Uniqueness results for the generators of the two-dimensional Euler and Navier-Stokes flows. (English) Zbl 1034.35086
The Euler and Navier-Stokes equations for an incompressible fluid in two dimensions with periodic boundary conditions are considered. Concerning the Euler equations, the associated (first order) Liouville operator \(L\) is considered as a symmetric linear operator in a Hilbert space \(L^2(\mu)\) with respect to a natural invariant Gaussian measure \(\mu\). For the Navier-Stokes equations with a suitable white noise forcing term the associated (second order) Kolmogorov operator \(K\) is considered. It is proved that both \(L\) and \(K\) are bounded by naturally positive Schrödinger-like operators. Some other uniqueness results for \(L\) and \(K\) are presented.
Reviewer: Il’ya Sh. Mogilevskij (Tver)
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B10 | Periodic solutions to PDEs |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
Euler and Navier-Stokes equations; invariant measures; Liouville and Kolmogorov generators; Navier-Stokes equations; white noise forcing term; uniquenessReferences:
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