Hydrodynamic limits of the nonlinear Schrödinger equation with the Chern-Simons gauge fields

<p style='text-indent:20px;'>We present two types of the hydrodynamic limit of the nonlinear Schrödinger-Chern-Simons (SCS) system. We consider two different scalings of the SCS system and show that each SCS system asymptotically converges towards the compressible and incompressible Euler system, coupled with the Chern-Simons equations and Poisson equation respectively, as the scaled Planck constant converges to 0. Our method is based on the modulated energy estimate. In the case of compressible limit, we observe that the classical theory of relative entropy method can be applied to show the hydrodynamic limit, with the additional quantum correction term. On the other hand, for the incompressible limit, we directly estimate the modulated energy to derive the desired asymptotic convergence.</p>

L2-type contraction of viscous shocks for scalar conservation laws

We study small shocks of 1D scalar viscous conservation laws with uniformly convex flux and nonlinear dissipation. We show that such shocks are [Formula: see text] stable independently of the strength of the dissipation, even with large perturbations. The proof uses the relative entropy method with a spatially-inhomogeneous pseudo-norm.

Relative Entropy Method Applied for the Fatigue Life Distribution of Carbon Fiber/Epoxy Composites

This paper verifies the feasibility of the relative entropy method in selecting the most suitable statistical distribution for the experimental data, which do not follow an exponential distribution. The efficiency of the relative entropy method is tested through the fractional order moment and the logarithmic moment in terms of the experimental data of carbon fiber/epoxy composites with different stress amplitudes. For better usage of the relative entropy method, the efficient range of its application is also studied. The application results show that the relative entropy method is not very fit for choosing the proper distribution for non-exponential random data when the heavy tail trait of the experimental data is emphasized. It is not consistent with the Kolmogorov–Smirnov test but is consistent with the residual sum of squares in the least squares method whenever it is calculated by the fractional moment or the logarithmic moment. Under different stress amplitudes, the relative entropy method has different performances.