Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows

By Davdi Gerard-Varet, Yasunori Maekawa, Nader Masmoudi. Duke mathematical journal, 2016

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We investigate the stability of boundary layer solutions of the two-dimensional incompressible Navier-Stokes equations. We consider shear flow solutions of Prandtl type : $$ u^\nu(t,x,y) \, = \, \big (U^E(t,y) + U^{BL}(t,\frac{y}{\sqrt{\nu}})\,, \, 0 \big )\, , \quad 0<\nu \ll 1\,. $$ We show that if $U^{BL}$ is monotonic and concave in $Y = y /\sqrt{\nu}$ then $u^\nu$ is stable over some time interval $(0,T)$, $T$ independent of $\nu$, under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$. We improve in this way the classical stability results of Sammartino and Caflisch in analytic class (both in $x$ and $y$). Moreover, in the case where $U^{BL}$ is steady and strictly concave, our Gevrey exponent for stability is optimal. The proof relies on new and sharp resolvent estimates for the linearized Orr-Sommerfeld operator.

Subfields: Applied Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/8feba6d273f4fbe2b2c9dfb47905ab58199dcdfa

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In Applied Mathematics

In Control and Systems Engineering

Citations in the corpus

The Inviscid Limit of Navier–Stokes Equations for Analytic Data on the Half-Space (2017, result): “) with remainder of order √ ν). A similar result is also obtained in [26, 37]. The Ansatz (1.3), with a speciﬁc boundary layer shear proﬁle, has been recently justiﬁed for data with Gevrey regularity [10]. When only data with Sobolev regularity are assumed, except for data with special symmetry data”
Well-posedness of the hydrostatic Navier–Stokes equations (2018, background): “n Sobolev spaces [12, 15]. Local in time wellposedness can be achieved when the initial datum is real analytic [39, 25], and even under the milder condition of Gevrey regularity in x[14]. We refer to [10, 42, 13, 20, 26, 9] and references therein for more results on the Prandtl system such as singul”
The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary (2019, result): “Lastly, we mention that in a recent remarkable result [16], Gerard-Varet-Maekawa-Masmoudi establish the stability in a Gevrey topology in x and a Sobolev topology in y, of Euler+Prandtl shear flows (cf.”