Equations Of Porous Medium Type Research Articles

Entropy solutions for time-fractional porous medium type equations

Entropy solutions for time-fractional porous medium type equations

A unified Bayesian inversion approach for a class of tumor growth models with different pressure laws

Abstract. In this paper, we use the Bayesian inversion approach to study the data assimilation problem for a family of tumor growth models described by porous-medium type equations. The models contain uncertain parameters and are indexed by a physical parameter m, which characterizes the constitutive relation between density and pressure. Based on these models, we employ the Bayesian inversion framework to infer parametric and nonparametric unknowns that affect tumor growth from noisy observations of tumor cell density. We establish the well-posedness and the stability theories for the Bayesian inversion problem and further prove the convergence of the posterior distribution in the so-called incompressible limit, m →∞. Since the posterior distribution across the index regime m ∈ [2,∞) can thus be treated in a unified manner, such theoretical results also guide the design of the numerical inference for the unknown. We propose a generic computational framework for such inverse problems, which consists of a typical sampling algorithm and an asymptotic preserving solver for the forward problem. With extensive numerical tests, we demonstrate that the proposed method achieves satisfactory accuracy in the Bayesian inference of the tumor growth models, which is uniform with respect to the constitutive relation.

Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium type equation

Classification of nonnegative traveling wave solutions for certain 1D degenerate parabolic equation and porous medium type equation

Interior Hölder continuity for singular-degenerate porous medium type equations with an application to a biofilm model

We show interior Hölder continuity for a class of quasi-linear degenerate reaction-diffusion equations. The diffusion coefficient in the equation has a porous medium type degeneracy and its primitive has a singularity. The reaction term is locally bounded except in zero. The class of equations we analyze is motivated by a model that describes the growth of biofilms. Our method is based on the original proof of interior Hölder continuity for the porous medium equation. We do not restrict ourselves to solutions that are limits in the weak topology of a sequence of approximate continuous solutions of regularized problems, which is a common assumption.

From random walks on networks to nonlinear diffusion.

Mathematical models of motility are often based on random-walk descriptions of discrete individuals that can move according to certain rules. It is usually the case that large masses concentrated in small regions of space have a great impact on the collective movement of the group. For this reason, many models in mathematical biology have incorporated crowding effects and managed to understand their implications. Here, we build on a previously developed framework for random walks on networks to show that in the continuum limit, the underlying stochastic process can be identified with a diffusion partial differential equation. The diffusion coefficient of the emerging equationis, in general, density dependent, and can be directly related to the transition probabilities of the random walk. Moreover, the relaxation time of the stochastic process is directly linked to the diffusion coefficient and also to the network structure, as it usually happens in the case of linear diffusion. As a specific example, we study the equivalent of a porous-medium-type equationon networks, which shows similar properties to its continuum equivalent. For this equation, self-similar solutions on a lattice and on homogeneous trees can be found, showing finite speed of propagation in contrast to commonly used linear diffusion equations. These findings also provide insights into reaction-diffusion systems with general diffusion operators, which have appeared recently in some applications.

On the Hölder regularity for obstacle problems to porous medium type equations

We show that signed weak solutions to parabolic obstacle problems with porous medium-type structure are locally Hölder continuous, provided that the obstacle is Hölder continuous.

Convergence rate for the incompressible limit of nonlinear diffusion–advection equations

The incompressible limit of nonlinear diffusion equations of porous medium type has attracted a lot of attention in recent years, due to its ability to link the weak formulation of cell-population models to free boundary problems of Hele-Shaw type. Although vast literature is available on this singular limit, little is known on the convergence rate of the solutions. In this work, we compute the convergence rate in a negative Sobolev norm and, upon interpolating with BV-uniform bounds, we deduce a convergence rate in appropriate Lebesgue spaces.

Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below

Given a complete, connected Riemannian manifold Mn with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium-type equation with respect to the 2-Wasserstein distance. We produce (sharp) stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm [35] and Otto-Westdickenberg [32]. The strategy of the proof mainly relies on a quantitative L1–L∞ smoothing property of the equation considered, combined with the Hamiltonian approach developed by Ambrosio, Mondino and Savaré in a metric-measure setting [4].

Existence, uniqueness and travelling waves to model an invasive specie interaction with heterogeneous reaction and non-linear diffusion

<abstract><p>It is the objective to provide a mathematical treatment of a model to predict the behaviour of an invasive specie proliferating in a domain, but with a certain hostile zone. The behaviour of the invasive is modelled in the frame of a non-linear diffusion (of Porous Medium type) equation with non-Lipschitz and heterogeneous reaction. First of all, the paper examines the existence and uniqueness of solutions together with a comparison principle. Once the regularity principles are shown, the solutions are studied within the Travelling Waves (TW) domain together with stability analysis in the frame of the Geometric Perturbation Theory (GPT). As a remarkable finding, the obtained TW profile follows a potential law in the stable connection that converges to the stationary solution. Such potential law suggests that the pressure induced by the invasive over the hostile area increases over time. Nonetheless, the finite speed, induced by the non-linear diffusion, slows down a possible violent invasion.</p></abstract>

