For the Boltzmann equation with an external force in the form of the gradientof a potential function in space variable, the stability of its stationary solutions as localMaxwellians was studied by S. Ukai et al. (2005) through the energy method. Based onthis stability analysis and some techniques on analyzing the convergence rates to station-ary solutions for the compressible Navier-Stokes equations, in this paper, we study theconvergence rate to the above stationary solutions for the Boltzmann equation which is afundamental equation in statistical physics for non-equilibrium rarefied gas. By combining the dissipation from the viscosity and heat conductivity on the fluid components andthe dissipation on the non-fluid component through the celebrated H-theorem, a convergence rate of the same order as the one for the compressible Navier-Stokes is obtained byconstructing some energy functionals.
The classical equations of a nonlinearly elastic plane membrane made of Saint Venant-Kirchhoff material have been justified by Fox, Raoult and Simo (1993) and Pantz (2000). We show that, under compression, the associated minimization problem admits no solution. The proof is based on a result of non-existence of minimizers of non-convex functionals due to Dacorogna and Marcellini (1995). We generalize the application of their result from plane elasticity to three-dimensional plane membranes.
Some uniqueness theorems of meromorphic mappings with moving targets are given under the inclusion relations between the zeros sets of meromorphic mappings.
This paper deals with the periodic solutions of Schrödinger flow from S 3 to S 2. It is shown that the periodic solution is related to the variation of some functional and there exist S1-invariant critical points to this functional. The proof makes use of the Principle of Symmetric Criticality of Palais.
A Riemannian manifold (M, g) is called Einstein manifold if its Ricci tensor satisfies r = c ⋅ g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.
This is a note on Abrams' paper "Modules, Comodules, and Cotensor Products over Frobenius Algebras, Journal of Algebras" (1999). With the application of Frobenius coordinates developed recently by Kadison, one has a direct proof of Abrams' characterization for Frobenius algebras in terms of comultiplication (see L. Kadison (1999)). For any Frobenius algebra, by using the explicit comultiplication, the explicit correspondence between the category of modules and the category of comodules is obtained. Moreover, with this we give very simplified proofs and improve Abrams' results on the Hom functor description of cotensor functor.
Abstract This paper deals with a kind of fourth degree systems with perturbations. Byusing the method of multi-parameter perturbation theory and qualitative analysis, it isproved that the system can have six limit cycles.
In this paper the authors use a modified Wirtinger presentation to give a lower bound on the unknotting number of a knot in S 3.
The Maslov P-index theory for a symplectic path is defined. Various properties of this index theory such as homotopy invariant, symplectic additivity and the relations with other Morse indices are studied. As an application, the non-periodic problem for some asymptotically linear Hamiltonian systems is considered.
Let {X m(t); t ∈ R +} be an m-Fold integrated Brownian motion. In this paper, with the help of small ball probability estimate, a functional law of the iterated logarithm (LIL) for X m(t) is established. This extends the classic Chung type liminf result for this process. Furthermore, a result about the weighted occupation measure for X m(t) is also obtained.