The structure of the solution to the generalized Stokes equations and a new method for solving them
Abstract
We show that the velocity solution u to the generalized Stokes equations with Dirichlet boundary conditions in dimension two and three is the superposition of two divergence-free velocity fields ω and thetaup, u=ω+thetaup. Both ω and thetaup solve screened Poisson vector equations with right hand side respectively the rotational component in the Helmholtz decomposition of the external force, and the gradient of a harmonic scalar function q called ``solid pressure". Similarly, the fluid pressure p is the superposition of the potential π in the Helmholtz decomposition of the external force, and of q, p=π+q. It turns out that regardless the pressure q, ω+thetaup is divergence-free and satisfies the generalized Stokes equations, except the normal component of the Dirichlet boundary condition. The role of q, which solves a boundary linear equation with a self-adjoint definite positive operator, is to constrain ω+ thetaup to satisfy even the normal component of the Dirichlet boundary condition. The method is attractive for solving numerically the time dependent Navier-Stokes equations in two and three dimensions. It decouples the velocity from the pressure and solves the incompressibility constraint as a byproduct of unconstrained screened Poisson vector equations for ω and thetaup, equipped with appropriate boundary conditions. In the two last sections we present two methods for solving numerically the generalized Stokes equations, and some results to demonstrate the efficiency of our approach.
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BibTeX
@misc{novruzi2026structure,
title = {{The structure of the solution to the generalized Stokes equations and a new method for solving them}},
author = {Arian Novruzi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.15685},
eprint = {2607.15685},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.15685},
}
RIS
TY - UNPB
TI - The structure of the solution to the generalized Stokes equations and a new method for solving them
AU - Arian Novruzi
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2607.15685
UR - https://arxiv.org/abs/2607.15685
AB - We show that the velocity solution u to the generalized Stokes equations with Dirichlet boundary conditions in dimension two and three is the superposition of two divergence-free velocity fields ω and thetaup, u=ω+thetaup. Both ω and thetaup solve screened Poisson vector equations with right hand side respectively the rotational component in the Helmholtz decomposition of the external force, and the gradient of a harmonic scalar function q called ``solid pressure". Similarly, the fluid pressure p is the superposition of the potential π in the Helmholtz decomposition of the external force, and of q, p=π+q. It turns out that regardless the pressure q, ω+thetaup is divergence-free and satisfies the generalized Stokes equations, except the normal component of the Dirichlet boundary condition. The role of q, which solves a boundary linear equation with a self-adjoint definite positive operator, is to constrain ω+ thetaup to satisfy even the normal component of the Dirichlet boundary condition. The method is attractive for solving numerically the time dependent Navier-Stokes equations in two and three dimensions. It decouples the velocity from the pressure and solves the incompressibility constraint as a byproduct of unconstrained screened Poisson vector equations for ω and thetaup, equipped with appropriate boundary conditions. In the two last sections we present two methods for solving numerically the generalized Stokes equations, and some results to demonstrate the efficiency of our approach.
ER -