On the stability of Prandtl boundary layer expansions of Navier-Stokes in the inviscid limit

On the stability of Prandtl boundary layer expansions of Navier-Stokes in the inviscid limit

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Toan Nguyen , Penn State
Fine Hall 322

I will present two recent results concerning the stability of boundary layer asymptotic expansions of solutions of Navier-Stokes with small viscosity. First, we show that the linearization around an arbitrary stationary shear flow (other than the Couette flow) admits an unstable eigenfunction with small wave number, when viscosity is sufficiently small. In boundary-layer variables, this yields an exponentially growing sublayer near the boundary and hence instability of the asymptotic expansions, within an arbitrarily small time, in the inviscid limit. The proof introduces a new functional-analytic approach to construct exact solutions of Orr-Sommerfeld in presence of critical layers (primitive Airy solutions). On the other hand, we show that the Prandtl asymptotic expansions hold for steady flows. Our proof involves delicate construction of approximate solutions (linearized Euler and Prandtl layers) and an introduction of a new positivity estimate for steady Navier-Stokes. This in particular establishes the inviscid limit of steady flows with prescribed boundary data up to order of square root of small viscosity. This is a joint work with Emmanuel Grenier and Yan Guo.