Nonlinear Diffusion Term Research Articles

An inverse problem for a quasilinear convection–diffusion equation

We study the inverse problem of recovering a semilinear diffusion term a(t,λ) as well as a quasilinear convection term B(t,x,λ,ξ) in a nonlinear parabolic equation ∂tu−div(a(t,u)∇u)+B(t,x,u,∇u)⋅∇u=0,in(0,T)×Ω, given the knowledge of the flux of the moving quantity associated with different sources applied at the boundary of the domain. This inverse problem that is modeled by the solution dependent parameters a and B has many physical applications related to various classes of cooperative interactions or complex mixing in diffusion processes. Our main result states that, under suitable assumptions, it is possible to fully recover the nonlinear diffusion term a as well as the nonlinear convection term B. The recovery of the diffusion term is based on the idea of solutions to the linearized equation with singularities near the boundary ∂Ω. Our proof of the recovery of the convection term is based on the idea of higher order linearization to reduce the inverse problem to a density property for certain anisotropic products of solutions to the linearized equation. We show this density property by constructing sufficiently smooth geometric optic solutions concentrating on rays in Ω.

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, {hat{u}}(hat{{mathbf {x}}},{hat{t}}) that is coupled to the concentration of an immobile extracellular substrate, {hat{s}}(hat{{mathbf {x}}},{hat{t}}). Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to {hat{s}}, while a logistic growth term models cell proliferation. The extracellular substrate {hat{s}} is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, c = c_{mathrm{min}}, as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, c > c_{mathrm{min}}. We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub.

Weak pullback mean random attractors for the stochastic convective Brinkman–Forchheimer equations and locally monotone stochastic partial differential equations

This work is concerned about the asymptotic behavior of the solutions of the two- and three-dimensional stochastic convective Brinkman–Forchheimer (SCBF) equations [Formula: see text] [Formula: see text] driven by white noise with nonlinear diffusion terms (for some [Formula: see text]). We prove the existence and uniqueness of weak pullback mean random attractors for the 2D SCBF equations (for [Formula: see text]) as well as 3D SCBF equations (for [Formula: see text], any [Formula: see text] and for [Formula: see text], [Formula: see text]) in Bochner spaces, when the diffusion terms are Lipschitz nonlinear functions. Furthermore, we establish the existence of weak pullback mean random attractors for a class of locally monotone stochastic partial differential equations.

Computer-assisted proofs for some nonlinear diffusion problems

In the last three decades, powerful computer-assisted techniques have been developed in order to validate a posteriori numerical solutions of semilinear elliptic problems of the form Δu+f(u,∇u)=0. By studying a well chosen fixed point problem defined around the numerical solution, these techniques make it possible to prove the existence of a solution in an explicit (and usually small) neighborhood of the numerical solution. In this work, we develop a similar approach for a broader class of systems, including nonlinear diffusion terms of the form ΔΦ(u). In particular, this enables us to obtain new results about steady states of a cross-diffusion system from population dynamics: the (non-triangular) SKT model. We also revisit the idea of automatic differentiation in the context of computer-assisted proof, and propose an alternative approach based on differential–algebraic equations.

Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces

<p style='text-indent:20px;'>In this paper we discuss the weak pullback mean random attractors for stochastic Ginzburg-Landau equations defined in Bochner spaces. We prove the existence and uniqueness of weak pullback mean random attractors for the stochastic Ginzburg-Landau equations with nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic Ginzburg-Landau equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time.</p>

Limiting behavior of invariant measures of highly nonlinear stochastic retarded lattice systems

<p style='text-indent:20px;'>This paper deals with the limiting behavior of invariant measures of the highly nonlinear stochastic retarded lattice systems. Although invariant measures of stochastic retarded lattice system has been studied recently, there is so far no result of invariant measure of stochastic retarded lattice systems with highly nonlinear drift or diffusion terms. We first show the existence of invariant measures of the systems. We then prove that any limit point of a tight sequence of invariant measures of the stochastic retarded lattice systems must be an invariant measure of the corresponding limiting system as the intensity of noise converges or the time-delay approaches zero.</p>

Wong-Zakai approximations and pathwise dynamics of stochastic fractional lattice systems

<p style='text-indent:20px;'>This paper is concerned with the pathwise dynamics of stochastic fractional lattice systems driven by Wong-Zakai type approximation noises. The existence and uniqueness of pullback random attractor are established for the approximate system with a wide class of nonlinear diffusion term. For system with linear multiplicative noise and additive white noise, the upper semicontinuity of random attractors for the corresponding approximate system are also proved when the step size of the approximation approaches zero.</p>

On convergence of explicit finite volume scheme for one-dimensional three-component two-phase flow model in porous media

Abstract In this work, we develop and analyze an explicit finite volume scheme for a one-dimensional nonlinear, degenerate, convection–diffusion equation having application in petroleum reservoir. The main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term. We analyze a numerical scheme corresponding to explicit discretization of the diffusion term and a Godunov scheme for the advection term. L ∞ {L}^{\infty } stability under appropriate CFL conditions and BV {\rm{BV}} estimates are obtained. It is shown that the scheme satisfies a discrete maximum principle. Then we prove convergence of the approximate solution to the weak solution of the problem, and we mount convergence results to a weak solution of the problem in L 1 {L}^{1} . Results of numerical experiments are presented to validate the theoretical analysis.

