Peter Knabner

Travelling waves in the transport of reactive solutes through porous media: Adsorption and binary ion exchange — Part 2

We study travelling wave solutions for the model developed in Part 1 of this paper. We develop and discuss a condition characterizing their existence. The possibility of finiteness is investigated. We consider the convergence to various limit cases and point out their different qualitative behaviour. Numerical examples are discussed.

Stabilization of ill-posed Cauchy problems for parabolic equations

SummaryIn this paper we study the noncharacteristic Cauchy problem, ut−(a(x)ux)x=0, x∈(0, l), t∈.(0, T], u(0, t)=ϕ(t), ux(0,t)=0, 0≦t≦T, assuming only L∞ for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.SuntoNel presente lavoro consideriamo il problema di Cauchy non ben posto ut= (a(x)ux)x, x∈(0, l), t∈(0, T), u(0, t)=ϕ(t), ux(0, t)=0, 0≦t≦T. Supponiamo che a sia misurabile e limitato inferiormente e superiormente da constanti positive. Introduciamo delle limitazioni a priori su u e dimostriamo la dipendenza continua di u rispetto al dato ϕ sia in (0, l)×(0, T) (di tipo hölderiano) sia per x=l (di tipo logaritmico). Consideriamo, inoltre, la dipendenza continua di u da a.

Travelling waves during the transport of reactive solute in porous media: combination of Langmuir and Freundlich isotherms

Recently, it has been shown that in the case of nonlinear solute adsorption the displacement may be in the form of a travelling wave. In this paper, we investigate whether a travelling wave type of behaviour can be expected when two different types of sorption sites can be distinguished with different isotherms and kinetics. Illustrations are given for cases where the overall isotherm comprises two contributions that follow the Langmuir and the Freundlich equations, respectively. Boundary conditions are chosen that ensure a decrease in concentration in the direction of flow. Depending on the value of the Freundlich power (p) the travelling wave may exist. For p = 1, the travelling wave always exists, whereas for 1 <p = 2 it depends on the values of the other adsorption parameters and whether a lower bound of the upstream concentration (at x = -8) is exceeded. For p = 2, the existence of the travelling wave requires that the upstream concentration does not exceed an (specified) upper bound. Besides illustrating some waves we show that two different rate functions that have the Freundlich isotherm as their limit for an infinite rate parameter result in qualitatively different travelling waves.

A transport model with micro- and macro-structure☆

Abstract In this paper we consider solute transport through porous media where the chemical species undergoes a chemical process through the surface of the porous skeleton. The problem is modeled by a system of differential equations for the macro-concentration u = u ( x , t ) and the micro-concentration u ′ = u ′( x , x ′, t ). We prove existence and uniqueness, and some properties of the set with positive concentration.

STABILITY ESTIMATES FOR ILL-POSED CAUCHY PROBLEMS FOR PARABOLIC EQUATIONS

Publisher Summary This chapter discusses stability estimates for ill-posed Cauchy problems for parabolic equations and focuses on the question of stability estimates. There is already an extensive literature about this problem, proving estimates of Holder type for x < l and of logarithmic type for x = l, but there are results lacking, showing the exact dependence of the Holder exponent on x and/or dealing with irregular coefficients. The chapter focuses on these questions for those semi-infinite cases, where Fourier transform techniques are applicable. The chapter also discusses a general linear parabolic equation with smooth space-dependent coefficients.

Control of stefan problems by means of linear-quadratic defect minimization

SummaryWe investigate the following problem: To influence a heat conduction process in such a way that the conductor melts in a prescribed manner. Since we treat a linear auxiliary problem, it suffices to deal with a linear-quadratic defect minimization problem with linear restrictions, where we use splines or polynomials as approximation spaces. In case of exact controllability we derive various order of convergence estimates, which we discuss for some numerical examples.

Enhanced Leaching of Organic Chemicals in Soils Due to Binding to Dissolved Organic Carbon

Sorption of organic chemicals in soils is considered here as an equilibrium of environmental chemicals in three phases: dissolved, sorbed to dissolved macromolecules, and sorbed to the bulk soil matrix. In the present study we demonstrate the binding of organic chemicals to water-soluble soil organic matter using an experimental acidic herbicide as an example. Sorption isotherms are determined for the sorption of organic chemicals to bulk soil materials (Ap horizons) and to water-soluble organic matter. From these data an effective equilibrium sorption isotherm to bulk soil can be calculated which takes into account the sorption of organic chemicals to dissolved organic carbon. This is compared to a more traditional two-phase model of the same type, where the mobile sorbent is ignored.

Regularization of the cauchy problem for the heat equation by norm bounds

We consider the non-characteristic Cauchy problem for the heat equation in a non-cylindrical domain. We impose norm bounds on the unknown boundary flux and study the amount of regularization by estimating the module of continuity. In addition to interior Holder estimates for temperature and gradient we also get uniform estimates of logarithmic type.

Regularizing the Cauchy Problem for the Heat Equation by Sign Restrictions

We consider the non-characteristic Cauchy problem for the heat equation. For certain a priori information about the solution the amount of regularization, i.e. the behaviour of the module of continuity is investigated. Especially for sign restrictions continuous dependence of Holder type in the interior is shown. This sharpens a result of Pucci (1959), who did not succeed in estimating the module of continuity. This estimate leads to some conjectures, which fit quite well within the results of numerical case studies.

Global Existence in a General Stefan-like Problem

Abstract We consider a one-dimensional two-phase Stefan-like free boundary problem for the heat equation with nonlinear Neumann boundary conditions. Local existence has been shown in 8, furthermore it is clear that cases exist, where global existence fails. The question arises, under which conditions the solution exists globally. In this paper it is shown that especially the natural sign conditions are sufficient for global existence.