Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels
Abstract The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of...
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Gespeichert in:
| Autor*in: |
Niethammer, B. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2012 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag 2012 |
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| Übergeordnetes Werk: |
Enthalten in: Communications in mathematical physics - Springer-Verlag, 1965, 318(2012), 2 vom: 01. Sept., Seite 505-532 |
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| Übergeordnetes Werk: |
volume:318 ; year:2012 ; number:2 ; day:01 ; month:09 ; pages:505-532 |
| Links: |
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| DOI / URN: |
10.1007/s00220-012-1553-5 |
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| Katalog-ID: |
OLC2038903468 |
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| 520 | |a Abstract The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree $${\gamma \in (-\infty,1)}$$ and satisfy K(x, y) ≤ C (xγ + yγ). More precisely, for any $${\rho \in (\gamma,1)}$$ we establish the existence of a continuous weak nonnegative self-similar profile with decay $${x^{-(1{+}\rho)}}$$ as x → ∞. For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem. | ||
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10.1007/s00220-012-1553-5 doi (DE-627)OLC2038903468 (DE-He213)s00220-012-1553-5-p DE-627 ger DE-627 rakwb eng 530 510 VZ Niethammer, B. verfasserin aut Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels 2012 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2012 Abstract The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree $${\gamma \in (-\infty,1)}$$ and satisfy K(x, y) ≤ C (xγ + yγ). More precisely, for any $${\rho \in (\gamma,1)}$$ we establish the existence of a continuous weak nonnegative self-similar profile with decay $${x^{-(1{+}\rho)}}$$ as x → ∞. For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem. Weak Solution Point Theorem Dual Problem Mild Solution Weak Topology Velázquez, J. J. L. aut Enthalten in Communications in mathematical physics Springer-Verlag, 1965 318(2012), 2 vom: 01. Sept., Seite 505-532 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:318 year:2012 number:2 day:01 month:09 pages:505-532 https://doi.org/10.1007/s00220-012-1553-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_30 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2006 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4318 AR 318 2012 2 01 09 505-532 |
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10.1007/s00220-012-1553-5 doi (DE-627)OLC2038903468 (DE-He213)s00220-012-1553-5-p DE-627 ger DE-627 rakwb eng 530 510 VZ Niethammer, B. verfasserin aut Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels 2012 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2012 Abstract The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree $${\gamma \in (-\infty,1)}$$ and satisfy K(x, y) ≤ C (xγ + yγ). More precisely, for any $${\rho \in (\gamma,1)}$$ we establish the existence of a continuous weak nonnegative self-similar profile with decay $${x^{-(1{+}\rho)}}$$ as x → ∞. For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem. Weak Solution Point Theorem Dual Problem Mild Solution Weak Topology Velázquez, J. J. L. aut Enthalten in Communications in mathematical physics Springer-Verlag, 1965 318(2012), 2 vom: 01. Sept., Seite 505-532 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:318 year:2012 number:2 day:01 month:09 pages:505-532 https://doi.org/10.1007/s00220-012-1553-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_30 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2006 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4318 AR 318 2012 2 01 09 505-532 |
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10.1007/s00220-012-1553-5 doi (DE-627)OLC2038903468 (DE-He213)s00220-012-1553-5-p DE-627 ger DE-627 rakwb eng 530 510 VZ Niethammer, B. verfasserin aut Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels 2012 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2012 Abstract The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree $${\gamma \in (-\infty,1)}$$ and satisfy K(x, y) ≤ C (xγ + yγ). More precisely, for any $${\rho \in (\gamma,1)}$$ we establish the existence of a continuous weak nonnegative self-similar profile with decay $${x^{-(1{+}\rho)}}$$ as x → ∞. For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem. Weak Solution Point Theorem Dual Problem Mild Solution Weak Topology Velázquez, J. J. L. aut Enthalten in Communications in mathematical physics Springer-Verlag, 1965 318(2012), 2 vom: 01. Sept., Seite 505-532 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:318 year:2012 number:2 day:01 month:09 pages:505-532 https://doi.org/10.1007/s00220-012-1553-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_30 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2006 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4318 AR 318 2012 2 01 09 505-532 |
