Random Dynamics of the Boussinesq System with Dynamical Boundary Conditions

Rare events in the Boussinesq system with fluctuating dynamical boundary conditions

“…As in [2], we have A 1 and A 2 are positive self-adjoint operators and can also define the following function spaces with respect to the operator A, defined in (2.4), this is reasonable because A −1 is compact and so the spectrum of A is discrete with finite multiplicities. The spectrum of A is denoted by (λ i ) i∈N and the appropriate eigenfunctions are (e i ) i∈N which form a complete orthonormal system in H .…”

“…This proof is a combination of [13] and [2]. Let Ω T = [0, T ] × Ω be endowed with the product measure ds ⊗ dP on B([0, T ]) ⊗ F .…”

“…We consider the Boussinesq system with random forcing and random dynamical boundary condition [2]: (2.1) γ θ ε = θ ε Γ , θ ε (0) = θ ε 0 , (2.2) with velocity u ε = u ε (t, x) = (u ε 1 (t, x), u ε 2 (t, x)) ∈ R 2 , salinity θ ε = θ ε (t, x) ∈ R, φ ε = (u ε , θ ε , θ ε Γ ), pressure p, x = (ξ, η) ∈ D and t > 0. Here is the Laplacian operator, γ is the trace operator with respect to the boundary, ∇ the gradient operator, div the divergence operator, Fr is the Froude number, Re is the Reynolds number and Pr is the Prandtl number.…”

“…For the dynamics of stochastic parabolic partial differential equations with dynamical boundary conditions we refer to [8,9,25,26]. However here we use the strategy in [2] by embedding the dynamical boundary condition into a stochastic evolution equation, i.e., the so-called stochastic Boussinesq equation, since the properties of the linear part of these equation, the boundary condition included, we can derive a positive form which gives us a linear symmetric operator, whose eigenfunctions form a complete orthonormal system. In [2], they derived a priori estimates for the existence of absorbing sets and showed that this random dynamical system generated by the stochastic Boussinesq system with additive noise had a random attractor.…”

Random Dynamics of the Stochastic Boussinesq Equations Driven by Lévy Noises

“…These coupled equations model a variety of phenomena arising in environmental, geophysical, and climate systems. The related Boussinesq fluid equations [3][4][5] under Gaussian fluctuations have been recently studied, for example, existence and uniqueness of solutions [6], stochastic flow, dynamical impact under random dynamical boundary conditions [7,8], and large deviation principles [9,10], among others.…”