Global existence for partially dissipative hyperbolic systems in the Lp framework, and relaxation limit
Here we investigate global strong solutions for a class of partially dissipative hyperbolic systems in the framework of critical homogeneous Besov spaces. Our primary goal is to extend the analysis of our previous paper (Crin-Barat and Danchin in Partially dissipative hyperbolic systems in the criti...
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| Published in | Mathematische annalen Vol. 386; no. 3-4; pp. 2159 - 2206 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.08.2023
Springer Nature B.V Springer Verlag |
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| ISSN | 0025-5831 1432-1807 |
| DOI | 10.1007/s00208-022-02450-4 |
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| Abstract | Here we investigate global strong solutions for a class of partially dissipative hyperbolic systems in the framework of critical homogeneous Besov spaces. Our primary goal is to extend the analysis of our previous paper (Crin-Barat and Danchin in
Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case
. Published online in Journal de Mathématiques Pures et Appliquées, 2022) to a functional framework where the low frequencies of the solution are only bounded in
L
p
-type spaces with
p
larger than 2. This unusual setting is in sharp contrast with the non-dissipative case (even linear), where well-posedness in
L
p
for
p
≠
2
fails (Brenner in Math Scand 19:27–37, 1966). Our new framework enables us to prescribe weaker smallness conditions for global well-posedness and to get a more accurate information on the qualitative properties of the constructed solutions. Our existence theorem applies to the multi-dimensional isentropic compressible Euler system with relaxation, and provide us with bounds that are
independent
of the relaxation parameter for general ill-prepared data, provided they are small enough. As a consequence, we justify rigorously the relaxation limit to the porous media equation and exhibit explicit rates of convergence in suitable norms, a completely new result to the best of our knowledge. |
|---|---|
| AbstractList | Here we investigate global strong solutions for a class of partially dissipative hyperbolic systems in the framework of critical homogeneous Besov spaces. Our primary goal is to extend the analysis of our previous paper (Crin-Barat and Danchin in Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case. Published online in Journal de Mathématiques Pures et Appliquées, 2022) to a functional framework where the low frequencies of the solution are only bounded in Lp-type spaces with p larger than 2. This unusual setting is in sharp contrast with the non-dissipative case (even linear), where well-posedness in Lp for p≠2 fails (Brenner in Math Scand 19:27–37, 1966). Our new framework enables us to prescribe weaker smallness conditions for global well-posedness and to get a more accurate information on the qualitative properties of the constructed solutions. Our existence theorem applies to the multi-dimensional isentropic compressible Euler system with relaxation, and provide us with bounds that are independent of the relaxation parameter for general ill-prepared data, provided they are small enough. As a consequence, we justify rigorously the relaxation limit to the porous media equation and exhibit explicit rates of convergence in suitable norms, a completely new result to the best of our knowledge. Here we investigate global strong solutions for a class of partially dissipative hyperbolic systems in the framework of critical homogeneous Besov spaces. Our primary goal is to extend the analysis of our previous paper [10] to a functional framework where the low frequencies of the solution are only bounded in L p-type spaces with p larger than 2. This enables us to prescribe weaker smallness conditions for global well-posedness and to get a more accurate information on the qualitative properties of the constructed solutions. Our existence theorem in particular applies to the multi-dimensional isentropic compressible Euler system with relaxation, and provide us with bounds that are independent of the relaxation parameter. As a consequence, we justify rigorously the relaxation limit to the porous media equation and exhibit explicit rates of convergence for suitable norms, a completely new result to the best of our knowledge. Here we investigate global strong solutions for a class of partially dissipative hyperbolic systems in the framework of critical homogeneous Besov spaces. Our primary goal is to extend the analysis of our previous paper (Crin-Barat and Danchin in Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case . Published online in Journal de Mathématiques Pures et Appliquées, 2022) to a functional framework where the low frequencies of the solution are only bounded in L p -type spaces with p larger than 2. This unusual setting is in sharp contrast with the non-dissipative case (even linear), where well-posedness in L p for p ≠ 2 fails (Brenner in Math Scand 19:27–37, 1966). Our new framework enables us to prescribe weaker smallness conditions for global well-posedness and to get a more accurate information on the qualitative properties of the constructed solutions. Our existence theorem applies to the multi-dimensional isentropic compressible Euler system with relaxation, and provide us with bounds that are independent of the relaxation parameter for general ill-prepared data, provided they are small enough. As a consequence, we justify rigorously the relaxation limit to the porous media equation and exhibit explicit rates of convergence in suitable norms, a completely new result to the best of our knowledge. |
| Author | Danchin, Raphaël Crin-Barat, Timothée |
| Author_xml | – sequence: 1 givenname: Timothée orcidid: 0000-0002-5012-3354 surname: Crin-Barat fullname: Crin-Barat, Timothée email: timothee.crin-barat@deusto.es organization: Chair of Computational Mathematics, Fundación Deusto – sequence: 2 givenname: Raphaël surname: Danchin fullname: Danchin, Raphaël organization: Univ Paris Est Creteil, Univ Gustave Eiffel, CNRS, LAMA UMR8050 |
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| Keywords | 35Q35 76N10 1991 Mathematics Subject Classification. 35Q35 relaxation limit critical regularity 76N10 Hyperbolic systems partially dissipative |
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| References | XuJKawashimaSDiffusive relaxation limit of classical solutions to the damped compressible Euler equationsJ. Differ. Equ.201425677179631217131329.3523810.1016/j.jde.2013.09.019 Coron, J.-M.: Control and nonlinearity, vol. 136. American Mathematical Society, Mathematical Surveys and Monographs (2007) MarcatiPRubinoBHyperbolic to parabolic relaxation theory for quasilinear first order systemsJ. Differ. Equ.200016235939917517100987.3510310.1006/jdeq.1999.3676 Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Doctoral Thesis (1983) Wasiolek, V.: Analyse asymptotique de systèmes hyperboliques quasi-linéaires du premier ordre. Thesis dissertation, Université Blaise Pascal - Clermont-Ferrand II (2015) BonyJ-MCalcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéairesAnn. Sci. École Norm. Sup.19814142092460495.3502410.24033/asens.1404 PengY-JWasiolekVParabolic limit with differential constraints of first-order quasilinear hyperbolic systemsAnn. I. H. Poincaré20163341103113035195341347.3502310.1016/j.anihpc.2015.03.006 Crin-BaratTDanchinRPartially dissipative one-dimensional hyperbolic systems in the critical regularity setting, and applicationsPure Appl. Anal.2022418512544193691489.3519310.2140/paa.2022.4.85 BrennerPThe cauchy problem for symmetric hyperbolic systems in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {L}^p$$\end{document}Math. Scand.19661927372124270154.1130410.7146/math.scand.a-10793 MasciaCNguyenTTLp- Lq decay estimates for dissipative linear hyperbolic systems in 1dJ. Differ. Equ.2017263618962301397.3514810.1016/j.jde.2017.07.011 ShizutaSKawashimaSSystems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equationHokkaido Math. J.1985142492757987560587.3504610.14492/hokmj/1381757663 Serre, D.: Systèmes de lois de conservation, tome 1. Diderot editeur, Arts et Sciences, Paris, New York (1996) YongW-AEntropy and global existence for hyperbolic balance lawsArch. Rational Mech. Anal200417247266205816510.1007/s00205-003-0304-3 Crin-Barat, T., Danchin, R.: Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case. Accepted in Journal de Mathématiques Pures et Appliquées (2022) BeauchardKZuazuaELarge time asymptotics for partially dissipative hyperbolic systemsArch. Rational Mech. Anal201119917722727543411237.3501710.1007/s00205-010-0321-y LiangZShuaiZConvergence rate from hyperbolic systems of balance laws to parabolic systemsAsymptot. Anal.202112241631981502.35137 Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften, vol. 343. Springer, Heidelberg (2011) MarcatiPMilaniAThe one-dimensional Darcy’s law as the limit of a compressible Euler flowJ. Differ. Equ.19908412914710426620715.3506510.1016/0022-0396(90)90130-H XuJWangZRelaxation limit in besov spaces for compressible Euler equationsJournal de Mathématiques Pures et