This thesis presents results from the author’s articles [26, 19, 20], which focus on studying
the transition threshold problem in different fluid flow profiles within the context of the
Navier-Stokes and Boussinesq equations. By utilizing linear stabilizing mechanisms like
enhanced dissipation (arising from shear flows) and dispersive effects (due to thermal
stratification), we determine the size of perturbations leading to an asymptotically stable
regime. This basin of attraction varies based on the background configuration and can
be quantified in terms of the fluid’s viscosity coefficient, ⌫: small perturbations stabilize,
while larger ones induce turbulence.
In homogeneous viscous fluids, we establish an upper bound for the threshold, proportional to ⌫ 2
3 , in the two-dimensional Poiseuille flow. This involves deriving linear
enhanced dissipation estimates and employing them in a nonlinear bootstrap argument
to address the full nonlinear problem. For inhomogeneous viscous and thermally diffusive
flows, we provide a comprehensive linear and nonlinear description of perturbations in
a three-dimensional stably stratified Couette flow. This system showcases the interplay
between linear effects from shearing movement and the oscillatory nature of stratification.
Dissipation enhancement estimates are derived through specific system symmetrization
and energy functional arguments. Additionally, we demonstrate how buoyant coupling
suppresses the lift-up effect for streak solutions, improving stability. We then employ
dispersion to establish an upper bound proportional to ⌫ 11
12 for the transition threshold
of the stably stratified Couette flow, focusing on dispersive estimates to control nonlinear
streak terms. Non-streak solutions are addressed using a bootstrap argument and an
energy functional adapted for nonlinear analysis through ghost multipliers, which incorporate linear behaviors and regularity of solutions. These methods play a crucial role in
studying interactions of nonlinear terms.