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.

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>

No abstract uploaded!

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.

This paper aims to mathematically advance the field of quantitative thermo-acoustic imaging. Given several electromagnetic data sets, we establish for the first time an analytical formula for reconstructing the absorption coefficient from thermal energy measurements. Since the formula involves derivatives of the given data up to the third order, it is unstable in the sense that small measurement noises may cause large errors. However, in the presence of measurement noise, the obtained formula, together with a noise regularization technique, provides a good initial guess for the true absorption coefficient. We finally correct the errors by deriving a reconstruction formula based on the least square solution of an optimal control problem and prove that this optimization step reduces the errors occurring and enhances the resolution.