---
_id: '53315'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>In a smoothly bounded two‐dimensional
    domain  and for a given nondecreasing positive unbounded , for each  and  the
    inequality\r\n<jats:disp-formula />is shown to hold for any positive  fulfilling\r\n<jats:disp-formula
    />This is thereafter applied to nonglobal solutions of the Keller–Segel system
    coupled to the incompressible Navier–Stokes equations through transport and buoyancy,
    and it is seen that in any such blow‐up event the corresponding population density
    cannot remain uniformly integrable over  near its explosion time.</jats:p>"
author:
- first_name: Yulan
  full_name: Wang, Yulan
  last_name: Wang
- first_name: Michael
  full_name: Winkler, Michael
  last_name: Winkler
citation:
  ama: Wang Y, Winkler M. An interpolation inequality involving $L\log L$ spaces and
    application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes
    system. <i>Journal of the London Mathematical Society</i>. 2024;109(3). doi:<a
    href="https://doi.org/10.1112/jlms.12885">10.1112/jlms.12885</a>
  apa: Wang, Y., &#38; Winkler, M. (2024). An interpolation inequality involving $L\log
    L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional
    Keller–Segel–Navier–Stokes system. <i>Journal of the London Mathematical Society</i>,
    <i>109</i>(3). <a href="https://doi.org/10.1112/jlms.12885">https://doi.org/10.1112/jlms.12885</a>
  bibtex: '@article{Wang_Winkler_2024, title={An interpolation inequality involving
    $L\log L$ spaces and application to the characterization of blow‐up behavior in
    a two‐dimensional Keller–Segel–Navier–Stokes system}, volume={109}, DOI={<a href="https://doi.org/10.1112/jlms.12885">10.1112/jlms.12885</a>},
    number={3}, journal={Journal of the London Mathematical Society}, publisher={Wiley},
    author={Wang, Yulan and Winkler, Michael}, year={2024} }'
  chicago: Wang, Yulan, and Michael Winkler. “An Interpolation Inequality Involving
    $L\log L$ Spaces and Application to the Characterization of Blow‐up Behavior in
    a Two‐dimensional Keller–Segel–Navier–Stokes System.” <i>Journal of the London
    Mathematical Society</i> 109, no. 3 (2024). <a href="https://doi.org/10.1112/jlms.12885">https://doi.org/10.1112/jlms.12885</a>.
  ieee: 'Y. Wang and M. Winkler, “An interpolation inequality involving $L\log L$
    spaces and application to the characterization of blow‐up behavior in a two‐dimensional
    Keller–Segel–Navier–Stokes system,” <i>Journal of the London Mathematical Society</i>,
    vol. 109, no. 3, 2024, doi: <a href="https://doi.org/10.1112/jlms.12885">10.1112/jlms.12885</a>.'
  mla: Wang, Yulan, and Michael Winkler. “An Interpolation Inequality Involving $L\log
    L$ Spaces and Application to the Characterization of Blow‐up Behavior in a Two‐dimensional
    Keller–Segel–Navier–Stokes System.” <i>Journal of the London Mathematical Society</i>,
    vol. 109, no. 3, Wiley, 2024, doi:<a href="https://doi.org/10.1112/jlms.12885">10.1112/jlms.12885</a>.
  short: Y. Wang, M. Winkler, Journal of the London Mathematical Society 109 (2024).
date_created: 2024-04-07T12:27:28Z
date_updated: 2024-04-07T12:36:25Z
doi: 10.1112/jlms.12885
intvolume: '       109'
issue: '3'
keyword:
- General Mathematics
language:
- iso: eng
publication: Journal of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6107
  - 1469-7750
publication_status: published
publisher: Wiley
status: public
title: An interpolation inequality involving $L\log L$ spaces and application to the
  characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes
  system
type: journal_article
user_id: '31496'
volume: 109
year: '2024'
...
---
_id: '63259'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>In a smoothly bounded two‐dimensional
    domain  and for a given nondecreasing positive unbounded , for each  and  the
    inequality\r\n<jats:disp-formula/>is shown to hold for any positive  fulfilling\r\n<jats:disp-formula/>This
    is thereafter applied to nonglobal solutions of the Keller–Segel system coupled
    to the incompressible Navier–Stokes equations through transport and buoyancy,
    and it is seen that in any such blow‐up event the corresponding population density
    cannot remain uniformly integrable over  near its explosion time.</jats:p>"
article_number: e12885
author:
- first_name: Yulan
  full_name: Wang, Yulan
  last_name: Wang
- first_name: Michael
  full_name: Winkler, Michael
  id: '31496'
  last_name: Winkler
citation:
  ama: Wang Y, Winkler M. An interpolation inequality involving LlogL$L\log L$ spaces
    and application to the characterization of blow‐up behavior in a two‐dimensional
    Keller–Segel–Navier–Stokes system. <i>Journal of the London Mathematical Society</i>.
