TY - JOUR

T1 - On nonlinear instability of Prandtl's boundary layers

T2 - The case of Rayleigh's stable shear flows

AU - Grenier, Emmanuel

AU - Nguyen, Toan T.

N1 - Publisher Copyright:
© 2024 Elsevier Masson SAS

PY - 2024/4

Y1 - 2024/4

N2 - In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to O(ν1/4) order terms in L∞ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when ν=0, are nonlinearly unstable when ν>0, provided ν is small enough, up to O(ν1/4) terms in L∞ norm.

AB - In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to O(ν1/4) order terms in L∞ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when ν=0, are nonlinearly unstable when ν>0, provided ν is small enough, up to O(ν1/4) terms in L∞ norm.

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U2 - 10.1016/j.matpur.2024.02.001

DO - 10.1016/j.matpur.2024.02.001

M3 - Article

AN - SCOPUS:85187014788

SN - 0021-7824

VL - 184

SP - 71

EP - 90

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

ER -