Qualitative mathematical analysis of the Richards equation

The Richards equation is widely used as a model for the flow of water in unsaturated soils. For modelling one-dimensional flow in a homogeneous soil, this equation can be cast in the form of a specific nonlinear partial differential equation with a time derivative and one spatial derivative. This paper is a survey of recent progress in the pure mathematical analysis of this last equation. The emphasis is on the interpretation of the results of the analysis. These are explained in terms of the qualitative behaviour of the flow of water in an unsaturated soil which is described by the Richards equation.Nomenclature a coefficient in second-order diffusion term of equation - b coefficient in first-order advection term of equation - D soil-moisture diffusivity [L²T^-1] - h pressure head [L] - H quarter-plane domain for Cauchy-Dirichlet problem [L] x [T] - K hydraulic conductivity scalar [LT^–1] - K hydraulic conductivity tensor [LT^–1] - q soil-moisture flux scalar [LT^–1] - q soil-moisture flux vector [LT^–1] - r dummy variable - R rectangle [L] x [T] - s dummy variable - s* representative value of dummy variable - S half-plane domain for Cauchy problem [L] x [T] - t time [T] - u unknown solution of partial differential equation - u₀ initial-value function - v soil-moisture velocity scalar [LT^–1] - v soil-moisture velocity vector [LT^–1]     

Upscaling in Subsurface Transport Using Cluster Statistics of Percolation

Transport/flow problems in soils have been treated in random resistor network representations (RRNs). Two lines of argument can be used to justify such a representation. Solute transport at the pore-space level may probably be treated using a system of linear, first-order differential equations describing inter-pore probability fluxes. This equation is equivalent to a random impedance network representation. Alternatively, Darcys law with spatially variable hydraulic conductivity is equivalent to an RRN. Darcys law for the hydraulic conductivity is applicable at sufficiently low pressure head in saturated soils, but only for steady-state flow in unsaturated soils. The result given here will have two contributions, one of which is universal to any linear conductance problem, i.e., requires only the applicability of Darcys (or Ohms) law. The second contribution depends on the actual distribution of linear conductances appropriate. Although nonlinear effects in RRNs (including changes in resistance values resulting from current, analogous to changes in matric potential resulting from flow) have been treated within the framework of percolation theory, the theoretical development lags the corresponding development of the linear theory, which is, in principle, on a solid foundation. In practice, calculations of the nonlinear conductivity in relatively (compared with soils) well characterized solid-state systems such as amorphous or impure semiconductors, do not agree with each other or with experiment. In semiconductors, however, experiments do at least appear consistent with each other.In the limit of infinite system size the transport properties of a sufficiently inhomogeneous medium are best calculated through application of critical rate analysis with the system resistivity related to the critical (percolating) resistance value, R_c. Here well-known cluster statistics of percolation theory are used to derive the variability, W (R,x) in the smallest maximal resistance, R of a path spanning a volume x³ as well as to find the dependence of the mean value of the conductivity, (x). The functional form of the cluster statistics is a product of a power of cluster size, and a scaling function, either exponential or Gaussian, but which, in either case, cuts off cluster sizes at a finite value for any maximal resistance other than R_c. Either form leads to a maximum in W (R,x) at R=R_c. When the exponential form of the cluster statistics is used, and when individual resistors are exponential functions of random variables (as in stochastic treatments of the unsaturated zone by the McLaughlin group [see Graham and MacLaughlin (1991), or the series of papers by Yeh et al. (1985, 1995), etc.], or as is known for hopping conduction in condensed matter physics), then W (R,x) has a power law decay in R/R_c (or R_c/R, the power being an increasing function of x. If the statistics of the individual resistors are given by power law functions of random variables (as in Poiseiulles Law), then an exponential decay in R for W (R,x) is obtained with decay constant an increasing function of x. Use, instead, of the Gaussian cluster statistics alters the case of power law decay in R to an approximate power, with the value of the power a function of both R and x.     

Global Nonlinear Stability for Steady Ideal Fluid Flow in Bounded Planar Domains

We prove stability of steady flows of an ideal fluid in a bounded, simply connected, planar region, that are strict maximisers or minimisers of kinetic energy on an isovortical surface. The proof uses conservation of energy and transport of vorticity for solutions of the vorticity equation with initial data in L^p for p>4/3. A related stability theorem using conservation of angular momentum in a circular domain is also proved.     

An Eigenvalue Criterion for Stability of a Steady Navier–Stokes Flow in {\mathbb{R}}^3

We study the resolvent equation associated with a linear operator L{\mathcal{L}} arising from the linearized equation for perturbations of a steady Navier–Stokes flow U^*{\mathbf{U^*}}. We derive estimates which, together with a stability criterion from [33], show that the stability of U^*{\mathbf{U^*}} (in the L²-norm) depends only on the position of the eigenvalues of L{\mathcal{L}}, regardless the presence of the essential spectrum.     

