Positivity and Carleman convergence for richer fluid systems

Establish positive semidefiniteness of the Hermitian part of the LCHS generator, together with accompanying Carleman convergence bounds, for advection-dominated Burgers regimes, incompressible Navier–Stokes systems, and magnetohydrodynamic couplings.

Background

The paper extends its PMR–LCHS construction from the one-dimensional viscous Burgers equation to broader polynomial fluid models, including advection-dominated regimes, incompressible Navier–Stokes equations, and magnetohydrodynamic systems. The LCHS method requires the Hermitian part of the linear generator to be positive semidefinite, potentially after applying a stabilizing spectral shift, while the finite Carleman truncation requires suitable convergence bounds.

For the dissipation-dominated Burgers system, the paper proves that a stabilizing shift suffices to enforce positivity and analyzes its postselection consequences. It does not establish analogous positivity results or Carleman convergence estimates for the richer systems and regimes described above, leaving these questions unresolved.

References

For advection-dominated regimes, the incompressible Navier--Stokes advection term, or magnetohydrodynamic couplings, positivity must be checked case by case, and any required shift feeds back into the amplitude-amplification cost {u_0}/{u(t)} in the same way it does for Burgers'. Establishing positivity, together with the accompanying Carleman convergence bounds, for these richer systems is part of what we leave to future work.

— Quantum algorithm for differential equations via permutation matrix representation with application to the Burgers equation  (2608.19508 - Sabharwal et al., 19 Aug 2026) in Section 6, final paragraph on extensions to more complex fluid equations