A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum

In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Shujuan Wang [verfasserIn] Jialin Ren [verfasserIn] Rijian Su [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2024

Schlagwörter:	compressible magnetohydrodynamic equations strong solution blow up vacuum

Übergeordnetes Werk:	In: Mathematics - MDPI AG, 2013, 12(2024), 5, p 687
Übergeordnetes Werk:	volume:12 ; year:2024 ; number:5, p 687

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.3390/math12050687

Katalog-ID:	DOAJ091248566

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520			\|a In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mo<∥</mo<<mi<ρ</mi<<msub<<mo<∥</mo<<mrow<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<;</mo<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mo<)</mo<</mrow<</mrow<</msub<<mo<+</mo<<mo<∥</mo<<mi mathvariant="bold"<u</mi<<msub<<mo<∥</mo<<mrow<<mi<C</mi<<mo<(</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<<mo<;</mo<<msup<<mi<L</mi<<mn<3</mn<</msup<<mo<)</mo<</mrow<</msub<<mo<<</mo<<mo<∞</mo<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<, then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<</semantics<</math<</inline-formula<. Notably, our analysis accommodates the presence of an initial vacuum.
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allfields	10.3390/math12050687 doi (DE-627)DOAJ091248566 (DE-599)DOAJ3ae79e4c36dd45cba1b1845c74983cef DE-627 ger DE-627 rakwb eng QA1-939 Shujuan Wang verfasserin aut A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mo<∥</mo<<mi<ρ</mi<<msub<<mo<∥</mo<<mrow<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<;</mo<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mo<)</mo<</mrow<</mrow<</msub<<mo<+</mo<<mo<∥</mo<<mi mathvariant="bold"<u</mi<<msub<<mo<∥</mo<<mrow<<mi<C</mi<<mo<(</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<<mo<;</mo<<msup<<mi<L</mi<<mn<3</mn<</msup<<mo<)</mo<</mrow<</msub<<mo<<</mo<<mo<∞</mo<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<, then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<</semantics<</math<</inline-formula<. Notably, our analysis accommodates the presence of an initial vacuum. compressible magnetohydrodynamic equations strong solution blow up vacuum Mathematics Jialin Ren verfasserin aut Rijian Su verfasserin aut In Mathematics MDPI AG, 2013 12(2024), 5, p 687 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:12 year:2024 number:5, p 687 https://doi.org/10.3390/math12050687 kostenfrei https://doaj.org/article/3ae79e4c36dd45cba1b1845c74983cef kostenfrei https://www.mdpi.com/2227-7390/12/5/687 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 12 2024 5, p 687
spelling	10.3390/math12050687 doi (DE-627)DOAJ091248566 (DE-599)DOAJ3ae79e4c36dd45cba1b1845c74983cef DE-627 ger DE-627 rakwb eng QA1-939 Shujuan Wang verfasserin aut A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mo<∥</mo<<mi<ρ</mi<<msub<<mo<∥</mo<<mrow<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<;</mo<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mo<)</mo<</mrow<</mrow<</msub<<mo<+</mo<<mo<∥</mo<<mi mathvariant="bold"<u</mi<<msub<<mo<∥</mo<<mrow<<mi<C</mi<<mo<(</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<<mo<;</mo<<msup<<mi<L</mi<<mn<3</mn<</msup<<mo<)</mo<</mrow<</msub<<mo<<</mo<<mo<∞</mo<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<, then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<</semantics<</math<</inline-formula<. Notably, our analysis accommodates the presence of an initial vacuum. compressible magnetohydrodynamic equations strong solution blow up vacuum Mathematics Jialin Ren verfasserin aut Rijian Su verfasserin aut In Mathematics MDPI AG, 2013 12(2024), 5, p 687 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:12 year:2024 number:5, p 687 https://doi.org/10.3390/math12050687 kostenfrei https://doaj.org/article/3ae79e4c36dd45cba1b1845c74983cef kostenfrei https://www.mdpi.com/2227-7390/12/5/687 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 12 2024 5, p 687
