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 Hs with s > 1 + d/2. The method follows that...

We introduce a new approach to decompose a system of viscous conservation laws with respect to each characteristic wave structures. Under this new decomposition, the global wave interactions of the system are reduced to coupling of nonlinear waves around constant states outside shock region and a scalar conservation law in the shock region to determine the behavior of local internal shock layer...

In this paper we propose a novel high resolution scheme for scalar nonlinear hyperbolic conservation laws. The aim of high resolution schemes is to provide at least second order accuracy in smooth regions and produce sharp solutions near the discontinuities. We prove that the proposed scheme that is derived by utilizing an appropriate flux limiter is nonlinear stable in the sense of total varia...

We analyze some recent developments in studying discontinuous solutions to nonlinear evolutionary partial differential equations. The central problems include the existence, compactness, and large-time behavior of discontinuous solutions. The nonlinear equations we discuss include nonlinear hyperbolic systems of conservation laws (especially the compressible Euler equations) and the compressibl...

We present a new formulation of the Runge-Kutta discontinuous Galerkin (RKDG) method [6, 5, 4, 3] for conservation Laws. The new formulation requires the computed RKDG solution in a cell to satisfy additional conservation constraint in adjacent cells and does not increase the complexity or change the compactness of the RKDG method. We use this new formulation to solve conservation laws on one-d...

An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An e...