This short note is to be published as the proceeding of a Laurent Schwartz PDE seminar talk that I gave last May at IHES, announcing our recent results (on channel flows and boundary layers), which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin 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 $R \to \infty$. 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. In fact, the material in this note is only the first half of what I spoke on that day, skipping the steady case!

Yesterday, Nov 17, Xu and Zhang posted a preprint on the ArXiv, entitled “Well-posedness of the Prandtl equation in Sobolev space without monotonicity” (arXiv:1511.04850), claiming to prove what the title says. This immediately causes some concern or possible contrary to what has been known previously! Here, monotonicity is of the horizontal velocity component in the normal direction to the boundary. It’s well-known that monotonicity implies well-posedness of Prandtl (e.g., Oleinik in the 60s; see this previous post for Prandtl equations). It is then first proved by Gerard-Varet and Dormy that without monotonicity, the Prandtl equation is linearly illposed (and some followed-up works on the nonlinear case that I wrote with Gerard-Varet, and then with Guo). Is there a contradictory to what it’s known and this new preprint of Xu and Zhang? The purpose of this blog post is to clarify this.

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