Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $L^\infty$ instability of Prandtl layers   In $1904$, Prandtl introduced his famous boundary layer in order to describe
the behavior of solutions of incompressible Navier Stokes equations near a
boundary as the viscosity goes to $0$. His Ansatz was that the solution of
Navier Stokes equations can be described as a solution of Euler equations, plus
a boundary layer corrector, plus a vanishing error term in $L^\infty$ in the
inviscid limit. In this paper we prove that, for a class of smooth solutions of
Navier Stokes equations, namely for shear layer profiles which are unstable for
Rayleigh equations, this Ansatz is false if we consider solutions with Sobolev
regularity, in strong contrast with the analytic case, pioneered by R.E.
Caflisch and M. Sammartino \cite{SammartinoCaflisch1,SammartinoCaflisch2}.
  Meanwhile we address the classical problem of the nonlinear stability of
shear layers near a boundary and prove that if a shear flow is spectrally
unstable for Euler equations, then it is non linearly unstable for the Navier
Stokes equations provided the viscosity is small enough.

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The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: cond-mat
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th