Existence and Uniqueness Proof for the Transport Equation and the Orthogonal Dynamics Equation for Zwanzig's Projection on Gaussian Spaces: A Vanishing Viscosity Method

Abstract

We prove the existence of unique solutions $u\in H^1(0,T;L_{\gamma }^2({\mathbb{R}}^n))\cap L^2(0,T;H_{\gamma }^1({\mathbb{R}}^n))$ of the transport equation $u_t=F\cdot \nabla u+f$ and the orthogonal dynamics equation $u_t=Q_{\text{Z}}(F\cdot \nabla )Q_{\text{Z}}u+f$, where $F$ is a time-independent vector field, $f$ is a forcing term, $P_{\text{Z}}=1-Q_{\text{Z}}$ is a Zwanzig projection (conditional expectation) on $L_{\gamma }^2({\mathbb{R}}^n)$ defined by $P_{\text{Z}}h\propto \int h(\cdot ,y)\rho (\cdot ,y)\mathrm{d}\mathrm{y}$, and $\gamma$ is the standard Gaussian measure with density $\rho$. To this end, we develop a vanishing viscosity method based on the maximal regularity theory for the Ornstein-Uhlenbeck operator on $L_{\gamma }^2({\mathbb{R}}^n)$. The following assumptions guarantee the existence and uniqueness of strong solutions: $f\in H^1(0,T;L_{\gamma }^2({\mathbb{R}}^n))\cap L^2(0,T;H_{\gamma }^1({\mathbb{R}}^n))$, $g\in H_{\gamma }^1({\mathbb{R}}^n)$, $F\in W^{1,\infty }({\mathbb{R}}^n,{\mathbb{R}}^n)$ and ${\mathrm{ess}\sup }_{{\mathbb{R}}^{\mathrm{n}}}\left(\mathrm{x}\cdot \mathrm{F}-\mathrm{div}\mathrm{F}\right)<\infty$, where $g$ denotes the initial value function.