On the nonlinear instability of Euler and Prandtl equations

By E. Grenier. 2000

306 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/d502503768621668f9f62da790373316d142eae0

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

Well-posedness of the Prandtl equation in Sobolev spaces (2012, background): “In [8], Grenier showed that the unstable Euler shear flow (us(y), 0) with us(y) having an inflection point (the well-known Rayleigh’s criterion) yields instability for the Prandtl equation.”
Linear Inviscid Damping for a Class of Monotone Shear Flow in Sobolev Spaces (2015, background): “We refer to [2, 10, 14, 24] and references therein for nonlinear instability.”
Well‐Posedness in Gevrey Function Space for 3D Prandtl Equations without Structural Assumption (2020, background): “Here we only mention the works [10] about the construction of blowup solutions (see also [5,6]), [17] on the unstable Euler shear flow that yields instability of Prandtl equation, [12, 16, 20] about the instability around a shear flow with a nondegenerate critical point, [18] on the instabilility ev”
On the local existence of analytic solutions to the Prandtl boundary layer equations (2013, unknown): “Additionally, in [12, 15] it is proven that if one linearizes the system ( 1.”
Nonlinear instability of ideal plane flows (2004, background): “In [11] Grenier proved nonlinear instability in the L(2) norm of velocity, for shear ows.”
The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary (2019, methodology): “We refer the reader to [1, 10, 17, 28, 35, 36, 37, 39, 47, 51, 52] for the well-posedness theory for the Prandtl equations, to [14, 26, 18, 38] for the identification of ill-posed regimes, and to [19, 20, 21, 22, 23] for recent works which show the invalidity of the Prandtl expansion at the level of”
A note on Prandtl boundary layers (2010, background): “In addition, the blow-up norm in [4] is H1 in the original variable y, whereas our result concerns the (weaker) norm in the stretched variable Y = y/ √ ν, upon noting that ‖u(y/ν)‖L2y(R+) = ν 1/4‖u(Y )‖L2Y (R+).”
On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half‐Plane (2012, background): “note here that the solvability of the Prandtl equations itself does not necessarily imply the desired asymptotic expansion, as in the counterexample by [18].”
The Inviscid Limit of Navier–Stokes Equations for Analytic Data on the Half-Space (2017, background): “ed, except for data with special symmetry data with special symmetry ([30, 13] and the references therein), the asymptotic expansions (1.3) are false due to the strong instability of boundary layers ([14, 15, 16, 17, 19]), including those boundary layers that are spectrally stable to the Euler equat”
Well-posedness of the Prandtl equation without any structural assumption (2018, background): “About the validity of boundary layer expansions in the unsteady setting, there are many possible references, among which [31, 14, 32, 11, 26, 16].”
Spectral instability of characteristic boundary layer flows (2014, background): “Available analyses (for instance, [3, 4]) do not appear applicable in the inviscid limit.”
Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations (2013, unknown): “). We also refer to [42, 51] for the use of the variational approach in the Vlasov case. Recently there were many mathematical studies of stability and instability of various ﬂows (see 2 for instance [32, 10, 41, 55]). We also refer to the following textbooks on the topic of hydrodynamic stability a”
Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs (2017, unknown): “[27] [31] [36] [50] [51]).”
Euler equations for incompressible ideal fluids (2007, background): “The following example, constructed by Grenier [56], shows that the Prandtl expansion cannot always be valid.”
Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations (2012, background): “Moreover, even when this is possible, some instabilities make impossible the control of the remainder on a suitable interval of time [22], [27].”
Global regularity for some classes of large solutions to the Navier-Stokes equations (2008, background): “Roughly speaking, this initial data indeed may give rise to instability in the inviscid case or for small viscosity, as for instance in the case of the Prandtl equation arising in the problem of vanishing viscosity in domains with boundaries (see [16]) and the hypothesis of analyticity can make the ”