Global expanding solutions of compressible Euler equations with small initial densities
Authors:
Shrish Parmeshwar, Mahir Hadžić and Juhi Jang
Journal:
Quart. Appl. Math. 79 (2021), 273-334
MSC (2010):
Primary 35L65, 35Q31, 76N10, 76N15
DOI:
https://doi.org/10.1090/qam/1580
Published electronically:
October 28, 2020
MathSciNet review:
4246494
Full-text PDF
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Additional Information
Abstract: We prove the existence of a large class of global-in-time expanding solutions to vacuum free boundary compressible Euler flows without relying on the existence of an underlying finite-dimensional family of special affine solutions of the flow.
References
- Daniel Coutand and Steve Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal. 206 (2012), no. 2, 515–616. MR 2980528, DOI https://doi.org/10.1007/s00205-012-0536-1
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- Magali Grassin, Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J. 47 (1998), no. 4, 1397–1432. MR 1687130, DOI https://doi.org/10.1512/iumj.1998.47.1608
- Mahir Hadžić and Juhi Jang, Nonlinear stability of expanding star solutions of the radially symmetric mass-critical Euler-Poisson system, Comm. Pure Appl. Math. 71 (2018), no. 5, 827–891. MR 3794516, DOI https://doi.org/10.1002/cpa.21721
- Mahir Hadžić and Juhi Jang, Expanding large global solutions of the equations of compressible fluid mechanics, Invent. Math. 214 (2018), no. 3, 1205–1266. MR 3878730, DOI https://doi.org/10.1007/s00222-018-0821-1
- Mahir Hadžić and J. Juhi Jang, A class of global solutions to the Euler-Poisson system, Comm. Math. Phys. 370 (2019), no. 2, 475–505. MR 3994577, DOI https://doi.org/10.1007/s00220-019-03525-1
- M. Ifrim and D. Tataru, The compressible Euler equations in a physical vacuum: a comprehensive Eulerian approach, Preprint, Arxiv: arXiv:2007.05668, (2020).
- Juhi Jang, Nonlinear instability theory of Lane-Emden stars, Comm. Pure Appl. Math. 67 (2014), no. 9, 1418–1465. MR 3245100, DOI https://doi.org/10.1002/cpa.21499
- Juhi Jang and Nader Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity, Comm. Pure Appl. Math. 62 (2009), no. 10, 1327–1385. MR 2547977, DOI https://doi.org/10.1002/cpa.20285
- Juhi Jang and Nader Masmoudi, Vacuum in gas and fluid dynamics, Nonlinear conservation laws and applications, IMA Vol. Math. Appl., vol. 153, Springer, New York, 2011, pp. 315–329. MR 2857004, DOI https://doi.org/10.1007/978-1-4419-9554-4_17
- Juhi Jang and Nader Masmoudi, Well and ill-posedness for compressible Euler equations with vacuum, J. Math. Phys. 53 (2012), no. 11, 115625, 11. MR 3026570, DOI https://doi.org/10.1063/1.4767369
- Juhi Jang and Nader Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, Comm. Pure Appl. Math. 68 (2015), no. 1, 61–111. MR 3280249, DOI https://doi.org/10.1002/cpa.21517
- Alois Kufner, Lech Maligranda, and Lars-Erik Persson, The Hardy inequality, Vydavatelský Servis, Plzeň, 2007. About its history and some related results. MR 2351524
- Tai-Ping Liu and Tong Yang, Compressible flow with vacuum and physical singularity, Methods Appl. Anal. 7 (2000), no. 3, 495–509. Cathleen Morawetz: a great mathematician. MR 1869301, DOI https://doi.org/10.4310/MAA.2000.v7.n3.a7
- L. V. Ovsyannikov, A new solution of the equations of hydrodynamics, Dokl. Akad. Nauk SSSR (N.S.) 111 (1956), 47–49 (Russian). MR 0085824
- C. Rickard, Global solutions to the compressible Euler equations with heat transport by convection around Dyson’s isothermal affine solutions, Preprint, Arxiv: arXiv:2007.03849 (2020).
- Olga S. Rozanova, Solutions with linear profile of velocity to the Euler equations in several dimensions, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003, pp. 861–870. MR 2053233
- Steve Shkoller and Thomas C. Sideris, Global existence of near-affine solutions to the compressible Euler equations, Arch. Ration. Mech. Anal. 234 (2019), no. 1, 115–180. MR 3981395, DOI https://doi.org/10.1007/s00205-019-01387-4
- Denis Serre, Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 1, 139–153 (French, with English and French summaries). MR 1437182
- Thomas C. Sideris, Spreading of the free boundary of an ideal fluid in a vacuum, J. Differential Equations 257 (2014), no. 1, 1–14. MR 3197239, DOI https://doi.org/10.1016/j.jde.2014.03.006
- Thomas C. Sideris, Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum, Arch. Ration. Mech. Anal. 225 (2017), no. 1, 141–176. MR 3634025, DOI https://doi.org/10.1007/s00205-017-1106-3
References
- Daniel Coutand and Steve Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal. 206 (2012), no. 2, 515–616. MR 2980528, DOI https://doi.org/10.1007/s00205-012-0536-1
- F. J. Dyson, Dynamics of a Spinning Gas Cloud, J. Math. Mech. 18(1): 91–101, (1968).
