arXiv subjects

Oussama Ben Said

Publications and source records attributed to Oussama Ben Said.

3 recordsLinked to original sources

arXiv2024

Stability and large-time behavior for the 2D Boussinesq system with vertical dissipation and horizontal thermal diffusion

This paper addresses the stability and large-time behavior problem on the perturbations near the hydrostatic balance of the two dimensional Boussinesq system, taking into account vertical dissipation and horizontal thermal diffusion. The spatial framework $Ω$ is defined as $ \mathbb{T}\times\mathbb{R}$, where $\mathbb{T}$ spans $[0, 1]$, representing the 1D periodic box, while $\mathbb{R}$ denotes the whole line. The results outlined in this article confirm the fact that the temperature can actually have a stabilizing effect on the buoyancy-driven fluids. The stability and long-time behavior issues discussed here are difficult due to the lack of the horizontal dissipation and vertical thermal diffusion. By formulating in the appropriate energy functional and implementing the orthogonal decomposition of the velocity and the temperature into their horizontal averages and oscillation parts, we are able to make up for the missing regularization and establish the nonlinear stability in the Sobolev space $H^2(Ω)$ and acheive the algebraic decay rates for the oscillation parts in the $H^1$-norm.

math.AP↗

arXiv2020

The stabilizing effect of the temperature on buoyancy-driven fluids

The Boussinesq system for buoyancy driven fluids couples the momentum equation forced by the buoyancy with the convection-diffusion equation for the temperature. One fundamental issue on the Boussinesq system is the stability problem on perturbations near the hydrostatic balance. This problem can be extremely difficult when the system lacks full dissipation. This paper solves the stability problem for a two-dimensional Boussinesq system with only vertical dissipation and horizontal thermal diffusion. We establish the stability for the nonlinear system and derive precise large-time behavior for the linearized system. The results presented in this paper reveal a remarkable phenomenon for buoyancy driven fluids. That is, the temperature actually smooths and stabilizes the fluids. If the temperature were not present, the fluid is governed by the 2D Navier-Stokes with only vertical dissipation and its stability remains open. It is the coupling and interaction between the temperature and the velocity in the Boussinesq system that makes the stability problem studied here possible. Mathematically the system can be reduced to degenerate and damped wave equations that fuel the stabilization.

math.AP↗

arXiv2019

Unique weak solutions of the d-dimensional micropolar equation with fractional dissipation

This article examines the existence and uniqueness of weak solutions to the d-dimensional micropolar equations ($d=2$ or $d=3$) with general fractional dissipation $(-Δ)^αu$ and $(-Δ)^βw$. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when $α\ge \frac12$ and $β\ge \frac12$, any initial data $(u_0, w_0)$ in the critical Besov space $u_0\in B^{1+\frac{d}{2}-2α}_{2,1}(\mathbb R^d)$ and $w_0\in B^{1+\frac{d}{2}-2β}_{2,1}(\mathbb R^d)$ yields a unique weak solution. For $α\ge 1$ and $β=0$, any initial data $u_0\in B^{1+\frac{d}{2}-2α}_{2,1}(\mathbb R^d)$ and $w_0\in B^{\frac{d}{2}}_{2,1}(\mathbb R^d)$ also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely $α=β=1$ have a unique weak solution for $(u_0, w_0)\in B^0_{2,1}$. The proof involves the construction of successive approximation sequences and extensive {\it a priori} estimates in Besov space settings.

math.AP↗