allfields_unstemmed	10.3390/math12050687 doi (DE-627)DOAJ091248566 (DE-599)DOAJ3ae79e4c36dd45cba1b1845c74983cef DE-627 ger DE-627 rakwb eng QA1-939 Shujuan Wang verfasserin aut A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mo<∥</mo<<mi<ρ</mi<<msub<<mo<∥</mo<<mrow<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<;</mo<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mo<)</mo<</mrow<</mrow<</msub<<mo<+</mo<<mo<∥</mo<<mi mathvariant="bold"<u</mi<<msub<<mo<∥</mo<<mrow<<mi<C</mi<<mo<(</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<<mo<;</mo<<msup<<mi<L</mi<<mn<3</mn<</msup<<mo<)</mo<</mrow<</msub<<mo<<</mo<<mo<∞</mo<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<, then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<</semantics<</math<</inline-formula<. Notably, our analysis accommodates the presence of an initial vacuum. compressible magnetohydrodynamic equations strong solution blow up vacuum Mathematics Jialin Ren verfasserin aut Rijian Su verfasserin aut In Mathematics MDPI AG, 2013 12(2024), 5, p 687 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:12 year:2024 number:5, p 687 https://doi.org/10.3390/math12050687 kostenfrei https://doaj.org/article/3ae79e4c36dd45cba1b1845c74983cef kostenfrei https://www.mdpi.com/2227-7390/12/5/687 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 12 2024 5, p 687
allfieldsGer	10.3390/math12050687 doi (DE-627)DOAJ091248566 (DE-599)DOAJ3ae79e4c36dd45cba1b1845c74983cef DE-627 ger DE-627 rakwb eng QA1-939 Shujuan Wang verfasserin aut A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mo<∥</mo<<mi<ρ</mi<<msub<<mo<∥</mo<<mrow<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<;</mo<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mo<)</mo<</mrow<</mrow<</msub<<mo<+</mo<<mo<∥</mo<<mi mathvariant="bold"<u</mi<<msub<<mo<∥</mo<<mrow<<mi<C</mi<<mo<(</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<<mo<;</mo<<msup<<mi<L</mi<<mn<3</mn<</msup<<mo<)</mo<</mrow<</msub<<mo<<</mo<<mo<∞</mo<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<, then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<</semantics<</math<</inline-formula<. Notably, our analysis accommodates the presence of an initial vacuum. compressible magnetohydrodynamic equations strong solution blow up vacuum Mathematics Jialin Ren verfasserin aut Rijian Su verfasserin aut In Mathematics MDPI AG, 2013 12(2024), 5, p 687 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:12 year:2024 number:5, p 687 https://doi.org/10.3390/math12050687 kostenfrei https://doaj.org/article/3ae79e4c36dd45cba1b1845c74983cef kostenfrei https://www.mdpi.com/2227-7390/12/5/687 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 12 2024 5, p 687
allfieldsSound	10.3390/math12050687 doi (DE-627)DOAJ091248566 (DE-599)DOAJ3ae79e4c36dd45cba1b1845c74983cef DE-627 ger DE-627 rakwb eng QA1-939 Shujuan Wang verfasserin aut A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum 2024 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mo<∥</mo<<mi<ρ</mi<<msub<<mo<∥</mo<<mrow<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<;</mo<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mo<)</mo<</mrow<</mrow<</msub<<mo<+</mo<<mo<∥</mo<<mi mathvariant="bold"<u</mi<<msub<<mo<∥</mo<<mrow<<mi<C</mi<<mo<(</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<<mo<;</mo<<msup<<mi<L</mi<<mn<3</mn<</msup<<mo<)</mo<</mrow<</msub<<mo<<</mo<<mo<∞</mo<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<, then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<</semantics<</math<</inline-formula<. Notably, our analysis accommodates the presence of an initial vacuum. compressible magnetohydrodynamic equations strong solution blow up vacuum Mathematics Jialin Ren verfasserin aut Rijian Su verfasserin aut In Mathematics MDPI AG, 2013 12(2024), 5, p 687 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:12 year:2024 number:5, p 687 https://doi.org/10.3390/math12050687 kostenfrei https://doaj.org/article/3ae79e4c36dd45cba1b1845c74983cef kostenfrei https://www.mdpi.com/2227-7390/12/5/687 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 12 2024 5, p 687
language	English
source	In Mathematics 12(2024), 5, p 687 volume:12 year:2024 number:5, p 687
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callnumber-first	Q - Science
author	Shujuan Wang
