Wellposedness in the Lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions
Abstract In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Ascoli, Davide [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2013 |
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| Anmerkung: |
© Springer Basel 2013 |
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| Übergeordnetes Werk: |
Enthalten in: Nonlinear differential equations and applications - Springer Basel, 1994, 21(2013), 2 vom: 27. Okt., Seite 263-287 |
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| Übergeordnetes Werk: |
volume:21 ; year:2013 ; number:2 ; day:27 ; month:10 ; pages:263-287 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00030-013-0246-x |
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OLC2069549844 |
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| 520 | |a Abstract In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in $${\mathbf{R}^3}$$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in $${L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}$$, where $${\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbf{R}^3}$$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to $${\partial\Omega}$$; they lie also in $${W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}$$, where $${q \in [1, \infty]}$$. | ||
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10.1007/s00030-013-0246-x doi (DE-627)OLC2069549844 (DE-He213)s00030-013-0246-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Ascoli, Davide verfasserin aut Wellposedness in the Lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Basel 2013 Abstract In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in $${\mathbf{R}^3}$$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in $${L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}$$, where $${\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbf{R}^3}$$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to $${\partial\Omega}$$; they lie also in $${W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}$$, where $${q \in [1, \infty]}$$. Selvaduray, Steave C. aut Enthalten in Nonlinear differential equations and applications Springer Basel, 1994 21(2013), 2 vom: 27. Okt., Seite 263-287 (DE-627)182292789 (DE-600)1194489-4 (DE-576)045287600 1021-9722 nnns volume:21 year:2013 number:2 day:27 month:10 pages:263-287 https://doi.org/10.1007/s00030-013-0246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4277 GBV_ILN_4323 AR 21 2013 2 27 10 263-287 |
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10.1007/s00030-013-0246-x doi (DE-627)OLC2069549844 (DE-He213)s00030-013-0246-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Ascoli, Davide verfasserin aut Wellposedness in the Lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Basel 2013 Abstract In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in $${\mathbf{R}^3}$$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in $${L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}$$, where $${\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbf{R}^3}$$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to $${\partial\Omega}$$; they lie also in $${W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}$$, where $${q \in [1, \infty]}$$. Selvaduray, Steave C. aut Enthalten in Nonlinear differential equations and applications Springer Basel, 1994 21(2013), 2 vom: 27. Okt., Seite 263-287 (DE-627)182292789 (DE-600)1194489-4 (DE-576)045287600 1021-9722 nnns volume:21 year:2013 number:2 day:27 month:10 pages:263-287 https://doi.org/10.1007/s00030-013-0246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4277 GBV_ILN_4323 AR 21 2013 2 27 10 263-287 |
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10.1007/s00030-013-0246-x doi (DE-627)OLC2069549844 (DE-He213)s00030-013-0246-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Ascoli, Davide verfasserin aut Wellposedness in the Lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Basel 2013 Abstract In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in $${\mathbf{R}^3}$$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in $${L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}$$, where $${\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbf{R}^3}$$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to $${\partial\Omega}$$; they lie also in $${W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}$$, where $${q \in [1, \infty]}$$. Selvaduray, Steave C. aut Enthalten in Nonlinear differential equations and applications Springer Basel, 1994 21(2013), 2 vom: 27. Okt., Seite 263-287 (DE-627)182292789 (DE-600)1194489-4 (DE-576)045287600 1021-9722 nnns volume:21 year:2013 number:2 day:27 month:10 pages:263-287 https://doi.org/10.1007/s00030-013-0246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4277 GBV_ILN_4323 AR 21 2013 2 27 10 263-287 |
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10.1007/s00030-013-0246-x doi (DE-627)OLC2069549844 (DE-He213)s00030-013-0246-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Ascoli, Davide verfasserin aut Wellposedness in the Lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Basel 2013 Abstract In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in $${\mathbf{R}^3}$$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in $${L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}$$, where $${\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbf{R}^3}$$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to $${\partial\Omega}$$; they lie also in $${W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}$$, where $${q \in [1, \infty]}$$. Selvaduray, Steave C. aut Enthalten in Nonlinear differential equations and applications Springer Basel, 1994 21(2013), 2 vom: 27. Okt., Seite 263-287 (DE-627)182292789 (DE-600)1194489-4 (DE-576)045287600 1021-9722 nnns volume:21 year:2013 number:2 day:27 month:10 pages:263-287 https://doi.org/10.1007/s00030-013-0246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4277 GBV_ILN_4323 AR 21 2013 2 27 10 263-287 |
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10.1007/s00030-013-0246-x doi (DE-627)OLC2069549844 (DE-He213)s00030-013-0246-x-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Ascoli, Davide verfasserin aut Wellposedness in the Lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions 2013 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Basel 2013 Abstract In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in $${\mathbf{R}^3}$$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in $${L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}$$, where $${\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbf{R}^3}$$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to $${\partial\Omega}$$; they lie also in $${W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}$$, where $${q \in [1, \infty]}$$. Selvaduray, Steave C. aut Enthalten in Nonlinear differential equations and applications Springer Basel, 1994 21(2013), 2 vom: 27. Okt., Seite 263-287 (DE-627)182292789 (DE-600)1194489-4 (DE-576)045287600 1021-9722 nnns volume:21 year:2013 number:2 day:27 month:10 pages:263-287 https://doi.org/10.1007/s00030-013-0246-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4277 GBV_ILN_4323 AR 21 2013 2 27 10 263-287 |
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| title_sort |
wellposedness in the lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions |
| title_auth |
Wellposedness in the Lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions |
| abstract |
Abstract In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in $${\mathbf{R}^3}$$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in $${L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}$$, where $${\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbf{R}^3}$$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to $${\partial\Omega}$$; they lie also in $${W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}$$, where $${q \in [1, \infty]}$$. © Springer Basel 2013 |
| abstractGer |
Abstract In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in $${\mathbf{R}^3}$$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in $${L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}$$, where $${\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbf{R}^3}$$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to $${\partial\Omega}$$; they lie also in $${W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}$$, where $${q \in [1, \infty]}$$. © Springer Basel 2013 |
| abstract_unstemmed |
Abstract In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in $${\mathbf{R}^3}$$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in $${L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}$$, where $${\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbf{R}^3}$$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to $${\partial\Omega}$$; they lie also in $${W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}$$, where $${q \in [1, \infty]}$$. © Springer Basel 2013 |
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| title_short |
Wellposedness in the Lipschitz class for a quasi-linear hyperbolic system arising from a model of the atmosphere including water phase transitions |
| url |
https://doi.org/10.1007/s00030-013-0246-x |
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| author2 |
Selvaduray, Steave C. |
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| up_date |
2024-07-03T22:35:41.078Z |
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