  This paper introduces a novel Lagrangian fluid solver based on covector flow
maps. We aim to address the challenges of establishing a robust flow-map solver
for incompressible fluids under complex boundary conditions. Our key idea is to
use particle trajectories to establish precise flow maps and tailor path
integrals of physical quantities along these trajectories to reformulate the
Poisson problem during the projection step. We devise a decoupling mechanism
based on path-integral identities from flow-map theory. This mechanism
integrates long-range flow maps for the main fluid body into a short-range
projection framework, ensuring a robust treatment of free boundaries. We show
that our method can effectively transform a long-range projection problem with
integral boundaries into a Poisson problem with standard boundary conditions --
specifically, zero Dirichlet on the free surface and zero Neumann on solid
boundaries. This transformation significantly enhances robustness and accuracy,
extending the applicability of flow-map methods to complex free-surface
problems.