Lp-approach to steady flows of viscous compressible fluids in exterior domains

We investigate steady compressible flows in three-dimensional exterior domains for small data and for both zero and nonzero (but constant) velocity at infinity. We prove existence and uniqueness of solutions in L ^p -spaces, p>3, and study their regularity as well as their decay at infinity.     

Stochastic Analysis of Macrodispersion in a Semi‐Confined Aquifer

In this paper, the macroscopic dispersion resulting from one and twodimensional flows through a semiconfined aquifer with spatially variable hydraulic conductivity K which is represented by a stationary (statistically homogeneous) random process is analyzed using the spectral representation technique. Stochastic fluctuation equations of the steady flow and solute transport are solved to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of the leakage factor and input covariances describing the hydraulic conductivity in a semiconfined aquifer bounded by a leaky layer above and an impervious stratum below. The macrodispersivity tensor is studied using some convenient forms of the log hydraulic conductivity process. The sensitivity of the resulting macrodispersivity to the input covariances is discussed along with the influence of the leakage factor for both one and twodimensional flows. It is found that the longitudinal macrodispersivities are increased due to the presence of leakage, while the transverse macrodispersivities are reduced due to leakage.     

A weighted least squares method for first-order hyperbolic systems

The paper presents a generalization of the classical L²-norm weighted least squares method for the numerical solution of a first-order hyperbolic system. This alternative least squares method consists of the minimization of the weighted sum of the L² residuals for each equation of the system. The order of accuracy of global conservation of each equation of the system is shown to be inversely proportional to the weight associated with the equation. The optimal relative weights between the equations are then determined in order to satisfy global conservation of the energy of the physical system. As an application of the algorithm, the shallow water equations on an irregular domain are first discretized in time and then solved using Laplace modification and the proposed least squares method.     

Global linear stability analysis of time‐averaged flows

The global linear stability analysis (LSA) of stationary/steady flows has been applied to various flows in the past and is fairly well understood. The LSA of time‐averaged flows is explored in this paper. It is shown that the LSA of time‐averaged flows can result in useful information regarding its stability. The method is applied to study flow past a cylinder at Reynolds number (Re) beyond the onset of vortex shedding. Compared with the direct numerical simulation, LSA of the Re=100 steady flow severely underpredicts the vortex shedding frequency. However, the LSA of the time‐averaged flow results in the correct value of the non‐dimensional frequency, St, of the associated instability. Copyright © 2007 John Wiley & Sons, Ltd.     

Fractional Dispersion,Lévy Motion,and the MADE Tracer Tests

The macrodispersion experiments (MADE) at the Columbus Air Force Base in Mississippi were conducted in a highly heterogeneous aquifer that violates the basic assumptions of local second-order theories. A governing equation that describes particles that undergo Lévy motion, rather than Brownian motion, readily describes the highly skewed and heavy-tailed plume development at the MADE site. The new governing equation is based on a fractional, rather than integer, order of differentiation. This order (), based on MADE plume measurements, is approximately 1.1. The hydraulic conductivity (K) increments also follow a power law of order =1.1. We conjecture that the heavy-tailed K distribution gives rise to a heavy-tailed velocity field that directly implies the fractional-order governing equation derived herein. Simple arguments lead to accurate estimates of the velocity and dispersion constants based only on the aquifer hydraulic properties. This supports the idea that the correct governing equation can be accurately determined before, or after, a contamination event. While the traditional ADE fails to model a conservative tracer in the MADE aquifer, the fractional equation predicts tritium concentration profiles with remarkable accuracy over all spatial and temporal scales.     

Decay and Continuity of the Boltzmann Equation in Bounded Domains

Boundaries occur naturally in kinetic equations, and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: in-flow, bounce-back reflection, specular reflection and diffuse reflection. We establish exponential decay in the L ^∞ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set at the boundary. Our contribution is based on a new L ² decay theory and its interplay with delicate L ^∞ decay analysis for the linearized Boltzmann equation in the presence of many repeated interactions with the boundary.     

A Note on the Apparent Conductivity of Stratified Porous Media in Unsaturated Steady Flow Above a Water Table

We aim at deriving the apparent unsaturated conductivity (AUC) K ^(ap), defined as the ratio between the mean flux and the mean head gradient in a stratified vadose zone above the water table. This is achieved for steady flow generated by a constant infiltrating flux applied at the soil surface. By adopting the first-order approximation in the two parameters of the conductivity curve, and under a few additional simplifying assumptions, we were able to analytically compute K ^(ap). It is shown that this latter varies between K ^(ap) ≡ K _H (the harmonic mean) at the water table, and K ^(ap) ≡ K _ef (the effective conductivity in gravitational flow) far above the water table. Profiles of the AUC are illustrated, and the impact of parameters values is discussed.     

Asymptotic Behavior of Solutions of the Compressible Navier�CStokes Equation Around the Plane Couette Flow

Asymptotic behavior of solutions to the compressible Navier–Stokes equation around the plane Couette flow is investigated. It is shown that the plane Couette flow is asymptotically stable for initial disturbances sufficiently small in some L ² Sobolev space if the Reynolds and Mach numbers are sufficiently small. Furthermore, the disturbances behave in large time in L ² norm as solutions of an n − 1 dimensional linear heat equation with a convective term.     

