Blowup Criterion for Viscous Baratropic Flows with Vacuum States

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional (3D) barotropic compressible Navier-Stokes equations. More precisely, if a solution of the 3D barotropic compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce’s criterion for 3-dimensional incompressible Euler equations (Ponce in Commun Math Phys 98:349–353, 1985). In addition, initial vacuum states are allowed in our cases.     

Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.     

Feedback coupling loss in single mode fiber lasers

Based on the expansion of the fundamental mode LP₀₁ in single mode fiber in terms of Laguerre-Gaussian free space modes, the feedback coupling losses for two different types of cavity mirror, i.e., a curvature mirror and a combing mirror of a lens and a plane mirror, are numerically calculated for the first time. The results show that, for the curvature mirror, the lowest coupling loss is obtained when its curvature radius matches the wavefront curvature. In particular, if a plane mirror is used as the cavity mirror, it has to be placed close to the fiber end to obtain the low coupling loss. For the combing mirror, the lowest coupling loss can be obtained when the plane mirror is placed at the back focal plane of the lens, and the variation of the coupling loss is insensitive to the mirror positions for the lens with longer focal length. Finally, the plane mirror and the combing mirror of a lens and a plane mirror are suggested to be the cavity mirror in the practical construction of the high power fiber lasers.     

Large depth-of-view portable three-dimensional laser scanner and its segmental calibration for robot vision

A portable 3D laser scanning system has been designed and built for robot vision. By tilting the charge coupled device (CCD) plane of portable 3D scanning system according to the Scheimpflug condition, the depth-of-view is successfully extended from less than 40 to 100 mm. Based on the tilted camera model, the traditional two-step camera calibration method is modified by introducing the angle factor. Meanwhile, a novel segmental calibration approach, i.e., dividing the whole work range into two parts and calibrating, respectively, with corresponding system parameters, is proposed to effectively improve the measurement accuracy of the large depth-of-view 3D laser scanner. In the process of 3D reconstruction, different calibration parameters are used to transform the 2D coordinates into 3D coordinates according to the different positions of the image in the CCD plane, and the measurement accuracy of is obtained experimentally. Finally, the experiment of scanning a lamina by the large depth-of-view portable 3D laser scanner used by an industrial robot IRB 4400 is also employed to demonstrate the effectiveness and high measurement accuracy of our scanning system.     

Global Weak Solutions to One-Dimensional Non-Conservative Viscous Compressible Two-Phase System

In this paper, we deal with global weak solutions of a non-conservative viscous compressible two-phase model in one space dimension. This work extends in some sense the previous work, [Bresch et al., Arch Rat Mech Anal 196:599–629, 2010], which provides the global existence of weak solutions in the multi-dimensional framework with 1 < γ_± < 6 assuming non-zero surface tension. In our study, we strongly improve the results by taking advantage of the one space dimension. More precisely, we obtain global existence of weak solutions without using capillarity terms and for pressure laws with the same range of coefficients as the degenerate barotropic mono-fluid system, namely γ_± > 1. Then we prove that any possible vacuum state has to vanish within finite time after which densities are always away from vacuum. This allows to prove that at least one phase corresponding to the global weak solution is a locally in time and space (in a sense to be defined) strong solution after the vacuum states vanish. Our paper may be understood as a non-straightforward generalization to the two-phase flow system of a previous paper [Li et al., Commun Math Phys 281(2):401–444, 2008], which treated the usual compressible barotropic Navier-Stokes equations for mono-fluid with a degenerate viscosity. Various important mathematical difficulties occur when we want to generalize those results to the two-phase flows system since the corresponding model is non-conservative. Far from vacuum, it involves a strong coupling between a nonlinear algebraic system and a degenerate PDE system under constraint linked to fractions. Moreover, fractional densities may vanish if densities or fractions vanish: A difficulty is to find estimates on the densities from estimates on fractional densities using the algebraic system. Original approximate systems have also to be introduced compared to the works on the degenerate barotropic mono-fluid system. Note that even if our result concerns “only” the one-dimensional case, it points out possible global weak solutions (for such a non-conservative system) candidates to approach for instance shock structures and to define an appropriate a priori family of paths in the phase space (in numerical schemes) at the zero dissipation limit.     

A Serrin criterion for compressible nematic liquid crystal flows

This paper is concerned with a simplified system, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. We establish a blowup criterion for three‐dimensional compressible nematic liquid crystal flows, which is analogous to the well‐known Serrin's blowup criterion for three‐dimensional incompressible viscous flows. Copyright © 2012 John Wiley & Sons, Ltd.     

Design of a single-polarization single-mode photonic crystal fiber double-core coupler

A simple single-polarization single-mode (SPSM) photonic crystal fiber (PCF) coupler with two cores is introduced. The full-vector finite-element method (FEM) is applied to analyze the modal interference phenomenon of the even and odd modes of two orthogonal polarizations and the power propagation within the two cores. Meanwhile, the SPSM coupling wavelength range and its corresponding coupling length for different structure parameters are numerically analyzed. The numerical results show that SPSM coupling can be realized with a broad range of wavelength, and the coupling length can be of millimeter magnitude. Moreover, the SPSM coupling wavelength range and the coupling length can be optimized by designing proper mirco-structure parameters of the coupler.     

Global Wellposedness for the 3D Inhomogeneous Incompressible Navier–Stokes Equations

This paper addresses the three-dimensional Navier–Stokes equations for an incompressible fluid whose density is permitted to be inhomogeneous. We establish a theorem of global existence and uniqueness of strong solutions for initial data with small ${\dot{H}^{\frac12}}$ -norm, which also satisfies a natural compatibility condition. A key point of the theorem is that the initial density need not be strictly positive.     

L∞ continuation principle to the non‐baratropic non‐resistive magnetohydrodynamic equations without heat conductivity

The aim of this paper is to establish a continuation principle for strong solutions to the full compressible magnetohydrodynamic system without resistivity and heat conductivity. We prove that if the solution loses its regularity in finite time, the dominated part is due to the hyperbolic effect. More precisely, it is essentially shown that the strong solution exists globally if the density, temperature, and magnetic field are bounded from above, where vacuum is allowed to exist. This verifies a problem proposed by D.Serre. Copyright © 2016 John Wiley & Sons, Ltd.     

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 that are of small energy but possibly large oscillations with constant state as far field, which could be either vacuum or nonvacuum. 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 that may have large oscillations and can contain vacuum states. © 2012 Wiley Periodicals, Inc.