Self-Similar Solutions to Coagulation Equations with Time-Dependent Tails: The Case of Homogeneity One
| Autor: | Juan J. L. Velázquez, Marco Bonacini, Barbara Niethammer |
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| Rok vydání: | 2019 |
| Předmět: |
Physics
self-similar solutions Mechanical Engineering Homogeneity (statistics) 010102 general mathematics Smoluchowski's equation 01 natural sciences Smoluchowski's equation Kernels with homogeneity one self-similar solutions 010101 applied mathematics Combinatorics Mathematics - Analysis of PDEs Mathematics (miscellaneous) Homogeneous FOS: Mathematics Differentiable function 0101 mathematics Kernels with homogeneity one Analysis Finite mass Analysis of PDEs (math.AP) |
| Zdroj: | Archive for Rational Mechanics and Analysis. 233:1-43 |
| ISSN: | 1432-0673 0003-9527 |
| DOI: | 10.1007/s00205-018-01353-6 |
| Popis: | We prove the existence of a one-parameter family of self-similar solutions with time dependent tails for Smoluchowski’s coagulation equation, for a class of kernels $${K(x,y)}$$ which are homogeneous of degree one and satisfy $${K(x,1) \to k_0 > 0}$$ as $${x\to 0}$$ . In particular, we establish the existence of a critical $${\rho_* > 0}$$ with the property that for all $${\rho\in(0,\rho_*)}$$ there is a positive and differentiable self-similar solution with finite mass M and decay $${A(t)x^{-(2+\rho)}}$$ as $${x\to\infty}$$ , with $${A(t)=e^{M(1+\rho)t}}$$ . Furthermore, we show that (weak) self-similar solutions in the class of positive measures cannot exist for large values of the parameter $${\rho}$$ . |
| Databáze: | OpenAIRE |
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