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∈ H1(0,T;L2γ(ℝn))∩ L2(0,T;H1γ(ℝn)) of the transport equation ut=F·∇ u+f and the orthogonal dynamics equation ut= QZ (F·∇) QZ u+f, where F is a time-independent vector field, f is a forcing term, PZ=1−QZ is a Zwanzig projection (conditional expectation) on L2γ(ℝn) defined by PZh ∝ ∫ h(· ,y) ρ(·,y)dy, and γ is the standard Gaussian measure with density ρ. To this end, we develop a vanishing viscosity method based on the maximal regularity theory for the Ornstein-Uhlenbeck operator on L2γ(ℝn). The following assumptions guarantee the existence and uniqueness of strong solutions: f∈ H1(0,T;L2γ(ℝn))∩ L2(0,T;H1γ(ℝn)), g∈ H1γ(ℝn), F∈ W1,∞(ℝn,ℝn) and ess supℝn(x· F−divF) < ∞, where g denotes the initial value function.
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BibTeX
@misc{widder2026existence,
title = {{Existence and Uniqueness Proof for the Transport Equation and the Orthogonal Dynamics Equation for Zwanzig's Projection on Gaussian Spaces: A Vanishing Viscosity Method}},
author = {Christoph Widder},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18647},
eprint = {2610.18647},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18647},
}
RIS
TY - UNPB
TI - Existence and Uniqueness Proof for the Transport Equation and the Orthogonal Dynamics Equation for Zwanzig's Projection on Gaussian Spaces: A Vanishing Viscosity Method
AU - Christoph Widder
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.18647
UR - https://arxiv.org/abs/2610.18647
AB - We prove the existence of unique solutions u∈ H^1(0,T;L^2_γ(ℝ^n))∩ L^2(0,T;H^1_γ(ℝ^n)) of the transport equation u_t=F·∇ u+f and the orthogonal dynamics equation u_t= Q_Z (F·∇) Q_Z u+f, where F is a time-independent vector field, f is a forcing term, P_Z=1−Q_Z is a Zwanzig projection (conditional expectation) on L^2_γ(ℝ^n) defined by P_Zh ∝ ∫ h(· ,y) ρ(·,y)dy, and γ is the standard Gaussian measure with density ρ. To this end, we develop a vanishing viscosity method based on the maximal regularity theory for the Ornstein-Uhlenbeck operator on L^2_γ(ℝ^n). The following assumptions guarantee the existence and uniqueness of strong solutions: f∈ H^1(0,T;L^2_γ(ℝ^n))∩ L^2(0,T;H^1_γ(ℝ^n)), g∈ H^1_γ(ℝ^n), F∈ W^1,∞(ℝ^n,ℝ^n) and ess sup_ℝ^n(x· F−divF) < ∞, where g denotes the initial value function.
ER -