Variational methods for PDEs aplied to stochastic partial differential equations
Gjermund Våge · MATHEMATICA SCANDINAVICA · 1998
During the last couple of years there has been a growing interest in stochastic partial di¡erential equations (SPDEs).Various methods have been used to study SPDEs, see [10] and the references therein.Here we apply white noise analysis to obtain abstract existence and uniqueness theorems.More speci¢cally we combine the ideas of Kondratiev spaces with variational methods for partial di¡erential equations.We show that this approach applies to elliptic, parabolic, as well as hyperbolic SPDEs.To illustrate our ideas on elliptic SPDEs, we prove in Section 4 that there exists a unique solution, u, satisfyingwhere F , f , and g are given stochastic processes and Å denotes the Wick product.If F is the Wick exponential of smoothed white noise, F exp Å W 0 x , we obtain the pressure equation in a stochastic medium, ¢rst solved in [6].This equation is a model for £ow in an (stochastic) isotropic porous medium where F is the permeability.If F , f are deterministic and g 0, the Wick product coincides with the ordinary product and existence of a unique variational solution of the deterministic problem (1)^(2) can be shown as follows.We apply both sides of (1) to a test function v P H H 1 0 D and integrate by parts to obtain the variational problem find u P H such that buY v Lv for all v P HY 3 where buY v F ruY rv L 2 D and Lv f Y v L 2 D .The Lax-Milgram theorem (see e.g.[4] or [14]) ensures (3) has a unique solution if LÁ is a con-MATH.SCAND.82 (1998), 113^137