    2024;109(3). doi:<a href="https://doi.org/10.1112/jlms.12885">10.1112/jlms.12885</a>
  apa: Wang, Y., &#38; Winkler, M. (2024). An interpolation inequality involving LlogL$L\log
    L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional
    Keller–Segel–Navier–Stokes system. <i>Journal of the London Mathematical Society</i>,
    <i>109</i>(3), Article e12885. <a href="https://doi.org/10.1112/jlms.12885">https://doi.org/10.1112/jlms.12885</a>
  bibtex: '@article{Wang_Winkler_2024, title={An interpolation inequality involving
    LlogL$L\log L$ spaces and application to the characterization of blow‐up behavior
    in a two‐dimensional Keller–Segel–Navier–Stokes system}, volume={109}, DOI={<a
    href="https://doi.org/10.1112/jlms.12885">10.1112/jlms.12885</a>}, number={3e12885},
    journal={Journal of the London Mathematical Society}, publisher={Wiley}, author={Wang,
    Yulan and Winkler, Michael}, year={2024} }'
  chicago: Wang, Yulan, and Michael Winkler. “An Interpolation Inequality Involving
    LlogL$L\log L$ Spaces and Application to the Characterization of Blow‐up Behavior
    in a Two‐dimensional Keller–Segel–Navier–Stokes System.” <i>Journal of the London
    Mathematical Society</i> 109, no. 3 (2024). <a href="https://doi.org/10.1112/jlms.12885">https://doi.org/10.1112/jlms.12885</a>.
  ieee: 'Y. Wang and M. Winkler, “An interpolation inequality involving LlogL$L\log
    L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional
    Keller–Segel–Navier–Stokes system,” <i>Journal of the London Mathematical Society</i>,
    vol. 109, no. 3, Art. no. e12885, 2024, doi: <a href="https://doi.org/10.1112/jlms.12885">10.1112/jlms.12885</a>.'
  mla: Wang, Yulan, and Michael Winkler. “An Interpolation Inequality Involving LlogL$L\log
    L$ Spaces and Application to the Characterization of Blow‐up Behavior in a Two‐dimensional
    Keller–Segel–Navier–Stokes System.” <i>Journal of the London Mathematical Society</i>,
    vol. 109, no. 3, e12885, Wiley, 2024, doi:<a href="https://doi.org/10.1112/jlms.12885">10.1112/jlms.12885</a>.
  short: Y. Wang, M. Winkler, Journal of the London Mathematical Society 109 (2024).
date_created: 2025-12-18T19:07:25Z
date_updated: 2025-12-18T20:14:39Z
doi: 10.1112/jlms.12885
intvolume: '       109'
issue: '3'
language:
- iso: eng
publication: Journal of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6107
  - 1469-7750
publication_status: published
publisher: Wiley
status: public
title: An interpolation inequality involving LlogL$L\log L$ spaces and application
  to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes
  system
type: journal_article
user_id: '31496'
volume: 109
year: '2024'
...
---
_id: '66358'
abstract:
- lang: eng
  text: <jats:title>Abstract</jats:title><jats:p>Affine Bruhat–Tits buildings are
    geometric spaces extracting the combinatorics of algebraic groups. The building
    of  parameterizes flags of subspaces/lattices in or, equivalently, norms on a
    fixed finite‐dimensional vector space, up to homothety. It has first been studied
    by Goldman and Iwahori as a piecewise‐linear analogue of symmetric spaces. The
    space of seminorms compactifies the space of norms and admits a natural surjective
    restriction map from the Berkovich analytification of projective space that factors
    the natural tropicalization map. Inspired by Payne's result that the analytification
    is the limit of all tropicalizations, we show that the space of seminorms is the
    limit of all tropicalized <jats:italic>linear</jats:italic> embeddings  and prove
    a faithful tropicalization result for compactified linear spaces. The space of
    seminorms is in fact the tropical linear space associated to the universal realizable
    valuated matroid.</jats:p>
article_number: e12850
author:
- first_name: Luca
  full_name: Battistella, Luca
  last_name: Battistella
- first_name: Kevin
  full_name: Kühn, Kevin
  last_name: Kühn
- first_name: Arne
  full_name: Kuhrs, Arne
  last_name: Kuhrs
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
- first_name: Alejandro
  full_name: Vargas, Alejandro
  last_name: Vargas
citation:
  ama: Battistella L, Kühn K, Kuhrs A, Ulirsch M, Vargas A. Buildings, valuated matroids,
    and tropical linear spaces. <i>Journal of the London Mathematical Society</i>.