A compositional Eulerian approach for modeling oil spills in the sea

We present a Compositional Eulerian model to forecast the evolution of oil spills in the sea. The model allows studying the fate of not only the oil concentration but also of each component (e.g., volatile, non-volatile, water in the oil). Therefore, the problem is formulated as a conservation equation for each component, plus an equation to estimate the age of the oil, which allows us to assess weathering processes (e.g., evaporation, natural dispersion, emulsion) and the associated changes in oil properties. We describe an efficient implementation, using second order numerical schemes for advection and nonlinear diffusion terms, to reduce numerical diffusion. We perform numerical experiments, based on real and synthetic cases, to illustrate and validate the capabilities of our model to forecast the evolution of oil spills and to perform environmental risk analysis in the case of a potential accident.

Long-time dynamics of fractional nonclassical diffusion equations with nonlinear colored noise and delay on unbounded domains

This paper deals with the long-time dynamics for a class of highly nonlinear fractional nonclassical diffusion equations with nonlinear colored noise and time delay defined on the whole space Rn. The existence and uniqueness of tempered pullback random attractors of the equations are established in C([−ρ,0],Hα(Rn)) (ρ>0 and α∈(0,1)) for polynomial growth drift and diffusion terms as well as Lipschitz time-delay terms. In a special case where the nonlinear diffusion term depends only on the space variable, the approximation of those random attractors is also investigated in C([−ρ,0],Hα(Rn)) when the correlation time of the colored approaches zero. The pullback asymptotical compactness of the solutions in C([−ρ,0],Hα(Rn)) is proved by virtue of the arguments of Arzela-Ascoli theorem, spectral decomposition as well as uniform tail-estimates developed by Wang (1999) [59] in order to surmount several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains as well as the weakly dissipative distinguishing structures of the equations.

Attractors for multi-valued lattice dynamical systems with nonlinear diffusion terms

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.

Stationary approximations of stochastic wave equations on unbounded domains with critical exponents

This paper is concerned with the stationary approximations of the stochastic wave equations defined on Rn with critical exponents. For a class of nonlinear diffusion terms, we prove the existence and uniqueness of tempered pullback attractors of the random wave equations driven by a stationary process as an approximation to the white noise. For a linear multiplicative noise, we prove the upper semicontinuity of these attractors as the step size of the approximations approaches zero. The asymptotic compactness of the solutions on Rn is established by the idea of energy equations and the uniform estimates on the tails of the solutions.

A Newton-Modified Weighted Arithmetic Mean Solution of Nonlinear Porous Medium Type Equations

The mathematical theory behind the porous medium type equation is well developed and produces many applications to the real world. The research and development of the fractional nonlinear porous medium models also progressed significantly in recent years. An efficient numerical method to solve porous medium models needs to be investigated so that the symmetry of the designed method can be extended to future fractional porous medium models. This paper contributes a new numerical method called Newton-Modified Weighted Arithmetic Mean (Newton-MOWAM). The solution of the porous medium type equation is approximated by using a finite difference method. Then, the Newton method is applied as a linearization approach to solving the system of nonlinear equations. As the system to be solved is large, high computational complexity is regulated by the MOWAM iterative method. Newton-MOWAM is formulated technically based on the matrix structure of the system. Some initial-boundary value problems with a different type of nonlinear diffusion term are presented. As a result, the Newton-MOWAM showed a significant improvement in the computation efficiency compared to the developed standard Weighted Arithmetic Mean iterative method. The analysis of efficiency, measured by the reduced number of iterations and computation time, is reported along with the convergence analysis.

Renormalization Group Approach to SDEs with Nonlinear Diffusion Terms

Renormalization Group Approach to SDEs with Nonlinear Diffusion Terms

Strong convergence of a GBM based tamed integrator for SDEs and an adaptive implementation

We introduce a tamed exponential time integrator which exploits linear terms in both the drift and diffusion for Stochastic Differential Equations (SDEs) with a one sided globally Lipschitz drift term. Strong convergence of the proposed scheme is proved, exploiting the boundedness of the geometric Brownian motion (GBM) and we establish order 1 convergence for linear diffusion terms. In our implementation we illustrate the efficiency of the proposed scheme compared to existing fixed step methods and utilize it in an adaptive time stepping scheme. Furthermore we extend the method to nonlinear diffusion terms and show it remains competitive. The efficiency of these GBM based approaches is illustrated by considering some well-known SDE models.

