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We present results concerning resolvent estimates for the linear operator associated with the system of differential equations governing 2 dimensional perturbations of plane Couette flow. We prove estimates on the L2 norm of the resolvent of this operator showing this norm to be proportional to the Reynolds number R for a region of the unstable half plane.… (More)
We discuss the problem of deriving estimates for the resolvent of the linear operator associated with three dimensional perturbations of plane Couette flow, and determining its dependence on the Reynolds number R. Depending on the values of the parameters involved, we derive estimates analytically. For the remaining values of the parameters, we prove that… (More)
This paper shows how the so called von Kármán model can be obtained as a singular limit of a modified Mindlin-Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth order dispersive operator is added. Introducing damping mechanisms, the authors also show that the energy of solutions for… (More)
We state and prove a theorem showing how iterates of the Volterra operator can be used to evaluate indeterminate forms of type 0/0. This general result allows one to recover the Boltzmann-Gibbs entropy as a limit of a wide class of generalized entropies, as considered in an earlier work.
We discuss the application of the resolvent technique to prove stability of plane Couette flow. Using this technique, we derive a threshold amplitude for perturbations that can lead to turbulence in terms of the Reynolds number. Our main objective is to show exactly how much control one should have over the perturbation to assure stability via this… (More)