Numerical blow-up analysis of the explicit L1-scheme for fractional ordinary differential equations

Numerical Algorithms - This paper deals with the blow-up behavior of numerical solutions to nonlinear fractional ordinary differential equations with a dissipative term. Based on the positivity...     

A Review of Results on Axially Symmetric Navier-Stokes Equations,with Addendum by X. Pan and Q. Zhang

In this paper, we give a brief survey of recent results on axially symmetric Navier-Stokes equations (ASNS) in the following categories: regularity criterion, Liouville property for ancient solutions, decay and vanishing of stationary solutions. Some discussions also touch on the full 3 dimensional equations. Two results, closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail.In the addendum, two new results in the 3rd category will also be presented, which are generalizations of recently published results by the author and coauthors.     

The critical exponent of degenerate parabolic systems

The Cauchy problemu _t=u +v ^p ,v _t =v +u ^q is studied, wherex R ^N , 0 <t < and ,,p andq, are positive exponents. It is proved that ifp,q 1 and 1 <pq < 1 + 2 max(p + ,q + )/n then every nontrivial non-negative solution is not global in time; whereaspq > 1 + 2 max(p + , q + )/n then there exist both positive global solutions and non-global solutions. In addition, the decaying in time of solutions tou _t,=u inR ⁿ × (0, ), an equation which occurs naturally in our study of above systems, is studied and solutions with the fastest decaying in time are constructed.     

Regularity of minimizers of non-isotropic integrals of the calculus of variations

The regularity of the minimizers of a special type of non-isotropic variational minimization problem is studied. The particularity of the potential of energy is that it has different growth rate with respect to different parts of the derivatives of the function. In particular, the model treated in this paper can be described as $$\Phi (Du) = |\partial _1 u|^2 + |\partial _2 u|^2 + |\partial _3 u|^2 + |\partial _3 u - |^p .$$ By using a result of P.Marcellini (cf. [4]) and perturbation method, it is proved that the minimizer of the Dirichlet boundary value problem is a function of W _loc ^{1, ∞} .     

Regio- and stereocontrolled synthesis and conformational analysis of benzimidazole nucleosides

Regio- and stereocontrolled synthesis and conformational analysis of a series of benzimidazole nucleosides were achieved. A simple method by ¹H NMR 1D NOE experiment was developed for estimation of syn or anti conformation of benzimidazole nucleosides. Substituents at C2 of benzimidazole demonstrated to play a key role both in the unexpected regioselectivity of the glycosidic reaction and in the conformation distributions of the final products.     

Aza-Baylis-Hillman reaction of salicyl N-tosylimines with methyl vinyl ketone, ethyl vinyl ketone or phenyl vinyl ketone

Reactions of salicyl N-tosylimines with methyl vinyl ketone, ethyl vinyl ketone or phenyl vinyl ketone proceeded smoothly under mild conditions to give the corresponding chromanes or aza-Baylis-Hillman adducts in moderate to excellent yields in the presence of phosphine or nitrogen Lewis base.