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.
KW - Boundary layers
KW - Hydrodynamic stability
KW - Navier-Stokes equation
KW - Nonlinear instability
KW - Prandtl layers
UR - https://www.scopus.com/pages/publications/85187014788
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 -