MAXIMAL ATTRACTORS FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS OF VISCOUS AND HEAT CONDUCTIVE FLUID

This article is concerned with the existence of maximal attractors in Hi(i=1,2,4) for the compressible Navier-Stokes equations for a polytropic viscous heat conduc-tive ideal gas in bounded annular domains Ωn in Rn(n =2,3).One of the important features is that the metric spaces H(1),H(2),and H(4) we work with are three incomplete metric spaces,as can be seen from the constraints θ0 and u0,with θ and u being absolute temperature and specific volume respectively.For any constants δ1,δ2,···,δ8 verifying some conditions,a sequence of closed subspaces Hδ(i)H(i)(i = 1,2,4) is found,and the existence of maximal(universal) attractors in Hδ(i)(i = 1,2,4) is established.     

HOMOCLINIC ORBITS FOR A CLASS OF THE SECOND ORDER HAMILTONIAN SYSTEMS

The existence of homoclinic orbits is obtained by the variational approach for a class of second order Hamiltonian systems q(t)+▽V(t,q(t))=0,where V(t,x)=-K(t,x)+W(t,x),K(t,x) is neither a quadratic form in x nor periodic in t and W(t,x) is superquadratic in x.     

PHASE RETRIEVAL OF TIME-LIMITED SIGNALS

The problem of reconstructing a signal ψ(x) from its magnitude |ψ(x)| is of considerable interest to engineers and physicists.This article concerns the problem of determining a time-limited signal f with period 2π when |f(eix)| is known for x ∈ [-π,π].It is shown that the conditions |g(eix)| = |f(eix)| and |g(ei(x+b))-g(eix)| = |f(ei(x+b))-f(eix)|,b = 2π,together imply that either g = wf or g = vf,where both w and v have period b.Furthermore,if 2bπ is irrational then the functions w and v reduce to some constants c1 and c2,respectively;if 2bπ is rational then w takes the form w=eiαB1(eix)B2(eix) and v takes the form ei(x2πN/b+α)B1(eix)B2(eix),where B1 and B2 are Blaschke products.     

RIGIDITY THEOREMS OF CLIFFORD TORUS

In this article, we prove that the Clifford torus S1(1-r2~(1/2)) × Sn-1(r) is the only closed hypersurface in the unit sphere Sn+1(1) with infinite fundamental group, which satisfy r2 ≥ (n-1)/n, RicM ≤ C-(H), and S ≤ S+(H). Moreover, we give a characterization of Clifford torus S1(1-r2~(1/2)) × Sn-1(r) with r2 = 2(n-1)+nH2±|H| n2H2+4(n-1)(1/2)/2n(1+H2) .     

VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION

In this article, using coordinate transformation and Gronwall inequality, we study the vortex motion law of the anisotropic Ginzburg-Landau equation in a smooth bounded domain Ω  R2, that is, ■tuε = 2Σ/j,k=1 (ajkj■xjkuε)xj + b(x)(1-ε|ε|2)uε/2u, x ∈Ω, and conclude that each vortex bj(t) (j=1, 2,···, N) satisfies dbdjt(t)= -(a1k(bj(t)b)■(xk))a(a (bj(t)), a2k(bj (t))/xk a(bj (t)) a(bj (t)) , where a(x) =(a11a22-a122(1/2)). We prove that all the vortices are pinned together to the critical points of a(x). Furthermore, we prove that these critical points can not be the maximum points.     

PERTURBATIONS OF ZEROS OF SOLUTIONS TO SECOND ORDER DIFFERENTIAL EQUATIONS WITH POLYNOMIAL COEFFICIENTS＊

Let P(z) and (P)(z) be polynomials of the same degree. We consider the equations u" =P(z)u and (u)" =(P)(z)(u) (z ∈ C) whose solutions are u(z) and (u)(z),respectively.Let zk(u) and zk((u)),k =1,2,…,be...