Appliquées201399436130032821258.3516310.1016/j.matpur.2012.06.002 BianchiniSHanouzetBNataliniRAsymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropyCommun. Pure Appl. Math.2007601559162223493491152.3500910.1002/cpa.20195 BoscarinoSRussoAOn a class of uniformly accurate IMEX Runge–Kutta schemes and applications to hyperbolic systems with relaxationSim J. Sci. Compt.20093131926194525161391193.6516210.1137/080713562 KawashimaSYongW-ADecay estimates for hyperbolic balance lawsJ. Anal. Appl.20092813324697131173.35365 ChenG-QLevermoreCDLiuT-PHyperbolic conservation laws with stiff relaxation terms and entropyCommun. Pure Appl. Math.199447678783012809890806.3511210.1002/cpa.3160470602 JuncaSRascleMStrong relaxation of the isothermal Euler system to the heat equationZ. Angew. Math. Phys.20025323926419006730997.3503510.1007/s00033-002-8154-7 Li, Y., Peng, Y-J., Zhao, L.: Convergence rate from hyperbolic systems of balance laws to parabolic systems. Appl. Anal. pp. 1079–1095 (2021) ChenQMiaoCZhangZGlobal well-posedness for compressible Navier–Stokes equations with highly oscillating initial velocityCommun. Pure Appl. Math.20106391173122426754851202.35002 MajdaACompressible fluid flow and systems of conservation laws in several space variable1984New-YorkSpringer0537.7600110.1007/978-1-4612-1116-7 DanchinRLocal theory in critical spaces for compressible viscous and heat-conductive gasesCommun. Partial Differ. Equ.2001267–81183123318552771007.3507110.1081/PDE-100106132 Zuazua, E.: Decay of partially dissipative hyperbolic systems. https://caa-avh.nat.fau.eu/enrique-zuazua-presentations/ (2020) BianchiniRUniform asymptotic and convergence estimates for the Jin-Xin model under the diffusion scalingSIAM J. Math. Anal.20185021877189937823991407.3513010.1137/17M1152395 HsiaoLQuasilinear Hyperbolic Systems and Dissipative Mechanisms1997SingaporeWorld Scientific Publishing0911.35003 CoulombelJ-FGoudonTThe strong relaxation limit of the multidimensional isothermal Euler equationsTrans. Am. Math. Soc.2007359263764822551901170.3547710.1090/S0002-9947-06-04028-1 Li, T-T.: Global classical solutions for quasilinear hyperbolic systems. Masson, Paris; John Wiley & Sons, Ltd., Chichester (1994) LinCCoulombelJ-FThe strong relaxation limit of the multidimensional Euler equationsNonlinear Differ. Equ. Appl.20132044746130571381268.3509910.1007/s00030-012-0159-0 CharveFDanchinRA global existence result for the compressible Navier–Stokes equations in the critical Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {L}^p$$\end{document} frameworkArch. Rational Mech. Anal201019823327126793721229.3516710.1007/s00205-010-0306-x XuJKawashimaSGlobal classical solutions for partially dissipative hyperbolic system of balance lawsArch. Rational Mech. Anal201421151355331490651293.3517310.1007/s00205-013-0679-8 Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. (2010) |
| References_xml | – reference: MarcatiPRubinoBHyperbolic to parabolic relaxation theory for quasilinear first order systemsJ. Differ. Equ.200016235939917517100987.3510310.1006/jdeq.1999.3676 – reference: MasciaCNguyenTTLp- Lq decay estimates for dissipative linear hyperbolic systems in 1dJ. Differ. Equ.2017263618962301397.3514810.1016/j.jde.2017.07.011 – reference: BianchiniRUniform asymptotic and convergence estimates for the Jin-Xin model under the diffusion scalingSIAM J. Math. Anal.20185021877189937823991407.3513010.1137/17M1152395 – reference: BonyJ-MCalcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéairesAnn. Sci. École Norm. Sup.19814142092460495.3502410.24033/asens.1404 – reference: XuJKawashimaSGlobal classical solutions for partially dissipative hyperbolic system of balance lawsArch. Rational Mech. Anal201421151355331490651293.3517310.1007/s00205-013-0679-8 – reference: XuJWangZRelaxation limit in besov spaces for compressible Euler equationsJournal de Mathématiques Pures et Appliquées201399436130032821258.3516310.1016/j.matpur.2012.06.002 – reference: Li, T-T.: Global classical solutions for quasilinear hyperbolic systems. Masson, Paris; John Wiley & Sons, Ltd., Chichester (1994) – reference: Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Doctoral Thesis (1983) – reference: Zuazua, E.: Decay of partially dissipative hyperbolic systems. https://caa-avh.nat.fau.eu/enrique-zuazua-presentations/ (2020) – reference: MarcatiPMilaniAThe one-dimensional Darcy’s law as the limit of a compressible Euler flowJ. Differ. Equ.19908412914710426620715.3506510.1016/0022-0396(90)90130-H – reference: Coron, J.