Homogenization for time-dependent two-dimensional incompressible Gaussian flows
The advection-diffusion equation \[ \partial_tC(t,x)= \kappa\Delta C(t,x)+ ({\mathbf v}(t,x)\nabla)U(t,x)\tag{1} \] is considered on the two-dimensional plane \(\mathbb{R}^2\), where \({\mathbf v}(t,x)\) \((=\Delta^\perp\varphi(t,x))\) is an incompressible random vector field. The authors study the diffusive scaling limit for time-dependent incompressible Gaussian flows under the assumption that the stream function \(\varphi\) is of the form \(\varphi(t,x)= \sum^n_{i=1}\{a_i(t)\cos(k_ix)+ b_i(t)\sin(k_i,x)\}\), where \(\{k_i\}\) are \(n\) points (Fourier modes) in \(\mathbb{R}^2\) and the \(a_i\)'s, \(b_i\)'s are mutually independent scalar Ornstein-Uhlenbeck processes satisfying \[ da_i(t)= -\alpha_ia_i(t)dt+ \sigma_idz^a_i(t),\quad db_i(t)= -\alpha_i(t)dt+ \sigma_idz^b_i(t), \] where \(\alpha_i\), \(\sigma_i\) are positive constants and \(z^a_i\), \(z^b_i\) are \(2n\) independent standard scalar Wiener processes. For each realization of the velocity field, the microscopic motion leading to (1) is given by the SDE \[ dX_t= \sqrt{2\kappa} dW(t)+{\mathbf v}(t,X_t)dt,\quad X_0= x, \] with a two-dimensional standard Wiener process \(\{W(t); t\geq 0\}\). Let \(Y_N(t)={1\over N} X_{N^2t}\) be the rescaled microscopic motion when \(N\) is large. The main result of the paper reads: The process \(\{Y_N(t); t\geq 0\}\) converges in distribution to a two-dimensional Brownian motion \(\{B(t); t\geq 0\}\) with covariance matrix \(D(\kappa)\) depending only upon the diffusivity constant \(\kappa\). The following is an immediate consequence of the above result: if the scalar function \(C_N(t,x)\) solves the parabolic problem \[ \partial_tC_N(t,x)=\kappa\Delta C_N(t,x)+ N({\mathbf v}(N^2t,x/N)\Delta)C_N(t,x)\tag{2} \] with \(C_N(0,x)= f(x)\in C^\infty_0(\mathbb{R}^2)\), then the function \(\mathbb{E}\{C_N(t,x)\}\) converges locally in the \(L^2\) sense to the solution \(C(t,x)\) of the deterministic parabolic equation \[ \partial_tC(t,x)= \nabla\cdot (D(\kappa)\nabla C(t,x)),\quad \text{with}\quad C(0,x)= f(x).\tag{3} \] For other related works, see e.g. \textit{T. Komorowski} and \textit{G. Papanicolaou} [ibid. 7, No. 1, 229-264 (1997)].
scientific article; zbMATH DE number 1014238