Global Classical and Weak Solutions to the Three-Dimensional Full Compressible Navier-Stokes System with Vacuum and Large Oscillations

We establish the global existence and uniqueness of classical solutions to the three-dimensional full compressible Navier-Stokes system with smooth initial data which are of small energy but possibly large oscillations where the initial density is allowed to vanish. Moreover, for the initial data which may be discontinuous and contain vacuum states, we also obtain the global existence of weak solutions. These results generalize previous ones on classical and weak solutions for initial density being strictly away from vacuum, and are the first for global classical and weak solutions which may have large oscillations and can contain vacuum states.Comment: 61 page

Existence and Blowup Behavior of Global Strong Solutions to the Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data

For periodic initial data with initial density allowed to vanish, we establish the global existence of strong and weak solutions for the two-dimensional compressible Navier-Stokes equations with no restrictions on the size of initial data provided the bulk viscosity coefficient is $\lambda = \rho^{\beta}$ with $\beta>4/3$. These results generalize and improve the previous ones due to Vaigant-Kazhikhov([Sib. Math. J. (1995), 36(6), 1283-1316]) which requires $\beta>3$. Moreover, both the uniform upper bound of the density and the large-time behavior of the strong and weak solutions are also obtained.Comment: 31 page

Serrin Type Criterion for the Three-Dimensional Viscous Compressible Flows

We extend the well-known Serrin's blowup criterion for the three-dimensional (3D) incompressible Navier-Stokes equations to the 3D viscous compressible cases. It is shown that for the Cauchy problem of the 3D compressible Navier-Stokes system in the whole space, the strong or smooth solution exists globally if the velocity satisfies the Serrin's condition and either the supernorm of the density or the $L^1(0,T;L^\infty)$-norm of the divergence of the velocity is bounded. Furthermore, in the case that either the shear viscosity coefficient is suitably large or there is no vacuum, the Serrin's condition on the velocity can be removed in this criteria.Comment: 16 page

Oblique Hanle Effect in Semiconductor Spin Transport Devices

Spin precession and dephasing ("Hanle effect") provides an unambiguous means to establish the presence of spin transport in semiconductors. We compare theoretical modeling with experimental data from drift-dominated silicon spin-transport devices, illustrating the non-trivial consequences of employing oblique magnetic fields (due to misalignment or intentional, fixed in-plane field components) to measure the effects of spin precession. Model results are also calculated for Hanle measurements under conditions of diffusion-dominated transport, revealing an expected Hanle peak-widening effect induced by the presence of fixed in-plane magnetic bias fields

Global Well-Posedness of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Isentropic Compressible Navier-Stokes Equations

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data which are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or non-vacuum. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum, and are the first for global classical solutions which may have large oscillations and can contain vacuum states.Comment: 30 page

On a Variant of the Elliott-Halberstam Conjecture and the Goldbach Conjecture

In this paper we prove that the binary Goldbach conjecture for sufficiently large even integers would follow under the assumptions of both the Elliott-Halberstam conjecture and a variant of the Elliott-Halberstam conjecture twisted by the M\"{o}bius function, provided that the sum of their level of distributions exceeds 1. This continues the work of Pan. An analogous result for the twin prime conjecture is obtained by Ram Murty and Vatwani