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Compressible Navier-Stokes flow in Schrödinger-type variables

James R. Beattie1,2,*, Max Sokolova3, Khush Negandhi3, and Bart Ripperda1,3,4,5

  • *Contact author: jbeattie@cita.utoronto.ca
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Phys. Rev. Research 8, 033383 – Published 30 September, 2026

DOI: https://doi.org/10.1103/6gnr-vkx3

Abstract

Fluid equations are nonlinear, dissipative, and non-Hamiltonian, which makes their relation to Schrödinger evolution and quantum algorithms nontrivial. We derive an exact Eulerian Cole-Hopf-type reformulation of isothermal compressible Navier-Stokes (NS) flow in Schrödinger-type amplitude variables. To our knowledge, this gives the first exact Cole-Hopf-type Schrödinger-variable reformulation of compressible NS flow. In two dimensions, a Helmholtz decomposition separates the velocity into compressive and vortical potentials, whose logarithmic transforms yield two scalar imaginary-time Schrödinger-type equations with nonlinear self-consistent potentials. We show that the mixed density-compressive amplitude Ψα=ραΘ1−2α, where ρ is the density, Θ is the compressive amplitude, and α≠0,1/2, satisfies a nonlinear Schrödinger-type equation with a vector-potential-coupled Laplacian. The transformed system is exactly equivalent to compressible NS and is nonlocal only through Helmholtz and Poisson projections. In three dimensions, the density-carrying equation retains the same vector-potential-coupled structure, while the solenoidal sector admits a compressible analog of Ohkitani’s [Phys. Rev. E 92, 033010 (2015)] incompressible NS Cole-Hopf formulation. Unlike unitary hydrodynamic Schrödinger-flow representations, the present equations are imaginary-time heat or drift-diffusion equations with self-consistent potentials, but they remain an exact change of variables for compressible NS. A two-dimensional Kelvin-Helmholtz shear-layer calculation verifies the transformed equations against a direct compressible NS simulation. The formulation exposes operator structures that may be useful for reduced flow descriptions, quantum algorithms for operator evolution, and quantum partial differential equation solvers.

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Phys. Rev. Research 8, 033383– Published 30 September, 2026
  • Received 29 April 2026
  • Accepted 14 July 2026
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