Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff

Abstract

For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz's space 𝒮(? ^N ) for all (strictly positive) time. The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role.     

Travelling wavefronts of reaction-diffusion equations in cylindrical domains

Abstract

Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This article is aimed at describing refined asymtotics in the Dirichlet problem in a ball. The beauty of explicit formulas actually highlights the structure of asymptotic expansions in the calculi on singular varieties.     

A Laurent Expansion for Regularized Integrals of Holomorphic Symbols

For a holomorphic family of classical pseudodifferential operators on a closed manifold we give exact formulae for all coefficients in the Laurent expansion of its Kontsevich–Vishik canonical trace. This generalizes to all higher-order terms a known result identifying the residue trace with a pole of the canonical trace. Received: July 2005 Revision: December 2005 Accepted: January 2006     

Inverse shape and surface heat transfer coefficient identification

In this paper, an inverse geometric problem for the modified Helmholtz equation arising in heat conduction in a fin is considered. This problem which consists of determining an unknown inner boundary of an annular domain and possibly its surface heat transfer coefficient from one or two pairs of boundary Cauchy data (boundary temperature and heat flux) is solved numerically using the meshless method of fundamental solutions (MFS). A nonlinear unconstrained minimisation of the objective function is regularised when noise is added to the input boundary data. The stability of the numerical results is investigated for several test examples with respect to noise in the input data and various values of the regularisation parameters.     

Trust-region and other regularisations of linear least-squares problems

We consider methods for regularising the least-squares solution of the linear system Ax=b. In particular, we propose iterative methods for solving large problems in which a trust-region bound ‖x‖≤Δ is imposed on the size of the solution, and in which the least value of linear combinations of ‖Ax−b‖₂ ^q and a regularisation term ‖x‖₂ ^p for various p and q=1,2 is sought. In each case, one or more “secular” equations are derived, and fast Newton-like solution procedures are suggested. The resulting algorithms are available as part of the ALAHAD optimization library. This work was partially supported by EPSRC grants EP/E053351/1 and EP/F005369/1.