On the nonlinear instability of Euler and Prandtl equations. (English) Zbl 1048.35081

The author proves that if the stationary solution of Euler equations is linearly unstable (in the sense that there exists an eigenvalue of the linearized Euler operator with nonnegative real part), then the time-dependent solution is nonlinearly unstable. This result is valid for the whole space, the half-space, for periodic and non-periodic strips, and for the periodic torus. An application to nonlinear stability of regularized vortex sheets is given. Similar results are also established for the nonlinear instability of Prandtl boundary layers, with application to the construction of asymptotic solutions for Navier-Stokes equations.


35Q35 PDEs in connection with fluid mechanics
76E30 Nonlinear effects in hydrodynamic stability
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
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