Controllability and ergodicity of three dimensional primitive equations driven by a finite-dimensional force. (English) Zbl 1504.35565
Summary: We study the problems of controllability and ergodicity of the system of three dimensional primitive equations modeling large-scale oceanic and atmospheric motions. The system is driven by an additive force acting only on a finite number of Fourier modes in the temperature equation. We first show that the velocity and temperature components of the equations can be simultaneously approximately controlled to arbitrary position in the phase space. The proof is based on Agrachev-Sarychev type geometric control approach. Next, we study the controllability of the linearization of primitive equations around a non-stationary trajectory of the randomly forced system. Assuming that the probability law of the forcing is decomposable and observable, we prove almost sure approximate controllability by using the same Fourier modes as in the nonlinear setting. Finally, combining the controllability of the linearized system with a criterion from S. Kuksin et al. [Geom. Funct. Anal. 30, No. 1, 126–187 (2020; Zbl 1442.35437)], we establish exponential mixing for the nonlinear primitive equations with a random force.
MSC:
| 35Q86 | PDEs in connection with geophysics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 86A05 | Hydrology, hydrography, oceanography |
| 86A10 | Meteorology and atmospheric physics |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76U60 | Geophysical flows |
| 76M35 | Stochastic analysis applied to problems in fluid mechanics |
| 93B05 | Controllability |
| 93C20 | Control/observation systems governed by partial differential equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 37A25 | Ergodicity, mixing, rates of mixing |
| 37L55 | Infinite-dimensional random dynamical systems; stochastic equations |
Citations:
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