Space–time chaos in a system of reaction–diffusion equations

We find conditions for the bifurcation of periodic spatially homogeneous and spatially inhomogeneous solutions of a three-dimensional system of nonlinear partial differential equations describing a soil aggregate model. We show that the transition to diffusion chaos in this model occurs via a subharmonic cascade of bifurcations of stable limit cycles in accordance with the universal Feigenbaum–Sharkovskii–Magnitskii bifurcation theory.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.     

On the Hölder regularity of the weak solution to a drift–diffusion system with pressure

In this paper we address the regularity issue of weak solution for the following linear drift–diffusion system with pressure

$$\begin{aligned} \partial _t u + b\cdot \nabla u -\Delta u + \nabla p = 0,\quad \mathrm {div}\,u=0,\quad u|_{t=0}(x)=u_0(x), \end{aligned}$$

where \(x\in \mathbb {R}^n\) and b is a given divergence-free vector field. Under some assumptions of the drift field b in the critical sense, and for the initial data \(u_0\in (L^2(\mathbb {R}^n))^n\), we prove that there exists a weak solution u(t) to this system such that u(t) for any time \(t>0\) is \(\alpha \)-Hölder continuous with \(\alpha \in (0,1)\). The proof of the Hölder regularity result utilizes a maximum-principle type method to improve the regularity of weak solution step by step.

     

The existence of global solution and “blow up” phenomenon for a system of multi dimensional symmetric regularized wave equations

The existence and uniqueness of the global smooth solution to the initial-boundary value problem of a system of multi-dimensions SRWE are proved. The sufficient conditions of blowing up of the solution are given.     

Invariant sets and the blow up threshold for coupled systems of reaction–diffusion

In this paper, we study a class of semilinear systems of reaction–diffusion on a bounded smooth domain with Dirichlet boundary condition. Applying the potential well method, we find invariant sets for the initial-boundary value problem and derive a threshold of blow up and global existence for its solution.     

Numeric solution of advection–diffusion equations by a discrete time random walk scheme

Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups, and discontinuities. Here we present an explicit numerical scheme for solving nonlinear advection–diffusion equations admitting shock solutions that is both easy to implement and stable. The numerical scheme is obtained by considering the continuum limit of a discrete time and space stochastic process for nonlinear advection–diffusion. The stochastic process is well posed and this guarantees the stability of the scheme. Several examples are provided to highlight the importance of the formulation of the stochastic process in obtaining a stable and accurate numerical scheme.     

Global existence to the initial–boundary value problem for a system of nonlinear diffusion and wave equations

We prove the global existence of weak solution pair to the initial boundary value problem for a system of m-Laplacian type diffusion equation and nonlinear wave equation. The interaction of two equations is given through nonlinear source terms $f(u,v)$ and $g(u,v)$.     

Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M _c such that if solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for . While characterising the possible infinite time blowing-up profile for M  =  M _c , we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two. This paper is under the Creative Commons licence Attribution-NonCommercial-ShareAlike 2.5.     

A numerical method for a system of singularly perturbed reaction–diffusion equations

A Dirichlet problem for a system of two coupled singularly perturbed reaction–diffusion ordinary differential equations is examined. A numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed. Numerical results are presented, which illustrate the theoretical results.     

Blowup of the solution to the Cauchy problem with arbitrary positive energy for a system of Klein–Gordon equations in the de Sitter metric

The ?⁴ model of a scalar (complex) field in the metric of an expanding universe, namely, in the de Sitter metric is considered. The initial energy of the system can have an arbitrarily high positive value. Sufficient conditions for solution blowup in a finite time are obtained. The existence of blowup is proved by applying H.A. Levine’s modified method is used.     

On the entropy method and exponential convergence to equilibrium for a recombination–drift–diffusion system with self-consistent potential

We consider a Shockley–Read–Hall recombination–drift–diffusion model coupled to Poisson’s equation and subject to boundary conditions, which imply conservation of the total charge. As main result, we derive an explicit functional inequality between relative entropy and entropy production rate, which implies exponential convergence to equilibrium with explicit constant and rate. We report that the key entropy–entropy production inequality ought rather not to be formulated on the states space of the parabolic–elliptic system, but on the reduced states space of the associated nonlocal drift–diffusion problem, where the Poisson equation is replaced by the corresponding Green function.     

Long-term coexistence for a competitive system of spatially varying gradient reaction–diffusion equations

Spatial distribution of interacting chemical or biological species is usually described by a system of reaction–diffusion equations. In this work we consider a system of two reaction–diffusion equations with spatially varying diffusion coefficients which are different for different species and with forcing terms which are the gradient of a spatially varying potential. Such a system describes two competing biological species. We are interested in the possibility of long-term coexistence of the species in a bounded domain. Such long-term coexistence may be associated either with a periodic in time solution (usually associated with a Hopf bifurcation), or with time-independent solutions. We prove that no periodic solution exists for the system. We also consider some steady states (the time-independent solutions) and examine their stability and bifurcations.     

Existence and stability of a stable stationary solution with a boundary layer for a system of reaction–diffusion equations with Neumann boundary conditions

Theoretical and Mathematical Physics - We consider an initial boundary value problem for a singularly perturbed parabolic system of two reaction–diffusion-type equations with Neumann...