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10.1007/s00220-012-1553-5 doi (DE-627)OLC2038903468 (DE-He213)s00220-012-1553-5-p DE-627 ger DE-627 rakwb eng 530 510 VZ Niethammer, B. verfasserin aut Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels 2012 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2012 Abstract The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree $${\gamma \in (-\infty,1)}$$ and satisfy K(x, y) ≤ C (xγ + yγ). More precisely, for any $${\rho \in (\gamma,1)}$$ we establish the existence of a continuous weak nonnegative self-similar profile with decay $${x^{-(1{+}\rho)}}$$ as x → ∞. For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem. Weak Solution Point Theorem Dual Problem Mild Solution Weak Topology Velázquez, J. J. L. aut Enthalten in Communications in mathematical physics Springer-Verlag, 1965 318(2012), 2 vom: 01. Sept., Seite 505-532 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:318 year:2012 number:2 day:01 month:09 pages:505-532 https://doi.org/10.1007/s00220-012-1553-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_30 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2006 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4318 AR 318 2012 2 01 09 505-532 |
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10.1007/s00220-012-1553-5 doi (DE-627)OLC2038903468 (DE-He213)s00220-012-1553-5-p DE-627 ger DE-627 rakwb eng 530 510 VZ Niethammer, B. verfasserin aut Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels 2012 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2012 Abstract The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree $${\gamma \in (-\infty,1)}$$ and satisfy K(x, y) ≤ C (xγ + yγ). More precisely, for any $${\rho \in (\gamma,1)}$$ we establish the existence of a continuous weak nonnegative self-similar profile with decay $${x^{-(1{+}\rho)}}$$ as x → ∞. For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem. Weak Solution Point Theorem Dual Problem Mild Solution Weak Topology Velázquez, J. J. L. aut Enthalten in Communications in mathematical physics Springer-Verlag, 1965 318(2012), 2 vom: 01. Sept., Seite 505-532 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:318 year:2012 number:2 day:01 month:09 pages:505-532 https://doi.org/10.1007/s00220-012-1553-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_30 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2006 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4318 AR 318 2012 2 01 09 505-532 |
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Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels |
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Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels |
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Niethammer, B. |
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Communications in mathematical physics |
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Niethammer, B. Velázquez, J. J. L. |
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self-similar solutions with fat tails for smoluchowski’s coagulation equation with locally bounded kernels |
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Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels |
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Abstract The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree $${\gamma \in (-\infty,1)}$$ and satisfy K(x, y) ≤ C (xγ + yγ). More precisely, for any $${\rho \in (\gamma,1)}$$ we establish the existence of a continuous weak nonnegative self-similar profile with decay $${x^{-(1{+}\rho)}}$$ as x → ∞. For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem. © Springer-Verlag 2012 |
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Abstract The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree $${\gamma \in (-\infty,1)}$$ and satisfy K(x, y) ≤ C (xγ + yγ). More precisely, for any $${\rho \in (\gamma,1)}$$ we establish the existence of a continuous weak nonnegative self-similar profile with decay $${x^{-(1{+}\rho)}}$$ as x → ∞. For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem. © Springer-Verlag 2012 |
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Abstract The existence of self-similar solutions with fat tails for Smoluchowski’s coagulation equation has so far only been established for the solvable kernels and the diagonal one. In this paper we prove the existence of such self-similar solutions for continuous kernels K that are homogeneous of degree $${\gamma \in (-\infty,1)}$$ and satisfy K(x, y) ≤ C (xγ + yγ). More precisely, for any $${\rho \in (\gamma,1)}$$ we establish the existence of a continuous weak nonnegative self-similar profile with decay $${x^{-(1{+}\rho)}}$$ as x → ∞. For the proof we consider the time-dependent problem in self-similar variables with the aim to use a variant of Tykonov’s fixed point theorem to establish the existence of a stationary profile. This requires to identify a weakly compact subset that is invariant under the evolution. In our case we define a set of nonnegative measures which encodes the desired decay behaviour in an integrated form. The main difficulty is to establish the invariance of the lower bound under the evolution. Our key idea is to choose as a test function in the time dependent problem the solution of the associated backward dual problem. © Springer-Verlag 2012 |
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Self-similar Solutions with Fat Tails for Smoluchowski’s Coagulation Equation with Locally Bounded Kernels |
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