Stable transport of information near essentially unstable localized structures

Localized mixing zone for Muskat bubbles and turned interfaces

We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh-Taylor and smoothness breakdown exhibited in [18,17]. At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat …

On the ill-posedness of the triple deck model

We analyze the stability properties of the so-called triple deck model, a classical refinement of the Prandtl equation to describe boundary layer separation. Combining the methodology introduced in [2], based on complex analysis tools, and stability estimates inspired from [3], we exhibit unstable linearizations of the triple deck equation. The growth rates of the corresponding unstable eigenmodes scale linearly with the tangential frequency. This shows that the recent result of Iyer and Vicol [11] of local well-posedness for analytic data is essentially optimal.

环形涡旋线的Lyapunov不稳定性

Lyapunov Unstable Motion Bifurcating from a Circular Vortex Filament

本文研究了由局部感应方程支配的三维闭合涡旋线动力学，证明了环形涡旋线存在一种称为轴向螺旋运动的闭合解，这些解从环形涡旋线分岔出来，虽然保持与环形涡旋线轨道的均匀接近，但因轴向速度不同而逐渐远离，揭示了轨道稳定性和Lyapunov稳定性之间的差距。

Compactness along the Branch of Semi-stable and Unstable Solutions for an Elliptic Problem with a Singular Nonlinearity

We study the branch of semi-stable and unstable solutions (i.e., those whose Morse index is at most one) of the Dirichlet boundary value problem $-Δu=\frac{λf(x)}{(1-u)^2}$ on a bounded domain $Ω\subset \R^N$, which models --among other things-- a simple electrostatic Micro-Electromechanical System (MEMS) device. We extend the results of [11] relating to the minimal branch, by obtaining compactness along unstable branches for $1\leq N \leq 7$ on any domain $Ω$ and for a large class of "permittivity profiles" $f$ . We also show the remarkable fact that power-like profiles $f(x) \simeq |x|^α$ ca…

Spinodal decomposition and coarsening fronts in the Cahn-Hilliard equation

We study spinodal decomposition and coarsening when initiated by localized disturbances in the Cahn-Hilliard equation. Spatio-temporal dynamics are governed by multi-stage invasion fronts. The first front invades a spinodal unstable equilibrium and creates a spatially periodic unstable pattern. Secondary fronts invade this unstable pattern and create a coarser pattern in the wake. We give linear predictions for speeds and wavenumbers in this process and show existence of corresponding nonlinear fronts. The existence proof is based on Conley index theory, a priori estimates, and Galerkin approx…

Diffusive stability of oscillations in reaction-diffusion systems

We study nonlinear stability of spatially homogeneous oscillations in reaction-diffusion systems. Assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase, we prove that spatially localized perturbations decay algebraically with the diffusive rate t^{-n/2} in space dimension n. We also compute the leading order term in the asymptotic expansion of the solution, and show that it corresponds to a spatially localized modulation of the phase. Our approach is based on a normal form transformation in the kinetics ODE which partially decouples the phase equation, a…