Global well-posedness for deconvolution magnetohydrodynamics models with fractional regularization

Abstract

In this paper, we consider two Approximate Deconvolution Magnetohydrodynamics models which are related to Large Eddy Simulation. We first study existence and uniqueness of solutions in the double viscous case. Then, we study existence and uniqueness of solutions of the Approximate Deconvolution MHD model with magnetic diffusivity, but without kinematic viscosity. In each case, we give the optimal value of regularizations where we can prove global existence and uniqueness of the solutions. The second model includes the Approximate Deconvolution Euler Model as a particular case. Finally, an asymptotic stability result is shown in the double viscous case with weaker condition on the regularization parameter. MSC: 76D05; 35Q30; 76F65; 76D03 $LaTeX: 2018/11/4 $ the triplet (w, B, q) fulfill T 0 ∂ t w, ϕ − D N,θ (w) ⊗ D N,θ (w), ∇ϕ + ν ∇w, ∇ϕ + ∇q, ϕ dt + T 0 (B) ⊗ (B), ∇ϕ dt = 0 for all ϕ ∈ L 2 (0, T ; H 3 2 −2θ ), (1.7) T 0 ∂ t B, ϕ + D N,θ (w) ⊗ B, ∇ϕ + µ ∇B, ∇ϕ dt − T 0

Global attractor for the Navier–Stokes equations with fractional deconvolution

“…We are convinced that the theorems proved in this paper can be extended to other models, even for magnetofluids, generalizing the results concerning the global attractor in magnetohydrodynamics (MHD), as in [7][8][9], to deconvolution and fractional deconvolution approximate MHD models, such as those in [1,3]. However, this would require several more computations, and it is postponed to future works.…”

“…This result has been proved by Berselli-Lewandowski [4] for θ > 3/4 and extended to θ ≥ 1/2 by Ali in [1], where the equations are coupled with the equations for the magnetic field (so the theorem above follows taking the magnetic field identically zero).…”

“…2.2 for the existence). This result has been generalized by Ali [1] to the case of magnetohydrodynamics requiring only θ ≥ 1/2. When the deconvolution order is 0, i.e.…”

“…In order to avoid overloading calculations, here, we derive the estimates by arguing directly, at a formal level, on the solution of the problem (3)- (5). We refer to [1] and [4] for a detailed discussion of the Galerkin approximation technique.…”