Smooth self-similar imploding profiles to 3D compressible Euler
Authors:
Tristan Buckmaster, Gonzalo Cao-Labora and Javier Gómez-Serrano
Journal:
Quart. Appl. Math. 81 (2023), 517-532
MSC (2020):
Primary 35Q31
DOI:
https://doi.org/10.1090/qam/1661
Published electronically:
March 20, 2023
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Abstract: The aim of this note is to present the recent results by Buckmaster, Cao-Labora, and Gómez-Serrano [Smooth imploding solutions for 3D compressible fluids, Arxiv preprint arXiv:2208.09445, 2022] concerning the existence of “imploding singularities” for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [Invent. Math. 227 (2022), pp. 247–413; Ann. of Math. (2) 196 (2022), pp. 567–778; Ann. of Math. (2) 196 (2022), pp. 779–889] and proves the existence of self-similar profiles for all adiabatic exponents $\gamma >1$ in the case of Euler; as well as proving asymptotic self-similar blow-up for $\gamma =\frac 75$ in the case of Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blow-up in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations.
References
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- Javier Gómez-Serrano and Rafael Granero-Belinchón, On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof, Nonlinearity 27 (2014), no. 6, 1471–1498. MR 3215843, DOI 10.1088/0951-7715/27/6/1471
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- Warwick Tucker, Validated numerics, Princeton University Press, Princeton, NJ, 2011. A short introduction to rigorous computations. MR 2807595
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References
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- T. Buckmaster, T. Drivas, S. Shkoller, and V. Vicol, Formation and development of singularities for the compressible Euler equations, Proceedings of the International Congress of Mathematicians.
- Tristan Buckmaster, Theodore D. Drivas, Steve Shkoller, and Vlad Vicol, Simultaneous development of shocks and cusps for 2D Euler with azimuthal symmetry from smooth data, Ann. PDE 8 (2022), no. 2, Paper No. 26, 199. MR 4514889, DOI 10.1007/s40818-022-00141-6
- Tristan Buckmaster, Steve Shkoller, and Vlad Vicol, Formation of shocks for 2D isentropic compressible Euler, Comm. Pure Appl. Math. 75 (2022), no. 9, 2069–2120. MR 4465909
- T. Buckmaster, S. Shkoller, and V. Vicol, Formation of point shocks for 3D compressible Euler, Communications on Pure and Applied Mathematics, to appear.
- T. Buckmaster, S. Shkoller, and V. Vicol, Shock formation and vorticity creation for 3d Euler, Communications on Pure and Applied Mathematics, to appear.
- Angel Castro, Diego Córdoba, and Javier Gómez-Serrano, Global smooth solutions for the inviscid SQG equation, Mem. Amer. Math. Soc. 266 (2020), no. 1292, v+89. MR 4126257, DOI 10.1090/memo/1292
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- Jiajie Chen, Thomas Y. Hou, and De Huang, Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations, Ann. PDE 8 (2022), no. 2, Paper No. 24, 75. MR 4508052, DOI 10.1007/s40818-022-00140-7
- Shuxing Chen and Liming Dong, Formation and construction of shock for $p$-system, Sci. China Ser. A 44 (2001), no. 9, 1139–1147. MR 1860832, DOI 10.1007/BF02877431
- R. F. Chisnell, An analytic description of converging shock waves, J. Fluid Mech. 354 (1998), 357–375. MR 1606487, DOI 10.1017/S0022112097007775
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- Demetrios Christodoulou and André Lisibach, Shock development in spherical symmetry, Ann. PDE 2 (2016), no. 1, Art. 3, 246. MR 3489205, DOI 10.1007/s40818-016-0009-1
- D. Christodoulou and S. Miao, Compressible flow and Euler’s equations, Surveys of Modern Mathematics, Vol. 9, International Press, Somerville, MA, Higher Education Press, Beijing, 2014.
- Diego Córdoba, Javier Gómez-Serrano, and Andrej Zlatoš, A note on stability shifting for the Muskat problem, II: From stable to unstable and back to stable, Anal. PDE 10 (2017), no. 2, 367–378. MR 3619874, DOI 10.2140/apde.2017.10.367
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- J. Dahne and J. Gómez-Serrano, Highest cusped waves for the Burgers-Hilbert equation, ArXiv preprint arXiv:2205.00802, 2022.
