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

DOI10.1215/00127094-2018-0020zbMATH Open1420.35187arXiv1607.06434OpenAlexW3103756802WikidataQ129352958 ScholiaQ129352958MaRDI QIDQ1615255

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Publication date: 2 November 2018

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Abstract: We investigate the stability of boundary layer solutions of the two-dimensional incompressible Navier-Stokes equations. We consider shear flow solutions of Prandtl type : ${\displaystyle }$ u^

u(t,x,y) , = , �ig (U^E(t,y) + U^{BL}(t,frac{y}{sqrt{ u}}),, , 0 �ig ), , quad 0<

u ll 1,. ${\displaystyle }$ We show that if $U^{BL}$ is monotonic and concave in $Y=y/sqrtu$ then $u^u$ is stable over some time interval $(0,T)$, $T$ independent of $u$, 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.

Full work available at URL: https://arxiv.org/abs/1607.06434

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