NECESSARY AND SUFFICIENT CONDITIONS OF EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF POLYNOMIAL SYSTEM

In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x^2+α_2xy+α_5x^2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx^j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.

THE INITIAL BOUNDARY VALUE PROBLEM FOR QUASI-LINEAR SCHRODINGER-POISSON EQUATIONS

In this article, the author studies the iuitial （Dirichlet.） boundary problem for a high field version of the Schroedinger-Poisson equations, which include a nonlinear term in the Poisson equation corresponding to a field-dependent dielectric constant and an effective potential in the Schroedinger equations on the unit cube. h global existence and uniqueness is established for a solution to this problem.

The Estimates L_(1)-L_(∞) for the Reduced Radial Equation of Schrodinger

Estimates of the type L1-L∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential elements in the study of the problems of initial values, the asymptotic times for large solutions and Scattering Theory for the Schrödinger equation and non-linear in general;for other equations of Non-linear Evolution. In general, the estimates Lp-Lp' express the dispersive nature of this equation. And its study plays an important role in problems of non-linear initial values;likewise, in the study of problems nonlinear initial values;see [1] [2] [3]. On the other hand, following a series of problems proposed by V. Marchenko [4], that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1], the main theorem (Theorem 1) of this article provides a transformation operator W?that transforms the Reduced Radial Schrödinger Equation (RRSE) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V(x)) in the Schrödinger Equation on Half-Line (RSEHL) under W. That is to say;W?eliminates the singular term of quadratic order of potential V(x) in the asymptotic development towards zero and adds to the potential V(x) a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential V(x) in the condition of the infinity boundary. Then the L1-L∞ estimates for the (RRSE) are preserved under the transformation operator , as in the case of (RSEHL) where they were established in [3]. Finally, as an open question, the possibility of extending the L1-L∞ estimates for the case (RSEHL), where added to the potential V(x) an analytical perturbation is mentioned.

Analysis of a class of globally divergence-free HDG methods for stationary Navier-Stokes equations

In this paper,we analyze a class of globally divergence-free(and therefore pressure-robust)hybridizable discontinuous Galerkin(HDG)finite element methods for stationary Navier-Stokes equations.The methods use the P_(k)/P_(k-1)(k≥1)discontinuous finite element combination for the velocity and pressure approximations in the interior of elements,piecewise Pm(m=k,k-1)for the velocity gradient approximation in the interior of elements,and piecewise P_(k)/P_(k) for the trace approximations of the velocity and pressure on the inter-element boundaries.We show that the uniqueness condition for the discrete solution is guaranteed by that for the continuous solution together with a sufficiently small mesh size.Based on the derived discrete HDG Sobolev embedding properties,optimal error estimates are obtained.Numerical experiments are performed to verify the theoretical analysis.