    2023;109(1). doi:<a href="https://doi.org/10.1112/jlms.12850">10.1112/jlms.12850</a>
  apa: Battistella, L., Kühn, K., Kuhrs, A., Ulirsch, M., &#38; Vargas, A. (2023).
    Buildings, valuated matroids, and tropical linear spaces. <i>Journal of the London
    Mathematical Society</i>, <i>109</i>(1), Article e12850. <a href="https://doi.org/10.1112/jlms.12850">https://doi.org/10.1112/jlms.12850</a>
  bibtex: '@article{Battistella_Kühn_Kuhrs_Ulirsch_Vargas_2023, title={Buildings,
    valuated matroids, and tropical linear spaces}, volume={109}, DOI={<a href="https://doi.org/10.1112/jlms.12850">10.1112/jlms.12850</a>},
    number={1e12850}, journal={Journal of the London Mathematical Society}, publisher={Wiley},
    author={Battistella, Luca and Kühn, Kevin and Kuhrs, Arne and Ulirsch, Martin
    and Vargas, Alejandro}, year={2023} }'
  chicago: Battistella, Luca, Kevin Kühn, Arne Kuhrs, Martin Ulirsch, and Alejandro
    Vargas. “Buildings, Valuated Matroids, and Tropical Linear Spaces.” <i>Journal
    of the London Mathematical Society</i> 109, no. 1 (2023). <a href="https://doi.org/10.1112/jlms.12850">https://doi.org/10.1112/jlms.12850</a>.
  ieee: 'L. Battistella, K. Kühn, A. Kuhrs, M. Ulirsch, and A. Vargas, “Buildings,
    valuated matroids, and tropical linear spaces,” <i>Journal of the London Mathematical
    Society</i>, vol. 109, no. 1, Art. no. e12850, 2023, doi: <a href="https://doi.org/10.1112/jlms.12850">10.1112/jlms.12850</a>.'
  mla: Battistella, Luca, et al. “Buildings, Valuated Matroids, and Tropical Linear
    Spaces.” <i>Journal of the London Mathematical Society</i>, vol. 109, no. 1, e12850,
    Wiley, 2023, doi:<a href="https://doi.org/10.1112/jlms.12850">10.1112/jlms.12850</a>.
  short: L. Battistella, K. Kühn, A. Kuhrs, M. Ulirsch, A. Vargas, Journal of the
    London Mathematical Society 109 (2023).
date_created: 2026-07-08T08:04:28Z
date_updated: 2026-07-08T08:04:43Z
doi: 10.1112/jlms.12850
intvolume: '       109'
issue: '1'
language:
- iso: eng
publication: Journal of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6107
  - 1469-7750
publication_status: published
publisher: Wiley
status: public
title: Buildings, valuated matroids, and tropical linear spaces
type: journal_article
user_id: '82981'
volume: 109
year: '2023'
...
---
_id: '40066'
author:
- first_name: Rodrigo
  full_name: Bañuelos, Rodrigo
  last_name: Bañuelos
- first_name: Krzysztof
  full_name: Bogdan, Krzysztof
  last_name: Bogdan
- first_name: Tomasz
  full_name: Luks, Tomasz
  id: '58312'
  last_name: Luks
citation:
  ama: Bañuelos R, Bogdan K, Luks T. Hardy–Stein identities and square functions for
    semigroups. <i>Journal of the London Mathematical Society</i>. 2016;94(2):462-478.
    doi:<a href="https://doi.org/10.1112/jlms/jdw042">10.1112/jlms/jdw042</a>
  apa: Bañuelos, R., Bogdan, K., &#38; Luks, T. (2016). Hardy–Stein identities and
    square functions for semigroups. <i>Journal of the London Mathematical Society</i>,
    <i>94</i>(2), 462–478. <a href="https://doi.org/10.1112/jlms/jdw042">https://doi.org/10.1112/jlms/jdw042</a>
  bibtex: '@article{Bañuelos_Bogdan_Luks_2016, title={Hardy–Stein identities and square
    functions for semigroups}, volume={94}, DOI={<a href="https://doi.org/10.1112/jlms/jdw042">10.1112/jlms/jdw042</a>},
    number={2}, journal={Journal of the London Mathematical Society}, publisher={Wiley},
    author={Bañuelos, Rodrigo and Bogdan, Krzysztof and Luks, Tomasz}, year={2016},
    pages={462–478} }'
  chicago: 'Bañuelos, Rodrigo, Krzysztof Bogdan, and Tomasz Luks. “Hardy–Stein Identities
    and Square Functions for Semigroups.” <i>Journal of the London Mathematical Society</i>
    94, no. 2 (2016): 462–78. <a href="https://doi.org/10.1112/jlms/jdw042">https://doi.org/10.1112/jlms/jdw042</a>.'
  ieee: 'R. Bañuelos, K. Bogdan, and T. Luks, “Hardy–Stein identities and square functions
    for semigroups,” <i>Journal of the London Mathematical Society</i>, vol. 94, no.