Theoretical and numerical analysis for a hybrid tumor model with diffusion depending on vasculature

In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of global in time weak-strong solutions using a regularization technique via an artificial diffusion in the ODE-system and a fixed point argument. In addition, stability results of the critical points are given under some constraints on parameters. Finally, we design a fully discrete finite element scheme for the model which preserves the pointwise and energy estimates of the continuous problem.

An Energy Stable Finite Element Scheme for the Three-Component Cahn–Hilliard-Type Model for Macromolecular Microsphere Composite Hydrogels

In this article, we present and analyze a finite element numerical scheme for a three-component macromolecular microsphere composite (MMC) hydrogel model, which takes the form of a ternary Cahn–Hilliard-type equation with Flory–Huggins–deGennes energy potential. The numerical approach is based on a convex–concave decomposition of the energy functional in multi-phase space, in which the logarithmic and the nonlinear surface diffusion terms are treated implicitly, while the concave expansive linear terms are explicitly updated. A mass lumped finite element spatial approximation is applied, to ensure the positivity of the phase variables. In turn, a positivity-preserving property can be theoretically justified for the proposed fully discrete numerical scheme. In addition, unconditional energy stability is established as well, which comes from the convexity analysis. Several numerical simulations are carried out to verify the accuracy and positivity-preserving property of the proposed scheme.

A benchmark study on the axial velocity profile of wave propagation in deformable blood vessels

Wave propagation models in the time domain have been extensively used in the available literature to study the flow characteristics in blood vessels. Most of the wave propagation models have considered flat or parabolic velocity profile functions to estimate the nonlinear convection and diffusion terms present in the conservation of momentum equation. There are only a few works available on the wave propagation analysis in which the velocity profile is approximated using different polynomial functions. In this study, a computationally efficient nonlinear axisymmetric formulation is presented without a priori assumed velocity profile function across the cross section to model the blood flow. Such a formulation in terms of axial velocity (u), pressure (p), and domain radius (R) facilitates the evolution/development of axial velocity profile as the flow progresses with time. The arterial mechanical behavior is modeled using a linear elastic constitutive relation. Partial differential equations are discretized using the finite element method and the Galerkin time integration technique in space and time domains, respectively. This study finds a phase difference between the shear stress at the wall and the flow rate. The flow characteristics and the velocity profile function are found to be in good agreement with the three-dimensional computational results available in the literature. The detailed investigation of the axial velocity across the cross section reveals neither flat nor parabolic profiles, as previously assumed in the literature.

On uniqueness and reconstruction of a nonlinear diffusion term in a parabolic equation

The problem of recovering coefficients in a diffusion equation is one of the basic inverse problems. Perhaps the most important term is the one that couples the length and time scales and is often referred to as the diffusion coefficient a in ut−∇(a∇u)=f. In this paper we seek the unknown a assuming that a=a(u) depends only on the value of the solution at a given point. Such diffusion models are the basic of a wide range of physical phenomena such as nonlinear heat conduction, chemical mixing and population dynamics. We shall look at two types of overposed data in order to effect recovery of a(u): the value of a time trace u(x0,t) for some fixed point x0 on the boundary of the region Ω; or the value of u on an interior curve Σ lying within Ω. As examples, these might represent a temperature measurement on the boundary or a census of the population in some subset of Ω taken at a fixed time T>0. In the latter case we shall show a uniqueness result that leads to a constructive method for recovery of a. Indeed, for both types of measured data we shall show reconstructions based on the iterative algorithms developed in the paper.

Analysis on Steady States of a Competition System with Nonlinear Diffusion Terms

Competition is a fundamental force shaping population size and structure as a result of limited availability of resources. In biomathematics, the biological models with competitive interactions exist widely. Furthermore, the nonlinear-diffusion (including self- and cross-diffusions) terms are incorporated to the biological models to better simulate the actual movement of species. Therefore, better compatibility with reality can be achieved by introducing nonlinear-diffusion into biological models with competitive interactions. As a result, a competition system with nonlinear-diffusion and nonlinear functional response is proposed and analyzed in this paper. We first briefly discuss the stability of trivial and semi-trivial solutions by spectrum analysis. Then the boundedness and the non-existence of steady states are studied. Based on the boundedness of the solutions, the existence of the steady states is also investigated by the fixed point index theory in a positive cone. The result shows that the two species can coexist when their diffusion and inter-specific competition pressures are controlled in a certain range.