We present a new a priori estimate for discrete coagulation–fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global ${L}^{2}$ bound on the mass density and was previously used, for instance, in the context of reaction–diffusion equations.In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case.

Keywords: Discrete coagulation–fragmentation systems, Mass conservation, Duality arguments

@article{AIHPC_2010__27_2_639_0, author = {Ca\~nizo, J.A. and Desvillettes, L. and Fellner, K.}, title = {Regularity and mass conservation for discrete coagulation{\textendash}fragmentation equations with diffusion}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {639--654}, publisher = {Elsevier}, volume = {27}, number = {2}, year = {2010}, doi = {10.1016/j.anihpc.2009.10.001}, mrnumber = {2595194}, zbl = {1193.35091}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2009.10.001/} }

TY - JOUR AU - Cañizo, J.A. AU - Desvillettes, L. AU - Fellner, K. TI - Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion JO - Annales de l'I.H.P. Analyse non linéaire PY - 2010 SP - 639 EP - 654 VL - 27 IS - 2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2009.10.001/ DO - 10.1016/j.anihpc.2009.10.001 LA - en ID - AIHPC_2010__27_2_639_0 ER -

%0 Journal Article %A Cañizo, J.A. %A Desvillettes, L. %A Fellner, K. %T Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion %J Annales de l'I.H.P. Analyse non linéaire %D 2010 %P 639-654 %V 27 %N 2 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2009.10.001/ %R 10.1016/j.anihpc.2009.10.001 %G en %F AIHPC_2010__27_2_639_0

Cañizo, J.A.; Desvillettes, L.; Fellner, K. Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 2, pp. 639-654. doi : 10.1016/j.anihpc.2009.10.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2009.10.001/

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