We establish a new on-diagonal lower estimate of continuous-time heat kernels for large time on graphs. To achieve the goal, we first introduce an upper estimate of heat kernels in natural graph metric, then use the upper estimate and the volume growth condition to show the validity of the on-diagonal lower estimate.

For systems of hyperbolic conservation laws, a new Glimm functional was recently constructed when the linearly degenerate manifold in each characteristic field is either the whole space or it consists of a finitely many smooth and transversal manifolds of codimension one. This new functional leads to the neat consistency and convergence rate estimation of the Glimm scheme. In this paper, by the motivation of the result in Ref. 2, we show that the corresponding new Glimm functional can be constructed for general systems only under the strict hyperbolicity assumption.

In this paper, we prove two existence results of solutions to mean field equations Δu+e^u=ρδ_0 and Δu=λe^u(e^u−1)+4π\sum_{j=1}^M δ_{p_j} on an arbitrary connected finite graph, where ρ>0 and λ>0 are constants, M is a positive integer, and p_1,…,p_M are arbitrarily chosen distinct vertices on the graph.

We give a generalized version of uncertainty principle, and apply it to the study of regularization properties of solutions to kinetic equations. In particular, both linearized and nonlinear space inhomogeneous Boltzmann equations without Grad's cutoff assumption are considered. <b><i>To cite this article: R. Alexandre et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).</i></b>

We characterize probability measure with finite moment of any order in terms of the symmetric difference operators of their Fourier transforms. By using our new characterization, we prove the continuity f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) , where f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) stands for the density of unique measure-valued solution f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) of the Cauchy problem for the homogeneous non-cutoff Boltzmann equation, with Maxwellian molecules, corresponding to a probability measure initial datum f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) satisfying\[\int| v|^{2k-2+ } dF_0 (v)&lt;\infty, 0\leq &lt; 2, k= 2, 3, 4,\cdots\] provided that f (t, v)\in C ((0,\infty), L^ 1_ {2k-2+ }) is not a single Dirac mass.