Analysis of stochastic groundwater flow problems. Part II: Stochastic partial differential equations in groundwater flow. A functional-analytic approach

Abstract

Following the scheme and concepts presented in Part I, Part II uses functional-analytic theory to analyze the problem of stochastic partial differential equations of the type appearing in groundwater flow. Equations are treated as abstract stochastic evolution equations for elliptic partial differential operators in an appropriate functional Sobolev space. Explicit forms of solutions are obtained by using a strongly continuous semigroup. The deterministic and the stochastic problem can then be treated under the same theoretical framework. Use of the theory is indicated in an application to the solution of the stochastic analogue of the regional groundwater flow problem studied in Part I. Two cases are solved: The randomly forced and the randomly initiated equation. The solution is obtained by applying the properties of semigroups and expressing the Wiener process as an infinite basis in a Hilbert space composed of independent unidimensional Wiener processes with incremental variance parameters. The first two moments of the solution as well as sample functions for different cases are derived.