doi: 10.1685/2010CAIM590

Rate of convergence to self-similarity for the fragmentation equation in L1 spaces

María J. Càceres, José A. Cañizo, Stéphane Mischler

Abstract


In a recent result by the authors [1] it was proved that solutions of the self-similar fragmentation equation converge to equilibrium exponentially fast. This was done by showing a spectral gap in weighted L2 spaces of the operator defining the time evolution. In the present work we prove that there is also a spectral gap in weighted L1 spaces, thus extending exponential convergence to a larger set of initial conditions. The main tool is an extension result in [2].

[1] M. J. Caceres, J.A. Canizo, and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, Journal de Mathemathiques Pures et Appliquees, to appear, 2011 (preprint arXiv:1010.546)

[2] M. P. Gualdani, S. Mischler, and C. Mouhot, Factorization for non-symmetric operators and exponential H-theorem, Preprint, Jun 2010.


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Communications in Applied and Industrial Mathematics
ISSN: 2038-0909