Temporal and Spatial Decays for the Stokes Flow

We estimate the time decay rates in L ¹, in the Hardy space and in L ^∞ of the gradient of solutions for the Stokes equations on the half spaces. For the estimates in the Hardy space we adopt the ideas in [7], and also use the heat kernel and the solution formula for the Stokes equations. We also estimate the temporal-spatial asymptotic estimates in L ^q , 1 < q < ∞, for the Stokes solutions. This work was supported by grant No. (R05-2002-000-00002-0(2002)) from the Basic Research Program of the Korea Science & Engineering Foundation.     

Non-gaussian diffusion modeling in composite porous media by homogenization: Tail effect

In the paper anomalous diffusion appearing in a porous medium composed of two porous components of considerably different diffusion characteristics is examined. The differences in diffusivities are supposed to result either from two medium types being present or from variations in pore size (double porosity media). The long-tail effect is predicted using the homogenization approach based on the application of multiple scale asymptotic developments. It is shown that, if the ratio of effective diffusion coefficients of two porous media is of the order of magnitude smaller or equal O( ²), where is a homogenization parameter, then the macroscopic behaviour of the composite may be affected by the presence of tail-effect. The results of the theoretical analysis were applied to a problem of diffusion in a bilaminate composite. Analytical calculations were performed to show the presence of the long-tail effect in two particular cases.Notations c _i the concentration of chemical species in water within the medium i - D _i the effective diffusion coefficient for the medium i - D _ij ^eff the macroscopic (or effective) diffusion tensor in the composite - ERV the elementary representative volume - h the thickness of the period - l a chracteristic length of the ERV or the periodic cell - L a characteristic macroscopic length - n the volumetric fraction of the material 2 - 1–n the volumetric fraction of the material 1 - N the unit vector normal to - t the time variable - x the macroscopic (or slow) space variable - y the microscopic (or fast) space variable - c _1c ,C _2c ,D _1c ,D _2c the characteristic quantities - T,T _1L ,T _2L ,T _1l ,T _2l the characteristic times - c ₁ ^* ,c ₂ ^* ,D ₁ ^* ,D ₂ ^* ,t ^* the non-dimensional variables - the homogenization parameter - ₁ the domain occupied by the material 1 - ₂ the domain occupied by the material 2 - the interface between the domains 1 and 2 - the total volume of the periodic cell - /x_i the gradient operator - the gradient operator     

Decay in Time of Incompressible Flows

In this paper we consider the Cauchy problem for incompressible flows governed by the Navier-Stokes or MHD equations. We give a new proof for the time decay of the spatial L₂ L_2 norm of the solution, under the assumption that the solution of the heat equation with the same initial data decays. By first showing decay of the first derivatives of the solution, we avoid some technical difficulties of earlier proofs based on Fourier splitting.     

Green's function,continual weighted composition operators along trajectories,and hyperbolicity of linear extensions for dynamical systems

The hyperbolicity of linear skew-product flows with infinite-dimensional fibersE over a dynamical system on a compact metric spaceX is described in terms of the existence and uniqueness of Green's function and in terms of the spectra for family of the semigroups of weighted composition operators acting inL ₂(R;E) and parametrized by the points ofX.     

Asymptotic Behavior of Stokes Solutions in 2D Exterior Domains

We estimate L ^p spatial-temporal decay rates of solutions of the linearized equations of incompressible flow in a 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we use a special form of a test function. The first author was supported by Korea Research Foundation Grant (KRF-2003-015-C00052), and the second author by KRF-2006-531-C0009.     

Asymptotic and other estimates for semilinear parabolic problem in semi-infinite cylinder

The spatial decay of solutions to initial-boundary value problems for a semilinear parabolic equation in a semi-infinite cylinder of variable cross-section subject to zero condition on the lateral boundaries is investigated. A second-order differential inequality that shows the spatial decay O(exp(−z ²/(4(t+t ₀)))) for an L ^2p cross-sectional measure of the solution is obtained. A first-order differential inequality leading to growth or decay is also derived. In the case of growth, an upper bound for blow-up in space is obtained, while in the case of decay an upper bound for the total energy in terms of data is obtained.     

Essential Spectrum of the Linearized 2D Euler Equation and Lyapunov–Oseledets Exponents

The linear stability of a steady state solution of 2D Euler equations of an ideal fluid is being studied. We give an explicit geometric construction of approximate eigenfunctions for the linearized Euler operator L in vorticity form acting on Sobolev spaces on two dimensional torus. We show that each nonzero Lyapunov–Oseledets exponent for the flow induced by the steady state contributes a vertical line to the essential spectrum of L. Also, we compute the spectral and growth bounds for the group generated by L via the maximal Lyapunov–Oseledets exponent. When the flow has arbitrarily long orbits, we show that the essential spectrum of L on L₂ is the imaginary axis.