Linear and Non-Linear Waves
http://hdl.handle.net/20.500.11824/4
2021-09-21T06:31:33ZEcho Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics
http://hdl.handle.net/20.500.11824/1315
Echo Chains as a Linear Mechanism: Norm Inflation, Modified Exponents and Asymptotics
Deng, Y.; Zillinger, C.
In this article we show that the Euler equations, when linearized around a low frequency perturbation to Couette flow, exhibit norm inflation in Gevrey-type spaces as time tends to infinity. Thus, echo chains are shown to be a (secondary) linear instability mechanism. Furthermore, we develop a more precise analysis of cancellations in the resonance mechanism, which yields a modified exponent in the high frequency regime. This allows us, in addition, to remove a logarithmic constraint on the perturbations present in prior works by Bedrossian, Deng and Masmoudi, and to construct solutions which are initially in a Gevrey class for which the velocity asymptotically converges in Sobolev regularity but diverges in Gevrey regularity.
2021-07-30T00:00:00ZStatic and Dynamical, Fractional Uncertainty Principles
http://hdl.handle.net/20.500.11824/1255
Static and Dynamical, Fractional Uncertainty Principles
Kumar, S.; Ponce-Vanegas, F.; Vega, L.
We study the process of dispersion of low-regularity solutions to the Schrödinger equation using fractional weights (observables). We give another proof of the uncertainty principle for fractional weights and use it to get a lower bound for the concentration of mass. We consider also the evolution when the initial datum is the Dirac comb in $\mathbb{R}$. In this case we find fluctuations that concentrate at rational times and that resemble a realization of a Lévy process. Furthermore, the evolution exhibits multifractality.
2021-03-01T00:00:00ZInvariant measures for the dnls equation
http://hdl.handle.net/20.500.11824/1211
Invariant measures for the dnls equation
Lucà, R.
We describe invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schr\"odinger equation (DNLS) constructed in \cite{MR3518561, Genovese2018}. The construction works for small $L^2$ data. The measures are absolutely continuous with respect to suitable weighted Gaussian measures supported on Sobolev spaces of increasing regularity. These results have been obtained in collaboration with Giuseppe Genovese (University of Z\"urich) and Daniele Valeri (University of Glasgow).
2020-10-02T00:00:00ZMagnetic domain-twin boundary interactions in Ni-Mn-Ga
http://hdl.handle.net/20.500.11824/1209
Magnetic domain-twin boundary interactions in Ni-Mn-Ga
Veligatla, M.; Garcia-Cervera, C.J.; Müllner, P.
The stress required for the propagation of twin boundaries in a sample with fine twins increases monotonically with ongoing deformation. In contrast, for samples with a single twin boundary, the stress exhibits a plateau over the entire twinning deformation range. We evaluate the twin boundary and magnetic domain boundary interactions for increasing twin densities. As the twinned regions get finer, these interaction regions result in additional magnetic domains that form magnetoelastic defects with high magnetostress concentrations. These magnetoelastic defects act as obstacles for twinning disconnections and, thus, harden the material. Whereas in a low twin density microstructure, these high-energy concentrations are absent or dilute and their effectiveness is reduced by the synergistic action of many twinning disconnections. Therefore, with increasing twin density, the interaction of twin boundary and magnetic domain boundaries reduces the twin boundary mobility. The defect strength has a distribution such that twinning disconnections overcome soft obstacles first and harder obstacles with ongoing deformation. The width of the distribution of obstacle strength and the density of obstacles increase with increasing twin density and, thus, the hardening coefficient increases with increasing twin density.
2020-04-01T00:00:00Z