LEADER 05498nam 22006253 450 001 9910953751903321 005 20231110232536.0 010 $a9781470466343 010 $a1470466341 035 $a(CKB)4940000000609985 035 $a(MiAaPQ)EBC6715035 035 $a(Au-PeEL)EBL6715035 035 $a(RPAM)22488171 035 $a(PPN)258258098 035 $a(OCoLC)1264680632 035 $a(EXLCZ)994940000000609985 100 $a20210901d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aUniqueness of Fat-Tailed Self-Similar Profiles to Smoluchowski's Coagulation Equation for a Perturbation of the Constant Kernel 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2021. 210 4$dİ2021. 215 $a1 online resource (118 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.271 311 08$a9781470447861 311 08$a147044786X 320 $aIncludes bibliographical references. 327 $aCover -- Title page -- Chapter 1. Introduction -- 1.1. Smoluchowski's equation -- 1.2. Long-time behaviour and self-similarity -- 1.3. The equation for self-similar profiles -- 1.4. Finite mass, fat-tailed profiles and scale invariance -- 1.5. Existence and uniqueness of self-similar profiles -- 1.6. The constant kernel =2 -- 1.7. Assumptions on the kernel -- 1.8. Preliminary work and main result -- 1.9. The boundary layer at zero -- 1.10. Outline of the main ideas and strategy of the proof -- Chapter 2. Functional setup and preliminaries -- 2.1. Function spaces and norms -- 2.2. Transforming the equation to Laplace variables -- 2.3. Notation and elementary properties of \T -- Chapter 3. Uniqueness of profiles -Proof of Theorem 1.12 -- 3.1. Key ingredients for the proof -- 3.2. Proof of Theorem 1.12 -- Chapter 4. Continuity estimates -- 4.1. Proof of \cref{Lem:est:Arho,Lem:est:B2} -- 4.2. Proof of Proposition 3.5 -- 4.3. Estimates for differences -Proof of Proposition 3.6 -- Chapter 5. Linearised coagulation operator -Proof of Proposition 3.7 -- Chapter 6. Uniform bounds on self-similar profiles -- 6.1. A priori estimates for self-similar profiles -- 6.2. Uniform convergence in Laplace variables -- 6.3. Proof of \cref{Prop:norm:boundedness,Prop:closeness:two:norm} -- Chapter 7. The boundary layer estimate -- 7.1. Boundary layer equation -- 7.2. Preliminary estimates -- 7.3. Proof of Proposition 3.10 -- Chapter 8. The representation formula for ?(?, ) -- 8.1. Analyticity properties -- 8.2. Proof of Proposition 7.11 -- Chapter 9. Integral estimate on \Qo?(?, ) -- 9.1. Proof of Proposition 7.12 -- Chapter 10. Asymptotic behaviour of several auxiliary functions -- 10.1. Bounds on moments -- 10.2. Asymptotic behaviour of _{ } and ? -- 10.3. Regularity properties close to zero -- Appendix A. Useful elementary results. 327 $aAppendix B. The representation formula for -- B.1. Proof of Proposition 1.2 -- B.2. Integral estimates on \Ker -- Appendix C. Existence of profiles -- Acknowledgments -- Bibliography -- Back Cover. 330 $a"This article is concerned with the question of uniqueness of self-similar profiles for Smoluchowski's coagulation equation which exhibit algebraic decay (fat tails) at infinity. More precisely, we consider a rate kernel which can be written as The perturbation is assumed to have homogeneity zero and might also be singular both at zero and at infinity. Under further regularity assumptions on we will show that for sufficiently small there exists, up to normalisation of the tail behaviour at infinity, at most one self-similar profile. Establishing uniqueness of self-similar profiles for Smoluchowski's coagulation equation is generally considered to be a difficult problem which is still essentially open. Concerning fat-tailed self-similar profiles this article actually gives the first uniqueness statement for a non-solvable kernel"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aStatistical mechanics 606 $aIntegro-differential equations 606 $aIntegral equations -- Integro-partial differential equations -- Integro-partial differential equations$2msc 606 $aStatistical mechanics, structure of matter -- Time-dependent statistical mechanics (dynamic and nonequilibrium) -- Interacting particle systems$2msc 606 $aIntegral transforms, operational calculus -- Integral transforms, operational calculus -- Laplace transform$2msc 615 0$aStatistical mechanics. 615 0$aIntegro-differential equations. 615 7$aIntegral equations -- Integro-partial differential equations -- Integro-partial differential equations. 615 7$aStatistical mechanics, structure of matter -- Time-dependent statistical mechanics (dynamic and nonequilibrium) -- Interacting particle systems. 615 7$aIntegral transforms, operational calculus -- Integral transforms, operational calculus -- Laplace transform. 676 $a515/.38 686 $a45K05$a82C22$a44A10$2msc 700 $aThrom$b Sebastian$01800877 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910953751903321 996 $aUniqueness of Fat-Tailed Self-Similar Profiles to Smoluchowski's Coagulation Equation for a Perturbation of the Constant Kernel$94345832 997 $aUNINA