spellingShingle	Shujuan Wang misc QA1-939 misc compressible magnetohydrodynamic equations misc strong solution misc blow up misc vacuum misc Mathematics A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum
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topic_title	QA1-939 A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum compressible magnetohydrodynamic equations strong solution blow up vacuum
topic	misc QA1-939 misc compressible magnetohydrodynamic equations misc strong solution misc blow up misc vacuum misc Mathematics
topic_unstemmed	misc QA1-939 misc compressible magnetohydrodynamic equations misc strong solution misc blow up misc vacuum misc Mathematics
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title	A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum
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title_full	A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum
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title_sort	blow-up criterion for 3d compressible isentropic magnetohydrodynamic equations with vacuum
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title_auth	A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum
abstract	In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mo<∥</mo<<mi<ρ</mi<<msub<<mo<∥</mo<<mrow<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<;</mo<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mo<)</mo<</mrow<</mrow<</msub<<mo<+</mo<<mo<∥</mo<<mi mathvariant="bold"<u</mi<<msub<<mo<∥</mo<<mrow<<mi<C</mi<<mo<(</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<<mo<;</mo<<msup<<mi<L</mi<<mn<3</mn<</msup<<mo<)</mo<</mrow<</msub<<mo<<</mo<<mo<∞</mo<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<, then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<</semantics<</math<</inline-formula<. Notably, our analysis accommodates the presence of an initial vacuum.
abstractGer	In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mo<∥</mo<<mi<ρ</mi<<msub<<mo<∥</mo<<mrow<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<;</mo<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mo<)</mo<</mrow<</mrow<</msub<<mo<+</mo<<mo<∥</mo<<mi mathvariant="bold"<u</mi<<msub<<mo<∥</mo<<mrow<<mi<C</mi<<mo<(</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<<mo<;</mo<<msup<<mi<L</mi<<mn<3</mn<</msup<<mo<)</mo<</mrow<</msub<<mo<<</mo<<mo<∞</mo<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<, then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<</semantics<</math<</inline-formula<. Notably, our analysis accommodates the presence of an initial vacuum.
abstract_unstemmed	In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mo<∥</mo<<mi<ρ</mi<<msub<<mo<∥</mo<<mrow<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<;</mo<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mo<)</mo<</mrow<</mrow<</msub<<mo<+</mo<<mo<∥</mo<<mi mathvariant="bold"<u</mi<<msub<<mo<∥</mo<<mrow<<mi<C</mi<<mo<(</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<<mo<;</mo<<msup<<mi<L</mi<<mn<3</mn<</msup<<mo<)</mo<</mrow<</msub<<mo<<</mo<<mo<∞</mo<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<, then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<</semantics<</math<</inline-formula<. Notably, our analysis accommodates the presence of an initial vacuum.
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title_short	A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum
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It is shown that if density and velocity satisfy <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<(</mo<<mo<∥</mo<<mi<ρ</mi<<msub<<mo<∥</mo<<mrow<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<;</mo<<msup<<mi<L</mi<<mo<∞</mo<</msup<<mo<)</mo<</mrow<</mrow<</msub<<mo<+</mo<<mo<∥</mo<<mi mathvariant="bold"<u</mi<<msub<<mo<∥</mo<<mrow<<mi<C</mi<<mo<(</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<<mo<;</mo<<msup<<mi<L</mi<<mn<3</mn<</msup<<mo<)</mo<</mrow<</msub<<mo<<</mo<<mo<∞</mo<<mo<)</mo<</mrow<</semantics<</math<</inline-formula<, then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mi<T</mi<<mo<]</mo<</mrow<</semantics<</math<</inline-formula<. 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A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum

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