Abstract
In the lines of the recent paper [J. Amer. Math. Soc. 23(2) (2010), 591-609], we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary non-monotonic shear flows, we show that for some C ∞ initial data, local in time H 1 solutions of the linearized Prandtl equation do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 71-88 |
| Number of pages | 18 |
| Journal | Asymptotic Analysis |
| Volume | 77 |
| Issue number | 1-2 |
| DOIs | |
| State | Published - Apr 11 2012 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
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Remarks on the ill-posedness of the Prandtl equation. / Gérard-Varet, D.; Nguyen, T.
In: Asymptotic Analysis, Vol. 77, No. 1-2, 11.04.2012, p. 71-88.Research output: Contribution to journal › Article
TY - JOUR
T1 - Remarks on the ill-posedness of the Prandtl equation
AU - Gérard-Varet, D.
AU - Nguyen, T.
PY - 2012/4/11
Y1 - 2012/4/11
N2 - In the lines of the recent paper [J. Amer. Math. Soc. 23(2) (2010), 591-609], we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary non-monotonic shear flows, we show that for some C ∞ initial data, local in time H 1 solutions of the linearized Prandtl equation do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.
AB - In the lines of the recent paper [J. Amer. Math. Soc. 23(2) (2010), 591-609], we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary non-monotonic shear flows, we show that for some C ∞ initial data, local in time H 1 solutions of the linearized Prandtl equation do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.
UR - http://www.scopus.com/inward/record.url?scp=84859471486&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84859471486&partnerID=8YFLogxK
U2 - 10.3233/ASY-2011-1075
DO - 10.3233/ASY-2011-1075
M3 - Article
AN - SCOPUS:84859471486
VL - 77
SP - 71
EP - 88
JO - Asymptotic Analysis
JF - Asymptotic Analysis
SN - 0921-7134
IS - 1-2
ER -