On the Stability of Symmetric Flows in a Two-Dimensional Channel

1. Auflage 2025
ISBN: 978-3-98547-099-0
Verlag: EMS Press

Buch, Englisch, 212 Seiten, Format (B × H): 170 mm x 240 mm, Gewicht: 393 g

Reihe: Memoirs of the European Mathematical Society

We consider the stability of symmetric flows in a two-dimensional channel (including the Poiseuille flow). In 2015 Grenier, Guo, and Nguyen have established instability of these flows in a particular region of the parameter space, affirming formal asymptotics results from the 1940's. We prove that these flows are stable outside this region in parameter space. More precisely we show that the Orr--Sommerfeld operator

{\mathcal B} =\Big(-\frac{d^2}{dx^2}+i\beta(U+i\lambda)\Big)\Big(\frac{d^2}{dx^2}-\alpha^2\Big) -i\beta U^{\prime\prime},

which is defined on

D({\mathcal B})=\{u\in H^4(0,1), u^\prime(0)=u^{(3)}(0)=0 \text{ and } u(1)=u^\prime(1)=0\}.

is bounded on the half-plane \Re \lambda \geq 0 for \alpha \gg \beta^{-1/10} or \alpha \ll \beta^{-1/6}.

Autoren/Hrsg.

Almog, Yaniv

Helffer, Bernard

Über Autor(innen)

Helffer, Bernard
CNRS and Nantes Université, France

Almog, Yaniv
Braude College of Engineering, Karmiel, Israel