On compressible Navier-Stokes equations on metric graphs
Abstract
We study a compatible linearization, about a zero-velocity rest equilibrium, of the one-dimensional barotropic compressible Navier-Stokes equations in Lagrangian variables on a finite directed metric graph. The junction closure is an abstract directed common-velocity law selected to preserve weighted-volume and mechanical-power balances. We give a finite-dimensional incidence characterization and a constructive sufficient condition for positive balanced cross-sectional weights. We also provide a common-pressure criterion that produces a nonempty class of compatible constant rest equilibria. For the resulting linear graph system, we prove maximal dissipativity of the homogeneous operator, finite-energy well-posedness for square-integrable boundary tractions, and existence and uniqueness for a linear-quadratic tracking problem with a weak adjoint and an L^2 boundary observation. Positive Lagrangian coordinate parameters are then pulled back to a unit reference graph. Uniform estimates and compactness for varying mass forms yield parameter-to-state continuity, continuity of the optimized L^2 -control value, and existence of an outer minimizer on a compact fixed-topology design set. Under the areal-mass normalization, an edge parameter is the extent of the material coordinate and A_e ℓ_e is the reference mass assigned to the edge. The fixed-reference-profile convention is therefore an abstract reference-mass redistribution model, not an Eulerian pipe-length or geometric shape derivative. A backward-Euler/P 1 section records exact finite-dimensional solvability, energy, envelope, and transpose-adjoint identities on fixed grids. We finally formulate a prospective GFDS program for the full nonlinear network, including the reference-domain sensitivity structure and the analytical results required for a future nonlinear projected gradient flow. The introduction relates this coordinate framework to the geometric gradient-flow analysis of Plotnikov and Sokołowski, with explicit distinctions between curve geometry, network parameters, and conditional descent. No nonlinear state theorem, continuous coordinate derivative, numerical convergence theorem, or continuous-to-discrete gradient claim is made.
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BibTeX
@misc{sokolowski2026compressible,
title = {{On compressible Navier-Stokes equations on metric graphs}},
author = {Jan Sokołowski},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05758667v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On compressible Navier-Stokes equations on metric graphs
AU - Jan Sokołowski
PY - 2026
DA - 2026/09/22
PB - HAL
UR - https://hal.science/hal-05758667v1
AB - We study a compatible linearization, about a zero-velocity rest equilibrium, of the one-dimensional barotropic compressible Navier-Stokes equations in Lagrangian variables on a finite directed metric graph. The junction closure is an abstract directed common-velocity law selected to preserve weighted-volume and mechanical-power balances. We give a finite-dimensional incidence characterization and a constructive sufficient condition for positive balanced cross-sectional weights. We also provide a common-pressure criterion that produces a nonempty class of compatible constant rest equilibria. For the resulting linear graph system, we prove maximal dissipativity of the homogeneous operator, finite-energy well-posedness for square-integrable boundary tractions, and existence and uniqueness for a linear-quadratic tracking problem with a weak adjoint and an L^2 boundary observation. Positive Lagrangian coordinate parameters are then pulled back to a unit reference graph. Uniform estimates and compactness for varying mass forms yield parameter-to-state continuity, continuity of the optimized L^2 -control value, and existence of an outer minimizer on a compact fixed-topology design set. Under the areal-mass normalization, an edge parameter is the extent of the material coordinate and A_e ℓ_e is the reference mass assigned to the edge. The fixed-reference-profile convention is therefore an abstract reference-mass redistribution model, not an Eulerian pipe-length or geometric shape derivative. A backward-Euler/P 1 section records exact finite-dimensional solvability, energy, envelope, and transpose-adjoint identities on fixed grids. We finally formulate a prospective GFDS program for the full nonlinear network, including the reference-domain sensitivity structure and the analytical results required for a future nonlinear projected gradient flow. The introduction relates this coordinate framework to the geometric gradient-flow analysis of Plotnikov and Sokołowski, with explicit distinctions between curve geometry, network parameters, and conditional descent. No nonlinear state theorem, continuous coordinate derivative, numerical convergence theorem, or continuous-to-discrete gradient claim is made.
ER -