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On the spectral instability of parallel shear flows. (English) Zbl 1353.76022
Summary: This short note is to announce our recent results [the authors, Adv. Math. 292, 52–110 (2016; Zbl 1382.76079); Duke Math. J. 165, No. 16, 3085–3146 (2016; Zbl 1359.35129)] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by W. Heisenberg [“Über Stabilität und Turbulenz von Flüssigkeitsströmen”, Ann. Phys. 379, No. 15, 577–627 (1924; doi:10.1002/andp.19243791502)], C. C. Lin [The theory of hydrodynamic stability. London, New York: Cambridge University Press (1955; Zbl 0068.39202)] and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number $$R \rightarrow \infty$$. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.
##### MSC:
 76E05 Parallel shear flows in hydrodynamic stability 35B35 Stability in context of PDEs 35Q53 KdV equations (Korteweg-de Vries equations) 76D05 Navier-Stokes equations for incompressible viscous fluids 76F10 Shear flows and turbulence
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##### References:
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