-M.: Control and nonlinearity, vol. 136. American Mathematical Society, Mathematical Surveys and Monographs (2007) – reference: HsiaoLQuasilinear Hyperbolic Systems and Dissipative Mechanisms1997SingaporeWorld Scientific Publishing0911.35003 – reference: BoscarinoSRussoAOn a class of uniformly accurate IMEX Runge–Kutta schemes and applications to hyperbolic systems with relaxationSim J. Sci. Compt.20093131926194525161391193.6516210.1137/080713562 – reference: Serre, D.: Systèmes de lois de conservation, tome 1. Diderot editeur, Arts et Sciences, Paris, New York (1996) – reference: JuncaSRascleMStrong relaxation of the isothermal Euler system to the heat equationZ. Angew. Math. Phys.20025323926419006730997.3503510.1007/s00033-002-8154-7 – reference: Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften, vol. 343. Springer, Heidelberg (2011) – reference: CoulombelJ-FGoudonTThe strong relaxation limit of the multidimensional isothermal Euler equationsTrans. Am. Math. Soc.2007359263764822551901170.3547710.1090/S0002-9947-06-04028-1 – reference: Crin-BaratTDanchinRPartially dissipative one-dimensional hyperbolic systems in the critical regularity setting, and applicationsPure Appl. Anal.2022418512544193691489.3519310.2140/paa.2022.4.85 – reference: LinCCoulombelJ-FThe strong relaxation limit of the multidimensional Euler equationsNonlinear Differ. Equ. Appl.20132044746130571381268.3509910.1007/s00030-012-0159-0 – reference: ChenQMiaoCZhangZGlobal well-posedness for compressible Navier–Stokes equations with highly oscillating initial velocityCommun. Pure Appl. Math.20106391173122426754851202.35002 – reference: PengY-JWasiolekVParabolic limit with differential constraints of first-order quasilinear hyperbolic systemsAnn. I. H. Poincaré20163341103113035195341347.3502310.1016/j.anihpc.2015.03.006 – reference: Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. (2010) – reference: DanchinRLocal theory in critical spaces for compressible viscous and heat-conductive gasesCommun. Partial Differ. Equ.2001267–81183123318552771007.3507110.1081/PDE-100106132 – reference: BeauchardKZuazuaELarge time asymptotics for partially dissipative hyperbolic systemsArch. Rational Mech. Anal201119917722727543411237.3501710.1007/s00205-010-0321-y – reference: MajdaACompressible fluid flow and systems of conservation laws in several space variable1984New-YorkSpringer0537.7600110.1007/978-1-4612-1116-7 – reference: Wasiolek, V.: Analyse asymptotique de systèmes hyperboliques quasi-linéaires du premier ordre. Thesis dissertation, Université Blaise Pascal - Clermont-Ferrand II (2015) – reference: CharveFDanchinRA global existence result for the compressible Navier–Stokes equations in the critical Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {L}^p$$\end{document} frameworkArch. Rational Mech. Anal201019823327126793721229.3516710.1007/s00205-010-0306-x – reference: XuJKawashimaSDiffusive relaxation limit of classical solutions to the damped compressible Euler equationsJ. Differ. Equ.201425677179631217131329.3523810.1016/j.jde.2013.09.019 – reference: Li, Y., Peng, Y-J., Zhao, L.: Convergence rate from hyperbolic systems of balance laws to parabolic systems. Appl. Anal. pp. 1079–1095 (2021) – reference: ShizutaSKawashimaSSystems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equationHokkaido Math. J.1985142492757987560587.3504610.14492/hokmj/1381757663 – reference: LiangZShuaiZConvergence rate from hyperbolic systems of balance laws to parabolic systemsAsymptot. Anal.202112241631981502.35137 – reference: YongW-AEntropy and global existence for hyperbolic balance lawsArch. Rational Mech. Anal200417247266205816510.1007/s00205-003-0304-3 – reference: Crin-Barat, T., Danchin, R.: Partially dissipative hyperbolic systems in the critical regularity setting: the multi-dimensional case. Accepted in Journal de Mathématiques Pures et Appliquées (2022) – reference: BrennerPThe cauchy problem for symmetric hyperbolic systems in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {L}^p$$\end{document}Math. Scand.19661927372124270154.1130410.7146/math.scand.a-10793 – reference: KawashimaSYongW-ADecay estimates for hyperbolic balance lawsJ. Anal. Appl.20092813324697131173.35365 – reference: BianchiniSHanouzetBNataliniRAsymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropyCommun. Pure Appl. Math.2007601559162223493491152.3500910.1002/cpa.20195 – reference: ChenG-QLevermoreCDLiuT-PHyperbolic conservation laws with stiff relaxation terms and entropyCommun. Pure Appl. Math.199447678783012809890806.3511210.1002/cpa.3160470602 |
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| Title | Global existence for partially dissipative hyperbolic systems in the Lp framework, and relaxation limit |
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