Well‐posedness and convergence results for strong solution to a 3D‐regularized Boussinesq system

Strong Solutions to 3D-Lagrangian Averaged Boussinesq System

“…The inviscid Lagrangian averaged Euler-α equations were originally derived as Euler-Poincaré equations in the framework of Hamilton's principle for geometric fluid mechanics [7]. The existence, uniqueness and continuous dependance of solutions to initial date, as well as convergence results of various α-models, as α vanishes can be found in [1,2,8,10,12] and references therein. Before stating our main results, let us introduce some notations that will be used throughout the paper.…”

“…we have already proved the statements 1 and 2 of Theorem 1.2.Now, we turn to the third statement of Theorem 1.2. For the first result, since (u k , θ k ) converges stronglyto (u, θ) in (L 2 ([0, T * ],Ḣ 1 ))2 , then by Cauchy-Schwarz inequality it converges weakly for almost every andd dt v k d dt u weakly in L 2 ([0, T * ],Ḣ −2 (T 3 )), as k → +∞.Let Λ ∈Ḣ 2 be a vector divergence free and Ξ ∈ L 2 a scaler test functions. Taking the inner product andintegrating over [0, t], for t ∈ [0, T * ], we obtain θ k (t), Ξ − θ k (0), Ξ − t (u k , θ k ), Ξ dτ = 0, v k (t), Λ − v k (0), Λ − t (u k , v k ), Λ dτ − t Λ dτ = 0.To handle the nonlinear terms, we use a standard compactness argument (thanks to the uniform boundsobtained with respect to α k above) so thatB(u k , v k ) → B(u, u) and B(u k , θ k ) → B(u, θ).…”

Analytical Study of a 3D-MHD System with Exponential Damping

“…This classical compactness argument is frequently used in ( [1]). Continuity in time of the solution can be proved in a standard way as in ( [4]), for example.…”

Global Well-Posedness and Convergence Results to a 3D Regularized Boussinesq System in Sobolev Spaces

“…Tis local aspect is due to the classical arguments based on a brutal application of Cauchy-Schwarz inequality while taking the scalar product of the Buoyancy force θe 3 with the velocity feld u. Tus, this was not a global in-time solution but it should be called a large time solution. Tis insufciency appeared widely in the literature and is still appearing both in the threedimensional case and the two-dimensional case which is supposed to be well understood, see among a wide literature [1,[7][8][9][10][11][12] and references therein. Here, we overcome this insufciency for the range of mean free initial temperature and we make two improvements that are interesting from an applicable point of view.…”