1,093 research outputs found
A brief summary of nonlinear echoes and Landau damping
In this expository note we review some recent results on Landau damping in
the nonlinear Vlasov equations, focusing specifically on the recent
construction of nonlinear echo solutions by the author [arXiv:1605.06841] and
the associated background. These solutions show that a straightforward
extension of Mouhot and Villani's theorem on Landau damping to Sobolev spaces
on is impossible and hence emphasize the
subtle dependence on regularity of phase mixing problems. This expository note
is specifically aimed at mathematicians who study the analysis of PDEs, but not
necessarily those who work specifically on kinetic theory. However, for the
sake of brevity, this review is certainly not comprehensive.Comment: Expository note for the Proceedings of the Journees EDP 2017, based
on a talk given at Journees EDP 2017 in Roscoff, France. Aimed at
mathematicians who study the analysis of PDEs, but not necessarily those who
work specifically on kinetic theory. 16 page
Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations
We prove asymptotic stability of shear flows close to the planar Couette flow
in the 2D inviscid Euler equations on \Torus \times \Real. That is, given an
initial perturbation of the Couette flow small in a suitable regularity class,
specifically Gevrey space of class smaller than 2, the velocity converges
strongly in L^2 to a shear flow which is also close to the Couette flow. The
vorticity is asymptotically driven to small scales by a linear evolution and
weakly converges as . The strong convergence of the
velocity field is sometimes referred to as inviscid damping, due to the
relationship with Landau damping in the Vlasov equations. This convergence was
formally derived at the linear level by Kelvin in 1887 and it occurs at an
algebraic rate first computed by Orr in 1907; our work appears to be the first
rigorous confirmation of this behavior on the nonlinear level.Comment: 78 page
Blister patterns and energy minimization in compressed thin films on compliant substrates
This paper is motivated by the complex blister patterns sometimes seen in
thin elastic films on thick, compliant substrates. These patterns are often
induced by an elastic misfit which compresses the film. Blistering permits the
film to expand locally, reducing the elastic energy of the system. It is
natural to ask: what is the minimum elastic energy achievable by blistering on
a fixed area fraction of the substrate? This is a variational problem involving
both the {\it elastic deformation} of the film and substrate and the {\it
geometry} of the blistered region. It involves three small parameters: the {\it
nondimensionalized thickness} of the film, the {\it compliance ratio} of the
film/substrate pair and the {\it mismatch strain}. In formulating the problem,
we use a small-slope (F\"oppl-von K\'arm\'an) approximation for the elastic
energy of the film, and a local approximation for the elastic energy of the
substrate.
For a 1D version of the problem, we obtain "matching" upper and lower bounds
on the minimum energy, in the sense that both bounds have the same scaling
behavior with respect to the small parameters. For a 2D version of the problem,
our results are less complete. Our upper and lower bounds only "match" in their
scaling with respect to the nondimensionalized thickness, not in the dependence
on the compliance ratio and the mismatch strain. The upper bound considers a 2D
lattice of blisters, and uses ideas from the literature on the folding or
"crumpling" of a confined elastic sheet. Our main 2D result is that in a
certain parameter regime, the elastic energy of this lattice is significantly
lower than that of a few large blisters
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