Recently, a lot of progress has been made on the study for the spatially homogeneous Boltzmann equation without angular cutoff, cf. [2, 3, 8, 22] and references therein, which shows that the singularity of collision cross-section yields some gain of regularity in the Sobolev space frame on weak solutions for Cauchy problem. That means, this gives the C regularity of weak solution for the spatia...

| Stochastic (Monte Carlo) solutions of the general balance ordinary diierential equations for measures are considered. For this purpose a wide class of n-particle processes is studied from the viewpoint of the main terms of asymptotical (n ! 1) bias and variance for the corresponding estimates. Boltzmann-like and linear equations are considered as the particular cases of balance equations, sev...

Searching minimizers of functions on the convenient level set of the constrain function we obtain weak solutions of a non-homogenuous p-Laplace equation in border case without using the regularity results of linear elliptic equations .

The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there ...

We describe the fundamental solution of the equation that is obtained by linearization of the coagulation equation with kernel K (x, y) = (xy)λ/2, around the steady state f (x) = x−(3+λ)/2 with λ ∈ (1, 2). Detailed estimates on its asymptotics are obtained. Some consequences are deduced for the flux properties of the particles distributions described by such models.

We consider some mathematical questions about Boltzmann equations for quantum particles, relativistic or non relativistic. Relevant particular cases such as Bose, Bose-Fermi, and photon-electron gases are studied. We also consider some simplifications such as the isotropy of the distribution functions and the asymptotic limits (systems where one of the species is at equilibrium). This gives ris...

In this paper, we give several remarks on Strichartz estimates for homogeneous wave equation. In particular, we show that the endpoint L4t L ∞ x estimate fail to be hold for n = 2 in general. When the data is radial, we prove the endpoint L2t L ∞ x estimate for n ≥ 3, and the L q tL ∞ x estimate with 2 < q < ∞ for n = 2.

The scheme developed by Hartle for describing slowly rotating bodies in 1967 was applied to the simple model of constant density by Chandrasekhar and Miller in 1974. The pivotal equation one has to solve turns out to be one of Heun’s equations. After a brief discussion of this equation and the chances of finding a closed form solution, a quickly converging series solution of it is presented. A ...

Let X be a regular irreducible variety in CP, Y the associated homogeneous variety in C, and N the restriction of the universal bundle of CP to X . In the present paper, we compute the obstructions to solving the ∂-equation in the L-sense on Y for 1 ≤ p ≤ ∞ in terms of cohomology groups H(X,O(N)). That allows to identify obstructions explicitly if X is specified more precisely, for example if i...