THE LYAPUNOV STABILITY OF NONLINEAR WAVE MOTION IN THE BAROTROPIC NONDIVERGENT ATMOSPHERE

Introducing the concept of pseudo-momentum, a generalized Arnold-Dikii functional is established, and then the sufficient condition for stability of nonlinear wave motion in the barotropic nondivergent atmosphere is derived by use of variational principle. It is found that the stability of nonlinear wave motion depends not only on its streamfield distri- bution, but also on its phase speed for the propagating nonlinear wave motion. Moreover, the stability criterion of trav- elling modon is also obtained, and it is shown that the travelling modon is stable if the scale of disturbance superimposed on the travelling modon remains to be less than that of the travelling modon.

Variable coefficient nonlinear systems derived from an atmospheric dynamical system

Variable coefficient nonlinear systems, the Korteweg de Vries （KdV）, the modified KdV （mKdV） and the nonlinear Schrǒdinger （NLS） type equations, are derived from the nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane by means of the multi-scale expansion method in two different ways, with and without the so-called y-average trick. The non-auto-Bǎcklund transformations are found to transform the derived variable coefficient equations to the corresponding standard KdV, mKdV and NLS equations. Thus, many possible exact solutions can be obtained by taking advantage of the known solutions of these standard equations. Further, many approximate solutions of the original model are ready to be yielded which might be applied to explain some real atmospheric phenomena, such as atmospheric blocking episodes.

INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).

Further generalization of Fermat's principle

From Maxwell’ s equations for electromagnetic fields, time-averaged energy flow density vector of stable monochromatic linearly polarized light in an isotropic insulative nonmagnetic medium is deduced. By the introduction of time-averaged energy flow density rays and the definition of new generalized refractive indexn G1, Fermat’s principle of geometric optics is further generalized and its application conditions are discussed. The generalized Fermat' s principle can be used to describe stable transmission of light in a medium with variable refractive index. The necessary and sufficient conditions of a nondivergent and nonfocusing light beam are derived from this Fermat’s principle.

Estimates of solutions of initial-irregular oblique derivative problems for linear parabolic equations of second order with measurable coefficients

The initial\|irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then some \%a priori\% estimates of solutions for the problems are given. By using the above estimates and the Leray\|Schauder theorem, the existence of solutions of the initial\|boundary value problems can be proved. The results are generalizations of corresponding theorems in literature.

Estimates for Green’s Functions of Elliptic Equations in Non-Divergence Form with Continuous Coefficients

We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients.We also present a sharp comparison with the corresponding Green5s function for constant coefficients equations.