Smooth imploding solutions for 3D compressible fluids
Building upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [67, 68, 69], we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents $\gamma>1$ . For the particular case $\gamm...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Pi |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S205050862400012X/type/journal_article |
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| author | Tristan Buckmaster Gonzalo Cao-Labora Javier Gómez-Serrano |
| author_facet | Tristan Buckmaster Gonzalo Cao-Labora Javier Gómez-Serrano |
| author_sort | Tristan Buckmaster |
| collection | DOAJ |
| description | Building upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [67, 68, 69], we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents
$\gamma>1$
. For the particular case
$\gamma =\frac 75$
(corresponding to a diatomic gas – for example, oxygen, hydrogen, nitrogen), akin to the result [68], we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability [67] and nonlinear stability [69], which allow us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the case
$\gamma =\frac 75$
. Moreover, unlike [69], the solutions constructed have density bounded away from zero and converge to a constant at infinity, representing the first example of singularity formation in such a setting. |
| format | Article |
| id | doaj-art-d6b8ecc290ce4039904f0f1d56b75124 |
| institution | Kabale University |
| issn | 2050-5086 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj-art-d6b8ecc290ce4039904f0f1d56b751242025-02-12T05:34:17ZengCambridge University PressForum of Mathematics, Pi2050-50862025-01-011310.1017/fmp.2024.12Smooth imploding solutions for 3D compressible fluidsTristan Buckmaster0https://orcid.org/0000-0001-6356-5699Gonzalo Cao-Labora1https://orcid.org/0000-0002-8426-8391Javier Gómez-Serrano2https://orcid.org/0000-0002-5962-0859Department of Mathematics, University of Maryland, 4176 Campus Dr, William E. Kirwan Hall, 20742, College Park, MD, USA; School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ, 08540, USA;Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08540, USA;Current address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA; E-mail:Department of Mathematics, Massachusetts Institute of Technology, 182 Memorial Drive, Cambridge, MA, 02139, USA; Current address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA; E-mail:Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, Barcelona, 08007, Spain; Centre de Recerca Matemàtica, Edifici C, Campus Bellaterra, Bellaterra, 08193, Spain;Current address: Department of Mathematics, Brown University, 314 Kassar House, 151 Thayer Street, Providence, RI, 02912, USA;Building upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [67, 68, 69], we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents $\gamma>1$ . For the particular case $\gamma =\frac 75$ (corresponding to a diatomic gas – for example, oxygen, hydrogen, nitrogen), akin to the result [68], we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability [67] and nonlinear stability [69], which allow us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the case $\gamma =\frac 75$ . Moreover, unlike [69], the solutions constructed have density bounded away from zero and converge to a constant at infinity, representing the first example of singularity formation in such a setting.https://www.cambridge.org/core/product/identifier/S205050862400012X/type/journal_article35Q3035Q3576N1065G30 |
| spellingShingle | Tristan Buckmaster Gonzalo Cao-Labora Javier Gómez-Serrano Smooth imploding solutions for 3D compressible fluids Forum of Mathematics, Pi 35Q30 35Q35 76N10 65G30 |
| title | Smooth imploding solutions for 3D compressible fluids |
| title_full | Smooth imploding solutions for 3D compressible fluids |
| title_fullStr | Smooth imploding solutions for 3D compressible fluids |
| title_full_unstemmed | Smooth imploding solutions for 3D compressible fluids |
| title_short | Smooth imploding solutions for 3D compressible fluids |
| title_sort | smooth imploding solutions for 3d compressible fluids |
| topic | 35Q30 35Q35 76N10 65G30 |
| url | https://www.cambridge.org/core/product/identifier/S205050862400012X/type/journal_article |
| work_keys_str_mv | AT tristanbuckmaster smoothimplodingsolutionsfor3dcompressiblefluids AT gonzalocaolabora smoothimplodingsolutionsfor3dcompressiblefluids AT javiergomezserrano smoothimplodingsolutionsfor3dcompressiblefluids |