Large deviation principles for 3D stochastic primitive equations

Abstract

In this paper, we establish the large deviation principle for 3D stochastic primitive equations with small perturbation multiplicative noise. The proof is mainly based on the weak convergence approach.

On the small‐time asymptotics of 3D stochastic primitive equations

“…Gao and Sun proved that the LDP holds for weak solutions of the 2D stochastic primitive equations. Dong et al obtained the same result for the strong solution of 3D stochastic primitive equations.…”

On the backward uniqueness of the stochastic primitive equations with additive noise

“…and Theorem 3.4. Given (v 1 (0), θ 1 (0)) and (v 2 (0), θ 2 (0)) ∈ H 1 × H 1 , let (v 1 , θ 1 ) and (v 2 , θ 2 ) be the two corresponding strong solutions to stochastic primitive equations with (14) holds. If we assume v 1 (t) = v 2 (t) and θ 1 (t) = θ 2 (t) for some t > 0, a.s., then P(v 1 (s) = v 2 (s), θ 1 (s) = θ 2 (s), ∀s ∈ [0, t]) = 1.…”

“…Under the periodic conditions, Glatt-Holtz, Kukavica, Vicol and Ziane considered the existence of invariant measure for the 3D PEs in [20]. The uniqueness of the invariant measure and large deviations for the 3D stochastic primitive equations were obtained by Dong, Zhai and Zhang in [14,15] under the periodic boundary conditions. Some analytical properties of weak solutions of 3D stochastic primitive equations with periodic boundary conditions were obtained in [13], in which the martingale problem associated to this model is shown to have a family of solutions satisfying the Markov property.…”

The asymptotic behavior of primitive equations with multiplicative noise

“…Debussche, Glatt-Holtz, Temam and Ziane concerned the global well-posedness of strong solution when the primitive equations are driven by multiplicative stochastic forcing in [12]. The ergodic theory and large deviations for the 3D stochastic primitive equations were studied by Dong, Zhai and Zhang in [13,14]. When the primitive equations are driven by an infinitedimensional fractional noise taking values in some Hilbert space, the latter author of this paper obtained the existence of random attractor in [30].…”