Degenerate Cahn–Hilliard and incompressible limit of a Keller–Segel model

The Keller-Segel model is a well-known system representing chemotaxis in living organisms. We study the convergence of a generalized nonlinear variant of the Keller-Segel to the degenerate Cahn-Hilliard system. This analysis is made possible from the observation that the Keller-Segel system is equivalent to a relaxed version of the Cahn-Hilliard system. Furthermore, this latter equivalent system has an interesting application in the modelling of living tissues. Indeed, compressible and incompressible porous medium type equations are widely used to describe the mechanical properties of living tissues. The relaxed degenerate Cahn-Hilliard system, can be viewed as a compressible living tissue model for which the movement is driven by Darcy's law and takes into account the effects of the viscosity as well as surface tension at the surface of the tissue. We study the convergence of the Keller-Segel system to the Cahn-Hilliard equation and some of the analytical properties of the model such as the incompressible limit of our model. Our analysis relies on a priori estimates, compactness properties, and on the equivalence between the Keller-Segel system and the relaxed degenerate Cahn-Hilliard system.

Well-posedness of singular-degenerate porous medium type equations and application to biofilm models

We show the well-posedness for a large class of degenerate parabolic equations with an additional singularity and mixed Dirichlet-Neumann boundary conditions on bounded Lipschitz domains. The proof is based on an L1-contraction result. In addition, we analyze systems where degenerate equations are coupled to semilinear reaction diffusion equations. This setting includes mathematical models for biofilm growth which are the motivation for our analysis.

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.

Traveling waves to one-dimensional Cauchy problems for scalar parabolic-hyperbolic conservation laws

In this paper, we introduce traveling waves to one-dimensional Cauchy problems (CP) for scalar parabolic-hyperbolic conservation laws. The equation is regarded as a linear combination of the scalar hyperbolic conservation laws and the porous medium type equations. Thus, this equation has both properties of hyperbolic equations and those of parabolic equations. Accordingly, it is difficult to investigate the behavior of solutions to (CP). Therefore, it is necessary to construct particular solutions and investigate their properties. In pure hyperbolic case, Riemann solutions are well studied because they are self-similar solutions. However, it cannot be expected in this paper. Hence, we focus on the traveling wave structure instead of the self-similar structure.At first, we construct traveling waves to (CP) and investigate their properties. Next, we discuss the asymptotic behavior of entropy solutions to (CP) using the constructed traveling waves. Finally, we estimate the propagation speed of support for entropy solutions to (CP) using the modified traveling waves.

The obstacle problem for singular doubly nonlinear equations of porous medium type

The obstacle problem for singular doubly nonlinear equations of porous medium type

Higher integrability in the obstacle problem for the fast diffusion equation

We prove local higher integrability of the spatial gradient for solutions to obstacle problems of porous medium type in the fast diffusion case m<1. The result holds for the natural range of exponents that is known from other regularity results for porous medium type equations. We also cover the case of signed solutions.

Some recent advances in nonlinear diffusion on negatively-curved Riemannian manifolds: from barriers to smoothing effects

In this survey paper we discuss a series of recent results concerning nonnegative solutions to nonlinear diffusion equations of porous-medium type on Cartan–Hadamard manifolds, a special class of negatively-curved Riemannian manifolds that generalize the Euclidean space. We focus on sharp barrier estimates, asymptotic convergence and smoothing effects, describing quantitatively how the curvature behavior at infinity affects the way solutions depart from having a Euclidean-like structure.

The obstacle problem for degenerate doubly nonlinear equations of porous medium type

We prove the existence of nonnegative variational solutions to the obstacle problem associated with the degenerate doubly nonlinear equation ∂tb(u)-div(Df(Du))=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\partial _t b(u) - {{\\,\\mathrm{div}\\,}}(Df(Du)) = 0, \\end{aligned}$$\\end{document}where the nonlinearity b :mathbb {R}_{ge 0} rightarrow mathbb {R}_{ge 0} is increasing, piecewise C^1 and satisfies a polynomial growth condition. The prototype is b(u) := u^m with m in (0,1). Further, f :mathbb {R}^n rightarrow mathbb {R}_{ge 0} is convex and fulfills a standard p-growth condition. The proof relies on a nonlinear version of the method of minimizing movements.

Moving singularities for nonlinear diffusion equations in two space dimensions

We construct solutions with prescribed moving singularities for equations of porous medium type in two space dimensions. This complements a previous study of the problem where only dimensions higher than two were considered.

Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour

We consider reaction–diffusion equations of porous medium type, with different kind of reaction terms, and nonnegative bounded initial data. For all the reaction terms under consideration there are initial data for which the solution converges to 1 uniformly in compact sets for large times. We will characterize for which reaction terms this happens for all nontrivial nonnegative initial data, and for which ones there are also solutions converging uniformly to 0. Problems in this family have a unique (up to translations) travelling wave with a finite front and we will see how its speed gives the asymptotic velocity of all the solutions with compactly supported initial data. We will also prove in the one-dimensional case that solutions with bounded compactly supported initial data converging to 1 do so approaching a translation of this unique travelling wave. We will prove a similar result for non-compactly supported initial data in a certain class.

Large deviations for stochastic porous media equations

In this paper, we establish the Freidlin-Wentzell type large deviation principle for porous medium-type equations perturbed by small multiplicative noise. The porous medium operator $\Delta (|u|^{m-1}u)$ is allowed. Our proof is based on weak convergence approach.