Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to ${4096}^3$. The results are analyzed in terms of the classical analyticity-strip method and Beale, Kato, and Majda (BKM) theorem. A well-resolved acceleration of the time decay of the width of the analyticity strip $\delta \left(t\right)$ is observed at the highest resolution for $3.7<t<3.85$ while preliminary three-dimensional visualizations show the collision of vortex sheets. The BKM criterion on the power-law growth of the supremum of the vorticity, applied on the same time interval, is not inconsistent with the occurrence of a singularity around $t\simeq 4$. These findings lead us to investigate how fast the analyticity-strip width needs to decrease to zero in order to sustain a finite-time singularity consistent with the BKM theorem. A simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the BKM theorem with the analyticity-strip method. It is shown that a finite-time blowup can exist only if $\delta \left(t\right)$ vanishes sufficiently fast at the singularity time. In particular, if a power law is assumed for $\delta \left(t\right)$ then its exponent must be greater than some critical value, thus providing a new test that is applied to our ${4096}^3$ Taylor-Green numerical simulation. Our main conclusion is that the numerical results are not inconsistent with a singularity but that higher-resolution studies are needed to extend the time interval on which a well-resolved power-law behavior of $\delta \left(t\right)$ takes place and check whether the new regime is genuine and not simply a crossover to a faster exponential decay. DOI: http://dx.doi.org/10.1103/PhysRevE.86.066302 ©2012 American Physical Society

Assuming that resonances play a major role in the transfer of energy among the Fourier modes, we apply the Wave Turbulence theory to describe the dynamics on nonlinear one-dimensional chains. We consider α and β Fermi-Pasta-Ulam-Tsingou (FPUT) systems, and the discrete nonlinear Klein-Gordon chain. We consider both the thermodynamic limit and the discrete regime and we conjecture that all the systems thermalize for large times, and that the equipartition time scales as a power-law of the strength of the nonlinearity, at least for a range of values of the nonlinear parameter. We perform state of the art numerical simulations and show that the results are mostly consistent with theoretical predictions. Some observed discrepancies are discussed. We suggest that the route to thermalization, based on the presence of exact resonance, has universal features. Moreover, a by-product of our analysis is the asymptotic integrability, up to four wave interactions, of the discrete nonlinear Klein-Gordon chain.

We investigate the ideal and incompressible magnetohydrodynamic (MHD) equations in three space dimensions for the development of potentially singular structures. The methodology consists in implementing the fourfold symmetries of the Taylor-Green vortex generalized to MHD, leading to substantial computer time and memory savings at a given resolution; we also use a regridding method that allows for lower-resolution runs at early times, with no loss of spectral accuracy. One magnetic configuration is examined at an equivalent resolution of ${6144}^3$ points and three different configurations on grids of ${4096}^3$ points. At the highest resolution, two different current and vorticity sheet systems are found to collide, producing two successive accelerations in the development of small scales. At the latest time, a convergence of magnetic field lines to the location of maximum current is probably leading locally to a strong bending and directional variability of such lines. A novel analytical method, based on sharp analysis inequalities, is used to assess the validity of the finite-time singularity scenario. This method allows one to rule out spurious singularities by evaluating the rate at which the logarithmic decrement of the analyticity-strip method goes to zero. The result is that the finite-time singularity scenario cannot be ruled out, and the singularity time could be somewhere between $t=2.33$ and $t=2.70.$ More robust conclusions will require higher resolution runs and grid-point interpolation measurements of maximum current and vorticity.