Science China Mathematics

In this paper, we work on compact quasi-Einstein metrics and prove several gap results. In the first part, we get a gap estimate for the first nonzero eigenvalue of the weighted Laplacian, by establishing a comparison theorem for the weighted heat kernel. In the second part, we establish two gap results for the Ricci curvature and the scalar curvature, based on which some rigid properties can be derived.

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H (X), H d (X), and H *,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide with L p (X) when p ∈ (1,∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H ∗,p (X) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1),1], it is proved that the space H *,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.

We study the large-time asymptotics of solutions toward the weak rarefaction wave of the quasineutral Euler system for a two-fluid plasma model in the presence of diffusions of velocity and temperature under small perturbations of initial data and also under an extra assumption $\frac{{\theta _{i, + } }} {{\theta _{e, + } }} = \frac{{\theta _{i, - } }} {{\theta _{e, - } }} \geqslant \frac{{m_i }} {{2m_e }}, $ , namely, the ratio of the thermal speeds of ions and electrons at both far fields is not less than one half. Meanwhile, we obtain the global existence of solutions based on energy method.

Let a ⩾ 1 be an integer. In this paper, we will prove the equation in the title has at most three positive integer solutions.

A second order explicit finite element scheme is given for the numerical computation to multi-dimensional scalar conservation laws.L p convergence to entropy solutions is proved under some usual conditions. For two-dimensional problems, uniform mesh, and sufficiently smooth solutions a second order error estimate inL 2 is proved under a stronger condition, Δt≤Ch 2/4

Using the observational data of the solar radio spectrometer with a broad bandwidth of the Chinese National Astronomical Observatories (CNAO), the Nobeyama Radio Heliograph (NoRH) and Polarimeters (NoRP), YOHKOH and SOHO satellites, it is found that there are two characteristics in an X3.3 flare event associated with coronal mass ejection (CME). (i) X-ray and EUV loop interaction and evolution appeared evidently in corona, followed by the formation of a twisted loop structure, which is consistent with the theoretical calculations of Amari et al. (1999a,b) for the origin of flare-CME events. (ii) The slow frequency drift was shown in the radio observations, corresponding to the relevant movement of the radio source observed by NoRH, at a speed of several tens kilometers per second, which may be associated with the speed of the shock waves resulting from the CME.