Chinese Annals of Mathematics

A two-dimensional nonlinear shell model "of Koiter's type" has recently been proposed by the first author. It is shown here that, according to two mutually exclusive sets of assumptions bearing on the associated manifold of admissible inextensional displacements, the leading term of a formal asymptotic expansion of the solution of this two-dimensional model, with the thickness as the "small" parameter, satisfies either the two-dimensional equations of a nonlinearly elastic "membrane" shell or those of a nonlinearly elastic "flexural" shell.

These conclusions being identical to those recently drawn by B. Miara, then by V. Lods and B. Miara, for the leading term of a formal asymptotic expansion of the solution of the equations of three-dimensional nonlinear elesticity, again with the thickness as the "small" parameter, the nonlinear shell model of Koiter's type considered here is thus justified, at least formally.

The authors establish the existence of a large class of mathematical entropies (the so-called weak entropies) associated with the Euler equations for an isentropic, compressible fluid governed by a general pressure law. A mild assumption on the behavior of the pressure law near the vacuum is solely required. The analysis is based on an asymptotic expansion of the fundamental soution (called here the entropy kernel) of a highly singular Euler-Poisson-Darboux equation. The entropy kernel is only Hölder continuous and its regularity is carefully investigated. Relying on a notion introduced earlier by the authors, it is also proven that, for the Euler equations, the set of entropy flux-splittings conincides with the set of entropies-entropy fluxes. These results imply the existence of a flux-splitting consistent with all of the entropy inequalities.

Following a recent paper of the authors in Communications in Partial Differential Equations, this paper establishes the global existence of weak solutions to a nonlinear variational wave equation under relaxed conditions on the initial data so that the solutions can contain singularities (blow-up). Propagation of local oscillations along one family of characteristics remains under control despite singularity formation in the other family of characteristics.

The Darboux tranformation of a 3×3 spectral problem which is associated with the higher-order nonlinear Schrödinger equation is given. Some solutions of the higher-order nonlinear Schrödinger equation are provided by taking different "seeds".

The authors characterize the existence of proper holomorphic mapping between a kind of nonsmooth Reinhardt domains defined as

Supersaturated designs are useful in screening experiments. This paper discusses the topic or multi-level supersaturated design. Two quantities, E(d²) and D_f, are proposed to evaluate the optimality of supersaturated designs. A lower bound of E(d²) is obtained with a necessary condition for achieving it. Some E(d²)-optimal supersaturated designs of 3, 4, and 5 levels are given.

The authors study the asymptotic behavior of the smooth solutions to the Cauchy problems for two macroscopic models (hydrodynamic and drift-diffusion models) for semiconductors and the related relaxation limit problem. First, it is proved that the solutions to these two systems converge to the unique stationary solution time asymptotically without the smallness assumption on doping profile. Then, very sharp estimates on the smooth solutions, independent of the relaxation time, are obtained and used to establish the zero relaxation limit.

This paper studies the long time behaviour and the blow up phenomenon of the heat flow for the Yang-Mills-Higgs field on a vector bundle over a compact 4-dimensional Riemannian manifold.

The author establishes a large deviation for k-dimensional Brownian motion B in stronger topology, by which the functional modulus of continuity for B in Hölder norm can be obtained.

Using the circle method and sieves, the author proves that a positive proportion of positive integers can be represented as the sum of four cubes of primes.

Let L be a simple Lie algebra with irreducible root system Φ having roots of different length, F be a field of characteristic different from 2, G=L(F) be a Chevalley group of type L over F. Denote by Φ^l the set of all long roots in Φ. Set G^l=<x_r(t); r∈Φ^l, t∈ F>. It is a subgroup of G generated by all the long root subgroups. This paper determines the pronormality of G^l in G when L is not of type G₂.

A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G that G=HK and H∩K=1. In this case, K is called a complement of H in G. In this note some results on complemented subgroups of finite groups are obtained.

Let M be an n-dimensional normal projective variety with only Gorenstein, terminal, Q-factorial singularities. Let L be an ample line bundle on M. Let τ denote the nef value of (M,L). The classification of (M,L) via the value morphism is given for the situations when τ satisfies n-5<τ<n-4, or n-6<τ<n-5.