Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems
Abstract. We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Mascia, Corrado [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2003 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag Berlin Heidelberg 2003 |
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| Übergeordnetes Werk: |
Enthalten in: Archive for rational mechanics and analysis - Springer-Verlag, 1957, 172(2003), 1 vom: 01. Dez., Seite 93-131 |
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| Übergeordnetes Werk: |
volume:172 ; year:2003 ; number:1 ; day:01 ; month:12 ; pages:93-131 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00205-003-0293-2 |
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OLC2056414422 |
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| 520 | |a Abstract. We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes. | ||
| 650 | 4 | |a Energy Estimate | |
| 650 | 4 | |a Nonlinear Stability | |
| 650 | 4 | |a Orbital Stability | |
| 650 | 4 | |a Characteristic Speed | |
| 650 | 4 | |a Spectral Stability | |
| 700 | 1 | |a Zumbrun, Kevin |4 aut | |
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10.1007/s00205-003-0293-2 doi (DE-627)OLC2056414422 (DE-He213)s00205-003-0293-2-p DE-627 ger DE-627 rakwb eng 530 510 VZ Mascia, Corrado verfasserin aut Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems 2003 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2003 Abstract. We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes. Energy Estimate Nonlinear Stability Orbital Stability Characteristic Speed Spectral Stability Zumbrun, Kevin aut Enthalten in Archive for rational mechanics and analysis Springer-Verlag, 1957 172(2003), 1 vom: 01. Dez., Seite 93-131 (DE-627)129519618 (DE-600)212130-X (DE-576)014933004 0003-9527 nnns volume:172 year:2003 number:1 day:01 month:12 pages:93-131 https://doi.org/10.1007/s00205-003-0293-2 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_32 GBV_ILN_40 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_100 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4314 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4700 AR 172 2003 1 01 12 93-131 |
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10.1007/s00205-003-0293-2 doi (DE-627)OLC2056414422 (DE-He213)s00205-003-0293-2-p DE-627 ger DE-627 rakwb eng 530 510 VZ Mascia, Corrado verfasserin aut Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems 2003 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2003 Abstract. We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes. Energy Estimate Nonlinear Stability Orbital Stability Characteristic Speed Spectral Stability Zumbrun, Kevin aut Enthalten in Archive for rational mechanics and analysis Springer-Verlag, 1957 172(2003), 1 vom: 01. Dez., Seite 93-131 (DE-627)129519618 (DE-600)212130-X (DE-576)014933004 0003-9527 nnns volume:172 year:2003 number:1 day:01 month:12 pages:93-131 https://doi.org/10.1007/s00205-003-0293-2 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_32 GBV_ILN_40 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_100 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4314 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4700 AR 172 2003 1 01 12 93-131 |
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10.1007/s00205-003-0293-2 doi (DE-627)OLC2056414422 (DE-He213)s00205-003-0293-2-p DE-627 ger DE-627 rakwb eng 530 510 VZ Mascia, Corrado verfasserin aut Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems 2003 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2003 Abstract. We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes. Energy Estimate Nonlinear Stability Orbital Stability Characteristic Speed Spectral Stability Zumbrun, Kevin aut Enthalten in Archive for rational mechanics and analysis Springer-Verlag, 1957 172(2003), 1 vom: 01. Dez., Seite 93-131 (DE-627)129519618 (DE-600)212130-X (DE-576)014933004 0003-9527 nnns volume:172 year:2003 number:1 day:01 month:12 pages:93-131 https://doi.org/10.1007/s00205-003-0293-2 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_32 GBV_ILN_40 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_100 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4314 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4700 AR 172 2003 1 01 12 93-131 |
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10.1007/s00205-003-0293-2 doi (DE-627)OLC2056414422 (DE-He213)s00205-003-0293-2-p DE-627 ger DE-627 rakwb eng 530 510 VZ Mascia, Corrado verfasserin aut Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems 2003 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2003 Abstract. We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes. Energy Estimate Nonlinear Stability Orbital Stability Characteristic Speed Spectral Stability Zumbrun, Kevin aut Enthalten in Archive for rational mechanics and analysis Springer-Verlag, 1957 172(2003), 1 vom: 01. Dez., Seite 93-131 (DE-627)129519618 (DE-600)212130-X (DE-576)014933004 0003-9527 nnns volume:172 year:2003 number:1 day:01 month:12 pages:93-131 https://doi.org/10.1007/s00205-003-0293-2 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_32 GBV_ILN_40 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_100 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4314 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4700 AR 172 2003 1 01 12 93-131 |
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10.1007/s00205-003-0293-2 doi (DE-627)OLC2056414422 (DE-He213)s00205-003-0293-2-p DE-627 ger DE-627 rakwb eng 530 510 VZ Mascia, Corrado verfasserin aut Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems 2003 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2003 Abstract. We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes. Energy Estimate Nonlinear Stability Orbital Stability Characteristic Speed Spectral Stability Zumbrun, Kevin aut Enthalten in Archive for rational mechanics and analysis Springer-Verlag, 1957 172(2003), 1 vom: 01. Dez., Seite 93-131 (DE-627)129519618 (DE-600)212130-X (DE-576)014933004 0003-9527 nnns volume:172 year:2003 number:1 day:01 month:12 pages:93-131 https://doi.org/10.1007/s00205-003-0293-2 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_32 GBV_ILN_40 GBV_ILN_62 GBV_ILN_65 GBV_ILN_70 GBV_ILN_100 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4314 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4700 AR 172 2003 1 01 12 93-131 |
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stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems |
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Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems |
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Abstract. We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes. © Springer-Verlag Berlin Heidelberg 2003 |
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Abstract. We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes. © Springer-Verlag Berlin Heidelberg 2003 |
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Abstract. We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes. © Springer-Verlag Berlin Heidelberg 2003 |
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