Vanishing Viscosity with Short Wave–Long Wave Interactions for Systems of Conservation Laws

Motivated by Benney’s general theory, we propose new models for short wave–long wave interactions when the long waves are described by nonlinear systems of conservation laws. We prove the strong convergence of the solutions of the vanishing viscosity and short wave–long wave interactions systems by using compactness results from compensated compactness theory and new energy estimates obtained for the coupled systems. We analyze several of the representative examples, such as scalar conservation laws, general symmetric systems, nonlinear elasticity and nonlinear electromagnetism.     

Hyperbolicity of the Nonlinear Models of Maxwell’s Equations

We consider the class of nonlinear models of electromagnetism that has been described by Coleman & Dill [7]. A model is completely determined by its energy density W(B,D). Viewing the electromagnetic field (B,D) as a 3×2 matrix, we show that polyconvexity of W implies the local well-posedness of the Cauchy problem within smooth functions of class H^s with s>1+d/2. The method follows that designed by Dafermos in his book [9] in the context of nonlinear elasticity. We use the fact that B×D is a (vectorial, non-convex) entropy, and we enlarge the system from 6 to 9 equations. The resulting system admits an entropy (actually the energy) that is convex. Since the energy conservation law does not derive from the system of conservation laws itself (Faradays and Ampères laws), but also needs the compatibility relations divB=divD=0 (the latter may be relaxed in order to take into account electric charges), the energy density is not an entropy in the classical sense. Thus the system cannot be symmetrized, strictly speaking. However, we show that the structure is close enough to symmetrizability, so that the standard estimates still hold true.     

Inequalities of Korn’s Type and Existence Results in the Theory of Cosserat Elastic Shells

This paper is concerned with the linear theory of anisotropic and inhomogeneous Cosserat elastic shells. We establish the inequalities of Korn’s type which hold on Cosserat surfaces. Using these inequalities, we prove the existence of the solution to the variational equations in the elastostatics of Cosserat shells. For the dynamic problems, we employ the semigroup of linear operators theory to obtain the existence, uniqueness and continuous dependence of solution.      

Circular Shallow Spherical Shells with Central Circular Hole Under Simultaneous Actions of Arbitrary Unsteady Temperature Field and Arbitrary Dynamic Normal Load

In this paper,under assumption that tempeature is linearly distributed along the thickness of theshell,we deal with problems as indicated in the title and obtain general solutions of them which areexpressed in analytic form.In the first part,we investigate free vibration of circular shallow spherical shells with circularholes at the center under usual arbitrary boundary conditions.As an example,we calculate fundamen-tal natural frequency of a circular shallow spherical shell whose edge is fixed(m=0).Results we get areexpressed in analytic form and check well with E.Reissner’s[1].Method for calculating frequencyequation is recently suggested by Chien Wei-zang and is to be introduced in appendix3.In the second part,we investigate forced vibration of shells as indicated in the title under arbitr-ary harmonic temperature field and arbitrary harmonic dynamic normal load.In the third part,we investigate forced vibration of the above mentioned shells with initialconditions under arbitrary unsteady temperature     

Dynamic homogenisation of Maxwell’s equations with applications to photonic crystals and localised waveforms on gratings

A two-scale asymptotic theory is developed to generate continuum equations that model the macroscopic behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. The resulting equations include tensors that represent effective refractive indices near band edge frequencies along all principal axes directions, and these govern scalar functions providing long-scale modulation of short-scale Bloch eigenstates, which can be used to predict the propagation of waves at frequencies outside of the long wavelength regime; these results are outside of the remit of typical homogenisation schemes.The theory we develop is applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in modelling photonic crystal fibres. Results of the asymptotic theory are verified against numerical simulations by comparing photonic band diagrams and evanescent decay rates for guided modes. The second example is the propagation of electromagnetic waves localised within a planar array of dielectric spheres; at certain frequencies strongly directional propagation is observed, commonly described as dynamic anisotropy. Computationally this is a challenging three-dimensional calculation, which we perform, and then demonstrate that the asymptotic theory captures the effect, giving highly accurate qualitative and quantitative comparisons as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour.