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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.
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
Full Text: DOI
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