We extend the framework and the convergence criteria of wavewise entropy inequalities of [H. Yang, Math. Comp., (1996), pp. 45-67] to a large class of semi-discrete high resolution schemes for hyperbolic conservation laws with source terms. This approach is based on an extended theory of Yang [22] on wave tracking and wave analysis and the theory of Vol'pert [21] on BV solutions. For the Cauchy problem of convex conservation laws with source terms, we use one of the criteria to prove the convergence to the entropy solution of generalized MUSCL schemes and a class of schemes using flux limiters previously discussed in 1984 by Sweby.

We construct the common and the ordered spectral representation for operators, generated as direct sums of self-adjoint extensions of quasi-differential minimal operators on a multiinterval set (self-adjoint vector-operators), acting in a Hilbert space. The structure of the ordered representation is investigated for the case of differential coordinate operators. Results, connected with other spectral properties of such vector-operators, such as the introduction of the identity resolution and the spectral multiplicity have also been obtained. Vector-operators have been mainly studied by W.N. Everitt, L. Markus and A. Zettl. Being a natural continuation of Everitt-Markus-Zettl theory, the presented results reveal the internal structure of self-adjoint differential vector-operators and are essential for the further study of their spectral properties.

A result concerning the existence of positive solutions for the Dirichlet boundary value problem-(φ_p (u'))' = f (t, u), t ∆ (0, 1), u(0) = c > 0 and u (1) = 0, is given in this paper. Here f (t, y) may change sign and may be singular at y = 0.

We study eigenfrequencies and propagator expansions for damped wave equations on compact manifolds. Under the assumption of geometric control, the propagator is shown to admit an expansion in terms of finitely many eigenmodes near the real axis, with an error term exponentially decaying in time. In the presence of a nondegenerate elliptic closed geodesic not meeting the support of the damping coefficient, we show that there exists a sequence of eigenfrequencies converging rapidly to the real axis. In the case of Zoll manifolds, we show that the propagator can be expanded in terms of the clusters of the eigenfrequencies in the entire spectral band.

In this paper we first give a general definition of a new kind of matrix products, called the semi-tensor product, which was firstly proposed in [4]. Certain new properties related to the later applications are proved. Using them, some problems in physics are investigated. First of all, the Carleman linearization of some dynamic physical systems is considered. It is used to investigate the invariants. A rigorous proof for the solvability is presented. Secondly, the problems of invariants of planar polynomial systems is converted to the solvability of a set of algebraic equations. Thirdly, we consider the contraction of a tensor field. A simple proof for general contraction is obtained.

The paper studies in less general scales of Banach spaces the dynamics generated by a Cahn-Hilliard type equation in a smooth open bounded domain of any space dimensions. The equation on the boundary satisfy nonlinear conditions. It establishes local well posedness of the problem and a priori uniform on the domain boundedness and existence in the large of the solutions is studied. It also discusses the asymptotic behaviour of the solutions in the form of existence of a global attractor. An adequate notion of upper semicontinuity of the attractor in the limit of high viscosity is considered and the limit attractor is found to correspond to finite dimensional processes. These processes are depicted by limits of the spatial average solutions of the problem.

In this paper, we prove the boundedness in the context of Triebel-Lizorkin spaces for multilinear Littlewood-Paley operators.