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On an inhomogeneous coagulation model with a differential sedimentation kernel (2404.11418v1)

Published 17 Apr 2024 in math.AP, math-ph, and math.MP

Abstract: We study an inhomogeneous coagulation equation that contains a transport term in the spatial variable modeling the sedimentation of clusters. We prove local existence of mass conserving solutions for a class of coagulation kernels for which in the space homogeneous case instantaneous gelation (i.e., instantaneous loss of mass) occurs. Our result holds true in particular for sum-type kernels of homogeneity greater than one, for which solutions do not exist at all in the spatially homogeneous case. Moreover, our result covers kernels that in addition vanish on the diagonal, which have been used to describe the onset of rain and the behavior of air bubbles in water.

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References (38)
  1. M. Aizenman and T. A. Bak “Convergence to equilibrium in a system of reacting polymers” In Communications in Mathematical Physics 65, 1979, pp. 203–230
  2. J. M. Ball and J. Carr “The discrete coagulation-fragmentation equations: Existence, uniqueness, and density conservation” In Journal of Statistical Physics 61, 1990, pp. 203–234
  3. J. Banasiak, W. Lamb and P. Laurençot “Analytic Methods for Coagulation-Fragmentation Models, Volume II” Boca Raton: CRC Press, 2019
  4. M. Bonacini, B. Niethammer and J. J. L. Velázquez “Self-similar solutions to coagulation equations with time-dependent tails: The case of homogeneity one” In Archive for Rational Mechanincs and Analysis 233, 2019, pp. 1–43
  5. A. V. Burobin “Existence and uniqueness of a solution of a Cauchy problem for the inhomogeneous three-dimensional coagulation equation” In Differential Equations 19, 1983, pp. 1568–1579
  6. J. Carr and F. P. Costa “Instantaneous gelation in coagulation dynamics” In Zeitschrift für angewandte Mathematik und Physik ZAMP 43.6, 1992, pp. 974–983
  7. J. A. Carrillo, L. Desvillettes and K. Fellner “Exponential decay towards equilibrium for the inhomogeneous Aizenman-Bak model” In Communications in Mathematical Physics 278, 2008, pp. 433–451
  8. J. A. Cañizo, L. Desvillettes and K. Fellner “Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion” In Annales de l’Institut Henri Poincaré (C), Analyse non linéaire 27, 2010, pp. 639–654
  9. J. A. Cañizo, S. Mischler and C. Mouhot “Rate of convergence to self-similarity for Smoluchowski’s coagulation equation with constant coefficients” In SIAM Journal on Mathematical Analysis 41.6, 2010, pp. 2283–2314
  10. “Existence and uniqueness for spatially inhomogeneous coagulation - condensation equation with unbounded kernels” In Journal of Integral Equations and Applications 9.3, 1997, pp. 219–236
  11. “Existence and uniqueness for spatially inhomogeneous coagulation equation with sources and effluxes” In Zeitschrift für angewandte Mathematik und Physik 47, 1995, pp. 580–594
  12. R. Clift, J. R. Grace and M. E. Weber “Bubbles, Drops, and Particles” Academic Press, 1978
  13. I. Cristian, B. Niethammer and J. J. L. Velázquez “A note on instantaneous gelation for coagulation kernels vanishing on the diagonal” In Preprint, 2024
  14. I. Cristian and J. J. L. Velázquez “Coagulation equations for non-spherical clusters” In Preprint arXiv: 2209.00644, 2023
  15. P. G. J. Dongen “On the possible occurrence of instantaneous gelation in Smoluchowski’s coagulation equation” In Journal of Physics A: Mathematical and General 20.7, 1987, pp. 1889
  16. P. Dubovskiǐ “An iterative method for solving the coagulation equation with spatially inhomogeneous velocity fields” In Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 30.11, 1990, pp. 1755–1757
  17. P. Dubovskiǐ “Solutions of a spatially inhomogeneous coagulation equation with particle fractionation taken into account” In Differential Equations 26.3, 1990, pp. 380–384
  18. “Gelation and mass conservation in coagulation-fragmentation models” In Journal of Differential Equations 195.1, 2003, pp. 143–174
  19. M. Escobedo, S. Mischler and B. Perthame “Gelation in coagulation and fragmentation models” In Communications in Mathematical Physics 231, 2002, pp. 157–188
  20. G. Falkovich, M. G. Stepanov and M. Vucelja “Rain initiation time in turbulent warm clouds” In Journal of Applied Meteorology and Climatology 45.4, 2006, pp. 591–599
  21. “Localization in stationary non-equilibrium solutions for multicomponent coagulation systems” In Communications in Mathematical Physics 388.1, 2021, pp. 479–506
  22. “Existence of self-similar solutions to Smoluchowski’s coagulation equation” In Communications in Mathematical Physics 256, 2005, pp. 589–609
  23. V. A. Galkin “Generalized solution of the Smoluchowski kinetic equation for spatially inhomogeneous systems” In Doklady Akademii Nauk SSSR 293.1, 1987, pp. 74–77
  24. V. A. Galkin “The Smoluchowski equation of the kinetic theory of coagulation for spatially non-uniform systems” In Doklady Akademii Nauk SSSR 285, 1985, pp. 1087–1091
  25. P. Horvai, S. V. Nazarenko and T. H. M. Stein “Coalescence of particles by differential sedimentation” In Journal of Statistical Physics 130, 2007, pp. 1177–1195
  26. P. Laurençot “On a class of continuous coagulation-fragmentation equations” In Journal of Differential Equations 167.2, 2000, pp. 245–274
  27. “The continuous coagulation-fragmentation equations with diffusion” In Archive for Rational Mechanincs and Analysis 162, 2002, pp. 45–99
  28. A. Majda “Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables” Springer New York, NY, 2012
  29. G. Menon and R. L. Pego “Approach to self-similarity in Smoluchowski’s coagulation equation” In Communications on Pure and Applied Mathematics 57.9, 2004, pp. 1197–1232
  30. G. Menon and R. L. Pego “Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence” In SIAM Review 48.4, 2006, pp. 745–768
  31. B. Niethammer and J. J. L. Velázquez “Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with locally bounded kernels” In Commununications in Mathematical Physics 318, 2013, pp. 505–532
  32. H. R. Pruppacher and J. D. Klett “Microphysics of Clouds and Precipitation” Springer Dordrecht, 1997
  33. I. W. Stewart “A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels” In Mathematical Methods in the Applied Sciences 11.5, 1989, pp. 627–648
  34. H. Tanaka, S. Inaba and K. Nakazawa “Steady-state size distribution for the self-similar collision cascade” In Icarus 123.2, 1996, pp. 450–455
  35. S. Throm “Stability and uniqueness of self-similar profiles in L1superscript𝐿1L^{1}italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT spaces for perturbations of the constant kernel in Smoluchowski’s coagulation equation” In Communications in Mathematical Physics 383, 2021, pp. 1361 –1407
  36. S. Throm “Uniqueness of fat-tailed self-similar profiles to Smoluchowski’s coagulation equation for a perturbation of the constant kernel” In Memoirs of the American Mathematical Society 271.1328, 2021
  37. “Coagulation-Fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel” In Communications on Pure and Applied Mathematics 75.6, 2019, pp. 1292–1331
  38. V. E. Zakharov and N. N. Filonenko “Weak turbulence of capillary waves” In Journal of Applied Mechanics and Technical Physics 8.5, 1967, pp. 37–40

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