- Ronald J. DiPerna, Global solutions to a class of nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 26 (1973), 1–28. MR 330788, DOI 10.1002/cpa.3160260102
- A. Enciso, J. Gómez-Serrano, and B. Vergara, Convexity of cusped Whitham waves, Arxiv preprint arXiv:1810.10935, 2018.
- C. Fefferman and R. de la Llave, Relativistic stability of matter. I, Rev. Mat. Iberoamericana 2 (1986), no. 1-2, 119–213. MR 864658, DOI 10.4171/RMI/30
- Jordi-Lluís Figueras and Rafael de la Llave, Numerical computations and computer assisted proofs of periodic orbits of the Kuramoto-Sivashinsky equation, SIAM J. Appl. Dyn. Syst. 16 (2017), no. 2, 834–852. MR 3633778, DOI 10.1137/16M1073790
- Jordi-Lluís Figueras, Marcio Gameiro, Jean-Philippe Lessard, and Rafael de la Llave, A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations, SIAM J. Appl. Dyn. Syst. 16 (2017), no. 2, 1070–1088. MR 3662023, DOI 10.1137/16M1073777
- Marcio Gameiro and Jean-Philippe Lessard, A posteriori verification of invariant objects of evolution equations: periodic orbits in the Kuramoto-Sivashinsky PDE, SIAM J. Appl. Dyn. Syst. 16 (2017), no. 1, 687–728. MR 3623202, DOI 10.1137/16M1073789
- James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 194770, DOI 10.1002/cpa.3160180408
- Javier Gómez-Serrano, Computer-assisted proofs in PDE: a survey, SeMA J. 76 (2019), no. 3, 459–484. MR 3990999, DOI 10.1007/s40324-019-00186-x
- Javier Gómez-Serrano and Rafael Granero-Belinchón, On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof, Nonlinearity 27 (2014), no. 6, 1471–1498. MR 3215843, DOI 10.1088/0951-7715/27/6/1471
- G. Guderley, Starke kugelige und zylindrische Verdichtungsstösse in der Nähe des Kugelmittelpunktes bzw. der Zylinderachse, Luftfahrtforschung 19 (1942), 302–311 (German). MR 8522
- Fredrik Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput. 66 (2017), no. 8, 1281–1292. MR 3681746, DOI 10.1109/TC.2017.2690633
- Fritz John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974), 377–405. MR 369934, DOI 10.1002/cpa.3160270307
- De-Xing Kong, Formation and propagation of singularities for $2\times 2$ quasilinear hyperbolic systems, Trans. Amer. Math. Soc. 354 (2002), no. 8, 3155–3179. MR 1897395, DOI 10.1090/S0002-9947-02-02982-3
- L. D. Landau and E. M. Lifshitz, Fluid mechanics, Course of Theoretical Physics, Vol. 6, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1959. Translated from the Russian by J. B. Sykes and W. H. Reid. MR 0108121
- Peter D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mathematical Phys. 5 (1964), 611–613. MR 165243, DOI 10.1063/1.1704154
- M.-P. Lebaud, Description de la formation d’un choc dans le $p$-système, J. Math. Pures Appl. (9) 73 (1994), no. 6, 523–565 (French, with French summary). MR 1309163
- Tai Ping Liu, Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations, J. Differential Equations 33 (1979), no. 1, 92–111. MR 540819, DOI 10.1016/0022-0396(79)90082-2
- Jonathan Luk and Jared Speck, Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent. Math. 214 (2018), no. 1, 1–169. MR 3858399, DOI 10.1007/s00222-018-0799-8
- J. Luk and J. Speck, The stability of simple plane-symmetric shock formation for 3D compressible Euler flow with vorticity and entropy, Analysis and PDE, 2021, to appear.
