Chinese Annals of Mathematics

The authors establish the Hilbertian invariance principle for the empirical process of a stationary Markov process, by extending the forward-backward martingale decomposition of Lyons-Meyer-Zheng to the Hilbert space valued additive functionals associated with general non-reversible Markov processes.

The authors consider the Euler equations for a compressible fluid in one space dimension when the equation of state of the fluid does not fulfill standard convexity assumption and viscosity and capillarity effects are taken into account. A typical example of nonconvex constitutive equation for fluids is Van der Waals' equation. The first order terms of these partial differential equations form a nonlinear system of mixed (hyperbolic -elliptic) type. For a class of nonconvex equations of state, an existence theorem of traveling waves solutions with arbitrary large amplitude is established here. The autors distinguish between classical (compressive) and nonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequalities, and are characterized by the so-called kinetic relation, whose properties are investigated in this paper.

Consider a backward heat equation in a bounded domain Ω ⊂ R² with the noisy data in the initial time geometry. The aim is to find the temperature for 0 < ε < t < T. For this ill-posed problem, the authors give a continuous dependence estimate of the solution. Moreover, the convergence rate of the approximate solution is also given.

The space-fractional telegraph equation is analyzed and the Fourier transform of its fundamental solution is obtained and discussed.

A symmetric process with discontinuous trajectories, whose transition function satisfies the space-fractional telegraph equation, is presented. Its limiting behaviour and the connection with symmetric stable processes is also examined.

The authors show that the self-similar set for a finite family of contractive similitudes (similarities, i.e., |f_i(x) - f_i(y)| = a_i|x - y|, x,y ∈ R^N, where 0 < a_i < 1) is uniformly perfect except the case that it is singleton. As a corollary, it is proved that this self-similar set has positive Hausdorff dimension provided that it is not a singleton. And a lower bound of the upper box dimension of the uniformly perfect sets is given. Meanwhile the uniformly perfect set with Hausdorff measure zero in its Hausdorff dimension is given.

The boundedness of high order Riesz-Bessel transformations generated by generalized shift operator on Ba Spaces is examined.

The author studies minimal surfaces in 3-dimensional solvable Lie groups with left invariant Riemannian metrics. A Weierstraß type integral representation formula for minimal surfaces is obtained.

The authors study the bifurcation problems of rough heteroclinic loop connecting three saddle points for the case β₁ > 1, β₂ > 1, β₃ < 1 and β₁ β₂ β₃ < 1. The existence, number, coexistence and incoexistence of 2-point-loop, 1-homoclinic orbit and 1-periodic orbit are studied. Meanwhile, the bifurcation surfaces and existence regions are given.

By using Darboux transformations, the authors give the explicit construction for local isometric immersions of space forms Mⁿ(c) into space forms M^2n-1(c + ε²) via purely algebraic algorithm.

The authors discuss the properties of solvable complete Lie superalgebra, proving that solvable Lie superalgebras of maximal rank are complete.

Let A be a unital C*-algebra, n ∈ N ∪ {∞}. It is proved that the isomorphism is isometric for some suitable distances. As an application, the author has the split exact sequence with i_A contractive (and isometric if n = ∞) under certain condition of A.

Some geometric behaviours of complete solutions to mean curvature flow before the singularities occur are studied. The author obtains the estimates of the rate of the distance between two fixed points and the derivatives of the second fundamental form. By use of a new maximum principle, some geometric properties at infinity are obtained.