- Magali Grassin, Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J. 47 (1998), no. 4, 1397–1432. MR 1687130, DOI https://doi.org/10.1512/iumj.1998.47.1608
- Mahir Hadžić and Juhi Jang, Nonlinear stability of expanding star solutions of the radially symmetric mass-critical Euler-Poisson system, Comm. Pure Appl. Math. 71 (2018), no. 5, 827–891. MR 3794516, DOI https://doi.org/10.1002/cpa.21721
- Mahir Hadžić and Juhi Jang, Expanding large global solutions of the equations of compressible fluid mechanics, Invent. Math. 214 (2018), no. 3, 1205–1266. MR 3878730, DOI https://doi.org/10.1007/s00222-018-0821-1
- Mahir Hadžić and J. Juhi Jang, A class of global solutions to the Euler-Poisson system, Comm. Math. Phys. 370 (2019), no. 2, 475–505. MR 3994577, DOI https://doi.org/10.1007/s00220-019-03525-1
- M. Ifrim and D. Tataru, The compressible Euler equations in a physical vacuum: a comprehensive Eulerian approach, Preprint, Arxiv: arXiv:2007.05668, (2020).
- Juhi Jang, Nonlinear instability theory of Lane-Emden stars, Comm. Pure Appl. Math. 67 (2014), no. 9, 1418–1465. MR 3245100, DOI https://doi.org/10.1002/cpa.21499
- Juhi Jang and Nader Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity, Comm. Pure Appl. Math. 62 (2009), no. 10, 1327–1385. MR 2547977, DOI https://doi.org/10.1002/cpa.20285
- Juhi Jang and Nader Masmoudi, Vacuum in gas and fluid dynamics, Nonlinear conservation laws and applications, IMA Vol. Math. Appl., vol. 153, Springer, New York, 2011, pp. 315–329. MR 2857004, DOI https://doi.org/10.1007/978-1-4419-9554-4_17
- Juhi Jang and Nader Masmoudi, Well and ill-posedness for compressible Euler equations with vacuum, J. Math. Phys. 53 (2012), no. 11, 115625, 11. MR 3026570, DOI https://doi.org/10.1063/1.4767369
- Juhi Jang and Nader Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, Comm. Pure Appl. Math. 68 (2015), no. 1, 61–111. MR 3280249, DOI https://doi.org/10.1002/cpa.21517
- Alois Kufner, Lech Maligranda, and Lars-Erik Persson, The Hardy inequality: About its history and some related results, Vydavatelský Servis, Plzeň, 2007. MR 2351524
- Tai-Ping Liu and Tong Yang, Compressible flow with vacuum and physical singularity: Cathleen Morawetz: a great mathematician, Methods Appl. Anal. 7 (2000), no. 3, 495–509. MR 1869301, DOI https://doi.org/10.4310/MAA.2000.v7.n3.a7
- L. V. Ovsyannikov, A new solution of the equations of hydrodynamics, Dokl. Akad. Nauk SSSR (N.S.) 111 (1956), 47–49 (Russian). MR 0085824
- C. Rickard, Global solutions to the compressible Euler equations with heat transport by convection around Dyson’s isothermal affine solutions, Preprint, Arxiv: arXiv:2007.03849 (2020).
- Olga S. Rozanova, Solutions with linear profile of velocity to the Euler equations in several dimensions, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003, pp. 861–870. MR 2053233
- Steve Shkoller and Thomas C. Sideris, Global existence of near-affine solutions to the compressible Euler equations, Arch. Ration. Mech. Anal. 234 (2019), no. 1, 115–180. MR 3981395, DOI https://doi.org/10.1007/s00205-019-01387-4
- Denis Serre, Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 1, 139–153 (French, with English and French summaries). MR 1437182
- Thomas C. Sideris, Spreading of the free boundary of an ideal fluid in a vacuum, J. Differential Equations 257 (2014), no. 1, 1–14. MR 3197239, DOI https://doi.org/10.1016/j.jde.2014.03.006
- Thomas C. Sideris, Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum, Arch. Ration. Mech. Anal. 225 (2017), no. 1, 141–176. MR 3634025, DOI https://doi.org/10.1007/s00205-017-1106-3
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Additional Information
Shrish Parmeshwar
Affiliation:
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, United Kingdom
ORCID:
0000-0002-1214-2896
Email:
shrish.parmeshwar@kcl.ac.uk
Mahir Hadžić
Affiliation:
Department of Mathematics, University College London, 25 Gordon Street, London WC1E 6XA, United Kingdom
Email:
m.hadzic@ucl.ac.uk
Juhi Jang
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, California; and Korea Institute for Advanced Study, Seoul, Republic of Korea
MR Author ID:
834174
Email:
juhi.jang@usc.edu
Received by editor(s):
June 12, 2020
Received by editor(s) in revised form:
July 19, 2020
Published electronically:
October 28, 2020
Additional Notes:
The first author acknowledges the support of the EPSRC studentship grant EP/N509498/1.
The second author acknowledges the support of the EPSRC grant EP/N016777/1 and the EPSRC Early Career Fellowship EP/S02218X/1.
The third author acknowledges the support of the NSF grant DMS-1608494 and the Simons Fellowship (grant number 616364).
Article copyright:
© Copyright 2020
Brown University