- Andrew Majda, The existence of multidimensional shock fronts, Mem. Amer. Math. Soc. 43 (1983), no. 281, v+93. MR 699241, DOI 10.1090/memo/0281
- Frank Merle, Pierre Raphaël, Igor Rodnianski, and Jeremie Szeftel, On blow up for the energy super critical defocusing nonlinear Schrödinger equations, Invent. Math. 227 (2022), no. 1, 247–413. MR 4359478, DOI 10.1007/s00222-021-01067-9
- Frank Merle, Pierre Raphaël, Igor Rodnianski, and Jeremie Szeftel, On the implosion of a compressible fluid I: Smooth self-similar inviscid profiles, Ann. of Math. (2) 196 (2022), no. 2, 567–778. MR 4445442, DOI 10.4007/annals.2022.196.2.3
- Frank Merle, Pierre Raphaël, Igor Rodnianski, and Jeremie Szeftel, On the implosion of a compressible fluid II: Singularity formation, Ann. of Math. (2) 196 (2022), no. 2, 779–889. MR 4445443, DOI 10.4007/annals.2022.196.2.4
- Ramon E. Moore, Methods and applications of interval analysis, SIAM Studies in Applied Mathematics, vol. 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1979. MR 551212
- Mitsuhiro T. Nakao, Michael Plum, and Yoshitaka Watanabe, Numerical verification methods and computer-assisted proofs for partial differential equations, Springer Series in Computational Mathematics, vol. 53, Springer, Singapore, [2019] ©2019. MR 3971222, DOI 10.1007/978-981-13-7669-6
- Olga Rozanova, Blow-up of smooth highly decreasing at infinity solutions to the compressible Navier-Stokes equations, J. Differential Equations 245 (2008), no. 7, 1762–1774. MR 2433485, DOI 10.1016/j.jde.2008.07.007
- Steve Shkoller and Vlad Vicol. Maximal development for Euler shock formation. preprint, 2022.
- Thomas C. Sideris, Delayed singularity formation in $2$D compressible flow, Amer. J. Math. 119 (1997), no. 2, 371–422. MR 1439554
- J. Meyer-ter-Vehn and C. Schalk, Self-similar spherical compression waves in gas dynamics, Z. Naturforsch. A 37 (1982), no. 8, 955–969. MR 676290
- Warwick Tucker, A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 (2002), no. 1, 53–117. MR 1870856, DOI 10.1007/s002080010018
- Warwick Tucker, Validated numerics, Princeton University Press, Princeton, NJ, 2011. A short introduction to rigorous computations. MR 2807595
- Jan Bouwe van den Berg, Maxime Breden, Jean-Philippe Lessard, and Lennaert van Veen, Spontaneous periodic orbits in the Navier-Stokes flow, J. Nonlinear Sci. 31 (2021), no. 2, Paper No. 41, 64. MR 4235780, DOI 10.1007/s00332-021-09695-4
- Zhouping Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math. 51 (1998), no. 3, 229–240. MR 1488513, DOI 10.1002/(SICI)1097-0312(199803)51:3$\langle$229::AID-CPA1$\rangle$3.3.CO;2-K
- Huicheng Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data, Nagoya Math. J. 175 (2004), 125–164. MR 2085314, DOI 10.1017/S002776300000893X
- Piotr Zgliczyński, Rigorous numerics for dissipative partial differential equations. II. Periodic orbit for the Kuramoto-Sivashinsky PDE—a computer-assisted proof, Found. Comput. Math. 4 (2004), no. 2, 157–185. MR 2049869, DOI 10.1007/s10208-002-0080-8
- Piotr Zgliczyński and Konstantin Mischaikow, Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation, Found. Comput. Math. 1 (2001), no. 3, 255–288. MR 1838755, DOI 10.1007/s102080010010
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35Q31
Additional Information
Tristan Buckmaster
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742
MR Author ID:
1093770
Email:
tristanb@umd.edu
Gonzalo Cao-Labora
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
MR Author ID:
1460043
ORCID:
0000-0002-8426-8391
Email:
gcaol@mit.edu
Javier Gómez-Serrano
Affiliation:
Department of Mathematics, Brown University, Providence, RI 02912
Email:
javier_gomez_serrano@brown.edu
Received by editor(s):
December 21, 2022
Received by editor(s) in revised form:
January 23, 2023
Published electronically:
March 20, 2023
Additional Notes:
The first author was supported by the NSF grants DMS-2243205 and DMS-1900149, a Simons Foundation Mathematical and Physical Sciences Collaborative Grant and a grant from the Institute for Advanced Study. The second author was supported by a grant from the Centre de Formació Interdisciplinària Superior, a MOBINT-MIF grant from the Generalitat de Catalunya and a Praecis Presidential Fellowship from the Massachusetts Institute of Technology. The second author would also like to thank the Department of Mathematics at Princeton University for partially supporting him during his stay at Princeton and for their warm hospitality. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program through the grant agreement 852741 (the second and third authors). The third author was partially supported by NSF through Grant DMS-1763356 and by the AGAUR project 2021-SGR-0087 (Catalunya). The second and third authors were partially supported by MICINN (Spain) research grant number PID2021–125021NA–I00.
Dedicated:
This review article is dedicated to Constantine Dafermos’s 80th birthday.
Article copyright:
© Copyright 2023
Brown University
