Instabilities

A roadmap to nonuniqueness of L^p weak solutions to Euler

ttn12 fluid dynamics, Instabilities

In this post, I discuss a rather classical roadmap to obtain the non-uniqueness of {L^p} weak solutions to the classical incompressible Euler equations; namely, focusing on the two-dimensional case, which reads in the vorticity formulation for vorticity function {\omega}:

\displaystyle \partial_t \omega + v \cdot \nabla \omega = 0, \qquad v = \nabla^\perp \Delta^{-1}\omega \ \ \ \ \ (1)

on {\mathbb{R}^2}, with initial vorticity in {L^p} (hence, vorticity remains in {L^p} for all times). It’s known, going far back to Yudovich ’63, that weak solutions with bounded vorticity are unique, leaving open the question of uniqueness of solutions whose vorticity is only in {L^p} for {p\in [1,\infty)}. This blog post is to discuss the possible quick roadmap to proving nonuniqueness arising from the instability nature of fluid models, focusing on the Euler equations (1).

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Invalidity of Prandtl’s boundary layers

ttn12 boundary layers, fluid dynamics, Instabilities, New papers , ,

I’ve just submitted this paper with Grenier (ENS Lyon) which studies Prandtl’s boundary layer asymptotic expansions for incompressible fluids on the half-space in the inviscid limit. In 1904, Prandtl introduced his well known boundary layers in order to describe the transition from Navier-Stokes to Euler equations in the inviscid limit.

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Green function for linearized Navier-Stokes around a boundary layer profile: near critical layers

ttn12 boundary layers, fluid dynamics, Instabilities , , ,

Emmanuel Grenier and I have just submitted this 84-page! long paper, also posted on arxiv (arXiv:1705.05323). This work is a continuation and completion of the program (initiated in Grenier-Toan1 and Grenier-Toan2) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a stationary boundary layer profile.

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Sublayer of Prandtl boundary layers

ttn12 boundary layers, fluid dynamics, Instabilities, Uncategorized , , ,

The aim of this paper (arXiv:1705.04672), with E. Grenier, is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: {\nu \rightarrow 0}.  In his CPAM2000 paper, Grenier proved that there exists no Prandtl’s asymptotic expansion involving one Prandtl’s boundary layer with thickness of order {\sqrt\nu}, which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order {\nu^{3/4}}. In this paper, we point out how the stability of the classical Prandtl’s layer is linked to the stability of this sublayer. In particular, we prove that the two layers cannot both be nonlinearly stable in {L^\infty}.  That is, either the Prandtl’s layer or the boundary sublayer is nonlinearly unstable in the sup norm.

Grenier’s nonlinear iterative scheme

ttn12 boundary layers, fluid dynamics, Instabilities ,

In his paper [Grenier, CPAM 2000], Grenier introduced a nonlinear iterative scheme to prove the instability of Euler and Prandtl equations. Recently, the scheme is also proved to be decisive in the study of water waves: [Ming-Rousset-Tzvetkov, SIAM J. Math. Anal., 2015], and plasma physics: [Han-Kwan & Hauray, CMP 2015] or my recent paper with Han-Kwan (see also my previous blog discussions). I am certain that it can be useful in other contexts as well. In this blog post, I’d like to give a sketch of the scheme to prove instability.

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Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits

ttn12 Instabilities, New papers, Plasma Physics

Daniel Han-Kwan and I have just submitted a paper entitled: “Nonlinear instability of Vlasov-Maxwell systems in the classical and quasineutral limits”, which is also available on arxiv: arXiv:1506.08537. In this paper, we study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit {\varepsilon \rightarrow 0}, with {\varepsilon} being the inverse of the speed of light, we construct a family of solutions that converge initially polynomially fast to a homogeneous solution {\mu} of Vlasov-Poisson in arbitrarily high Sobolev norms, but become of order one away from {\mu} in arbitrary negative Sobolev norms within time of order {|\log \varepsilon|}. Second, we deduce the invalidity of the quasineutral limit in {L^2} in arbitrarily short time.

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The onset of instability in first-order systems

ttn12 Instabilities, New papers

Nicolas Lerner, Ben Texier and I have just submitted to arxiv our long paper on “The onset of instability in first-order systems”, in which we prove the Hadamard’s instability for first-order quasilinear systems that lose its hyperbolicity in positive times. Precisely, we consider the Cauchy problem for the following first-order systems of partial differential equations:

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