Averaging for fundamental solutions of parabolic equations

[摘要] Herein, an averaging theory for the solutions to Cauchy initial value problems of arbitrary order, E-dependent parabolic partial differential equations is developed. Indeed, by directly developing bounds between the derivatives of the fundamental solution to such an equation and derivatives of the fundamental solution of an ''averaged'' parabolic equation, we bring forth a novel approach to comparing x-derivatives of partial derivative(1)u(epsilon)(x, t) = (\k\less than or equal to 2p) Sigma A(k)(x, t/epsilon) partial derivative(x)(k) u(epsilon)(x, t) + f(epsilon)(x, t), u(epsilon)(x, 0) = phi(epsilon)(x) on R-d x [0, T] to like derivaties of partial derivative(1)u(x, t) = (\k\less than or equal to 2p)Sigma A(k)(0) (x) partial derivative(x)(k)u(epsilon)(x, t) + f(x, t), u(x) = phi(x) (as epsilon --> 0) under general regularity conditions and our basic hypothesis that \\integral(0)(l) A(k)(x, s/epsilon) - A(k)(0)(x) ds\\-->(epsilon-->0)0 for each x, t (i.e., pointwise). The flexibility afforded by studing fundamental vis-a-vis specific solutions of these equations not only permits epsilon-dependent Cauchy data and provides a unified method of treating all x-derivatives of u(epsilon) up to order 2p - 1 but also proves an invaluable tool when considering related problems of stochastic averaging. Our development was motivated by and retains a strong resemblance to the classical theory of parabolic partial differential equations. However, it will turn out that the classical conditions under which fundamental solutions are known to exist are somewhat unsuitable for our purposes and a modified set of conditions must be used. (C) 1991 Academic Press.

[发布日期] 1997-05-01