrable) solutions and the rigorous fast-reaction limit are proved. The limiting system inherits the entropy structure with an entropy that is not the sum of the entropies of the components. Uniform estimates are derived from the entropy inequality and the duality method.

Does repulsion-type directional preference in chemotactic migration continue to regularize Keller–Segel systems when coupled to the Navier–Stokes equations?

The repulsive Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{llll} n_t + u\cdot \nabla n &{}=&{} \Delta n + \nabla \cdot (n\nabla c), &{}\quad x\in \Omega , \; t>0, \\ c_t + u\cdot \nabla c &{}=&{} \Delta c -c+n, &{}\quad x\in \Omega , \; t>0, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \quad \nabla \cdot u=0, &{}\quad x\in \Omega , \; t>0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$is considered in smoothly bounded planar domains, where $$\Phi \in W^{2,\infty }(\Omega )$$ is given. It is well-known that the corresponding fluid-free analogue, when posed under homogeneous no-flux boundary conditions, admits global classical solutions for arbitrarily large initial data, thus substantially differing from the classical two-dimensional Keller–Segel system featuring chemoattraction-driven finite-time blow-up for some initial data. The literature on such chemorepulsion systems, however, strongly relies on the presence of an associated energy structure which is apparently destroyed by the fluid interaction mechanism in ($$\star $$). By making use of appropriate functional inequalities involving certain logarithmic expressions arising due to the planarity of the considered setting, it is shown that nevertheless an initial-boundary value problem for ($$\star $$) admits globally defined classical solutions for all reasonably regular initial data.

Stationary solutions to a chemotaxis-consumption model with realistic boundary conditions for the oxygen

Previous studies of chemotaxis models with consumption of the chemoattractant (with or without fluid) have not been successful in explaining pattern formation even in the simplest form of concentration near the boundary, which had been experimentally observed. Following the suggestions that the main reason for that is the usage of inappropriate boundary conditions, in this paper we study the solutions to the stationary chemotaxis system [Formula: see text] in bounded domains [Formula: see text], [Formula: see text], under the no-flux boundary conditions for [Formula: see text] and the physically meaningful condition [Formula: see text] on [Formula: see text], with the given parameter [Formula: see text] and [Formula: see text], [Formula: see text], satisfying [Formula: see text], [Formula: see text] on [Formula: see text]. We prove the existence and uniqueness of solutions for any given mass [Formula: see text]. These solutions are nonconstant.

On a Keller–Segel–Stokes system with logistic type growth: blow-up prevention enforced by sublinear signal production

In this paper, the coupled chemotaxis–fluid system $$\begin{aligned} \left\{ \begin{array}{lll} n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (n\nabla c)+rn-\mu n^2,\ \ \ &{}x\in \Omega ,\ t>0,\\ c_t+u\cdot \nabla c=\Delta c-c+f(n),\ \ &{}x\in \Omega ,\ t>0,\\ u_t=\Delta u+\nabla P+n\nabla \phi ,\ \ &{}x\in \Omega ,\ t>0,\\ \nabla \cdot u=0,\ \ &{}x\in \Omega ,\ t>0\\ \end{array}\right. \end{aligned}$$ is considered associated with no-flux boundary conditions in a smooth bounded domain $$\Omega \subset \mathbb {R}^3$$ , where $$r\ge 0$$ and $$\mu >0$$ are given parameters. $$f\in C^1([0, \infty ))$$ is a given function satisfying $$f(s)\le Ks^{\alpha }$$ with $$K>0, \alpha >0$$ for all $$s>0$$ and $$f(0)\ge 0$$ . It is proved that whenever $$\begin{aligned} \frac{1}{2}<\alpha <1, \end{aligned}$$ for all $$r\ge 0$$ and $$\mu >0$$ , there exists at least a global classical solution that is uniformly bounded, which exhibits the mild saturation effect of signal production.

Global renormalized solutions to reaction-cross-diffusion systems with self-diffusion

The global-in-time existence of renormalized solutions to reaction-cross-diffusion systems for an arbitrary number of variables in bounded domains with no-flux boundary conditions is proved. The cross-diffusion part describes the segregation of population species and is a generalization of the Shigesada-Kawasaki-Teramoto model. The diffusion matrix is not diagonal and generally neither symmetric nor positive semi-definite, but the diffusion operator possesses a formal gradient-flow or entropy structure. The reaction part includes reversible reactions of mass-action kinetics and does not obey any growth condition. The existence result generalizes both the condition on the reaction part required in the boundedness-by-entropy method and the proof of J. Fischer for reaction-diffusion systems with diagonal diffusion matrices. Moreover, we do not assume local Lipschitz continuity of the reaction terms but only continuity.