Homogenization of a periodic parabolic Cauchy problem

Abstract

We study homogenization for a periodic parabolic Cauchy problem in Rd. Coefficients are assumed to be periodic in x ∈ Rd and independent of time. We prove that the solutions converge in L2(R) to the solution of the ”homogenized” problem, as the small period e tends to zero. For the L2(R)-norm of the difference, we obtain an order-sharp estimate which is uniform with respect to the data of the problem. Convergence of the solutions in different functional classes is also studied. §0. Introduction 0.1. We study homogenization of the parabolic Cauchy problem Q(e−1x) ∂ue(x, τ) ∂τ = −Âeue(x, τ) + F(x, τ), Q(ex)ue(x, 0) = φ(x), (0.1) x ∈ R, τ ∈ (0, T ), e > 0, in the small period limit e → 0. Here ue(x, τ) is a C-valued function of x and τ , Âe = b(D)∗g(e−1x)b(D) is a matrix elliptic second order operator and Q is a positive matrix-valued function. We assume that the matrix-valued functions g and Q are periodic with respect to some lattice Γ ⊂ R. Let Ω ⊂ R be the elementary cell of the lattice Γ. For the precise definition of the operator Âe, see §3 below. Our goal is to prove convergence of the solutions ue of problem (0.1) to the solution u0 of the ”homogenized” problem and to give the error estimate. The ”homogenized” problem has the form Q ∂u0(x, τ) ∂τ = −Âu0(x, τ) + F(x, τ), Qu0(x, 0) = φ(x), (0.2) where  = b(D)∗g0b(D) is the effective operator with the constant effective matrix of coefficients g, and Q is the mean value of the matrix Q. The homogenization problem for parabolic equations has been intensively studied by traditional methods of homogenization theory (e. g., see the books [BaPa, BeLP, ZhKO, Sa]). The main attention has been paid to the diffusion equation (where n = 1 and Âe = −div g(e−1x)∇). The corresponding results give convergence (in an appropriate functional class) of the solutions ue to u0. For instance, in [BeLP] it is proved that, if φ ∈ L2(R) and F ∈ L2((0, T );H−1(Rd)), then the solutions ue of the diffusion equation converge to u0 weakly in the class L2((0, T );H(R)).