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Strong fragmentation and coagulation with power-law rates

Banasiak, Jacek and Lamb, Wilson and Langer, Matthias (2013) Strong fragmentation and coagulation with power-law rates. Journal of Engineering Mathematics, 82 (1). pp. 199-215. ISSN 0022-0833

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Abstract

Existence of global classical solutions to fragmentation and coagulation equations with unbounded coagulation rates has been recently proved for initial conditions with finite higher order moments. These results cannot be directly generalized to the most natural space of solutions with finite mass and number of particles due to the lack of precise characterization of the domain of the generator of the fragmentation semigroup. In this paper we show that such a generalization is possible in the case when both fragmentation and coagulation are described by power-law rates which are commonly used in the engineering practice. This is achieved through direct estimates of the resolvent of the fragmentation operator, which in this case is explicitly known, proving that it is sectorial and carefully intertwining the corresponding intermediate spaces with appropriate weighted L1 spaces.