55 research outputs found
On the stability of symmetric flows in a two-dimensional channel
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
which is defined on
D({\mathcal B})=\{u\in H^4(0,1)\,,\, u^\prime(0)=u^{(3)}(0)=0 \mbox{ and }\,
u(1)=u^\prime(1)=0\}. is bounded on the half-plane for
or
On the spectrum of some Bloch-Torrey vector operators
We consider the Bloch-Torrey operator in where
. In contrast with the (as well
as the ) case considered in previous works.
We obtain that is in the continuous spectrum for as well as discrete spectrum outside the real line. For a finite interval
we find the left margin of the spectrum. In addition, we prove that the
Bloch-Torrey operator must have an essential spectrum for a rather general
setup in , and find an effective description for its domain
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