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

We prove the existence of unique solutions ${u\in H^1(0,T;L^2_γ(\mathbb{R}^n))\cap L^2(0,T;H^1_γ(\mathbb{R}^n))}$ 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, $f$