Spatial Decay Bounds for the Brinkman Fluid Equations in Double-Diffusive Convection

In this paper, we consider the Brinkman equations pipe flow, which includes the salinity and the temperature. Assuming that the fluid satisfies nonlinear boundary conditions at the finite end of the cylinder, using the symmetry of differential inequalities and the energy analysis methods, we establish the exponential decay estimates for homogeneous Brinkman equations. That is to prove that the solutions of the equation decay exponentially with the distance from the finite end of the cylinder. To make the estimate of decay explicit, the bound for the total energy is also derived.

Uniform and weak stability of Bresse system with one infinite memory in the shear angle displacements

In this paper, we consider a one-dimensional linear Bresse system in a bounded open interval with one infinite memory acting only on the shear angle equation. First, we establish the well posedness using the semigroup theory. Then, we prove two general (uniform and weak) decay estimates depending on the speeds of wave propagations and the arbitrary growth at infinity of the relaxation function.