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Article Dans Une Revue Archive for Rational Mechanics and Analysis Année : 2023

Controllability and Ergodicity of three dimensional Primitive Equations Driven by a Finite-Dimensional Force

Résumé

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 Kuksin et al. (Geom Funct Anal 30(1):126–187, 2020), we establish exponential mixing for the nonlinear primitive equations with a random force.
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Dates et versions

hal-04122500 , version 1 (18-09-2023)

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Pierre Marie Boulvard, Peng Gao, Vahagn Nersesyan. Controllability and Ergodicity of three dimensional Primitive Equations Driven by a Finite-Dimensional Force. Archive for Rational Mechanics and Analysis, 2023, 247 (1), ⟨10.1007/s00205-022-01835-8⟩. ⟨hal-04122500⟩

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