    2, pp. 462–478, 2016, doi: <a href="https://doi.org/10.1112/jlms/jdw042">10.1112/jlms/jdw042</a>.'
  mla: Bañuelos, Rodrigo, et al. “Hardy–Stein Identities and Square Functions for
    Semigroups.” <i>Journal of the London Mathematical Society</i>, vol. 94, no. 2,
    Wiley, 2016, pp. 462–78, doi:<a href="https://doi.org/10.1112/jlms/jdw042">10.1112/jlms/jdw042</a>.
  short: R. Bañuelos, K. Bogdan, T. Luks, Journal of the London Mathematical Society
    94 (2016) 462–478.
date_created: 2023-01-25T15:43:14Z
date_updated: 2023-01-26T17:20:19Z
department:
- _id: '555'
doi: 10.1112/jlms/jdw042
extern: '1'
intvolume: '        94'
issue: '2'
keyword:
- General Mathematics
language:
- iso: eng
page: 462-478
publication: Journal of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6107
  - 1469-7750
publication_status: published
publisher: Wiley
status: public
title: Hardy–Stein identities and square functions for semigroups
type: journal_article
user_id: '58312'
volume: 94
year: '2016'
...
---
_id: '66352'
author:
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: Ulirsch M. Tropical geometry of moduli spaces of weighted stable curves. <i>Journal
    of the London Mathematical Society</i>. 2015;92(2):427-450. doi:<a href="https://doi.org/10.1112/jlms/jdv032">10.1112/jlms/jdv032</a>
  apa: Ulirsch, M. (2015). Tropical geometry of moduli spaces of weighted stable curves.
    <i>Journal of the London Mathematical Society</i>, <i>92</i>(2), 427–450. <a href="https://doi.org/10.1112/jlms/jdv032">https://doi.org/10.1112/jlms/jdv032</a>
  bibtex: '@article{Ulirsch_2015, title={Tropical geometry of moduli spaces of weighted
    stable curves}, volume={92}, DOI={<a href="https://doi.org/10.1112/jlms/jdv032">10.1112/jlms/jdv032</a>},
    number={2}, journal={Journal of the London Mathematical Society}, publisher={Wiley},
    author={Ulirsch, Martin}, year={2015}, pages={427–450} }'
  chicago: 'Ulirsch, Martin. “Tropical Geometry of Moduli Spaces of Weighted Stable
    Curves.” <i>Journal of the London Mathematical Society</i> 92, no. 2 (2015): 427–50.
    <a href="https://doi.org/10.1112/jlms/jdv032">https://doi.org/10.1112/jlms/jdv032</a>.'
  ieee: 'M. Ulirsch, “Tropical geometry of moduli spaces of weighted stable curves,”
    <i>Journal of the London Mathematical Society</i>, vol. 92, no. 2, pp. 427–450,
    2015, doi: <a href="https://doi.org/10.1112/jlms/jdv032">10.1112/jlms/jdv032</a>.'
  mla: Ulirsch, Martin. “Tropical Geometry of Moduli Spaces of Weighted Stable Curves.”
    <i>Journal of the London Mathematical Society</i>, vol. 92, no. 2, Wiley, 2015,
    pp. 427–50, doi:<a href="https://doi.org/10.1112/jlms/jdv032">10.1112/jlms/jdv032</a>.
  short: M. Ulirsch, Journal of the London Mathematical Society 92 (2015) 427–450.
date_created: 2026-07-08T07:24:26Z
date_updated: 2026-07-09T07:01:53Z
doi: 10.1112/jlms/jdv032
intvolume: '        92'
issue: '2'
language:
- iso: eng
page: 427-450
publication: Journal of the London Mathematical Society
publication_identifier:
  issn:
  - 0024-6107
  - 1469-7750
publication_status: published
publisher: Wiley
status: public
title: Tropical geometry of moduli spaces of weighted stable curves
type: journal_article
user_id: '82981'
volume: 92
year: '2015'
...