Article
| Title: | Analytical solutions to the cylindrically symmetric compressible Navier-Stokes equations with free boundary (English) |
| Author: | Dong, Jianwei |
| Author: | Xia, Junchao |
| Author: | Yuen, Manwai |
| Author: | Zhang, Lijun |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 1 |
| Year: | 2026 |
| Pages: | 123-139 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | The compressible Navier-Stokes equations with Coriolis force are usually used to describe the large-scale flow motions in a thin layer of viscous fluids under the influence of the Coriolis rotational force, such as the motions of some geophysical flows and oceanic flows. We consider the vacuum free boundary problem of this system with cylindrical symmetry. We take the viscosity coefficients as $\mu (\rho )=\rho ^{\theta }$, $\lambda (\rho )=(\theta -1)\rho ^{\theta }$, where $\rho $ denotes the density of the fluid and $\theta $ is a constant. We construct some self-similar analytical solutions when $\gamma =\theta >1$ or $\gamma >1$, $\theta =\frac {1}{2}$, where $\gamma $ is the adiabatic exponent. Compared with the analytical solution to the system without the Coriolis force, the free boundary of solution constructed in this paper does not spread out infinitely in time. This indicates that the Coriolis rotational force plays a crucial role in preventing the free boundary from spreading out. Moreover, when $\theta =1$ and $\gamma =2$, under the stress-free boundary condition, we construct an analytical solution for the problem without the Coriolis force. (English) |
| Keyword: | compressible Navier-Stokes equation |
| Keyword: | density-dependent viscosity |
| Keyword: | analytical solution |
| Keyword: | free boundary |
| MSC: | 35C06 |
| MSC: | 35Q30 |
| MSC: | 35R35 |
| DOI: | 10.21136/CMJ.2026.0200-25 |
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| Date available: | 2026-03-13T09:30:16Z |
| Last updated: | 2026-03-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153564 |
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| Reference: | [1] Bresch, D., Desjardins, B.: Existence of global weak solutions for 2D viscous shallow water equations and convergence to the quasi-geostrophic model.Commun. Math. Phys. 238 (2003), 211-223. Zbl 1037.76012, MR 1989675, 10.1007/s00220-003-0859-8 |
| Reference: | [2] Bresch, D., Desjardins, B.: On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models.J. Math. Pures Appl. (9) 86 (2006), 362-368. Zbl 1121.35094, MR 2257849, 10.1016/j.matpur.2006.06.005 |
| Reference: | [3] Bresch, D., Desjardins, B., Lin, C.-K.: On some compressible fluid models: Korteweg, lubrication, and shallow water systems.Commun. Partial Differ. Equations 28 (2003), 843-868. Zbl 1106.76436, MR 1978317, 10.1081/PDE-120020499 |
| Reference: | [4] Chen, P., Zhang, T.: A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients.Commun. Pure Appl. Anal. 7 (2008), 987-1016. Zbl 1144.35042, MR 2393409, 10.3934/cpaa.2008.7.987 |
| Reference: | [5] Dong, J., Cui, H.: Analytical solutions to the cylindrically symmetric compressible Navier-Stokes equations with density-dependent viscosity and vacuum-free boundary.Bull. Braz. Math. Soc. (N.S.) 55 (2024), Article ID 8, 19 pages. Zbl 1532.35340, MR 4693283, 10.1007/s00574-023-00382-4 |
| Reference: | [6] Dong, J., Xue, H., Zhang, Q.: Analytical solutions to the pressureless Navier-Stokes equations with density-dependent viscosity coefficients.Commun. Contemp. Math. 26 (2024), Article ID 2350022, 18 pages. Zbl 1539.35162, MR 4731313, 10.1142/S0219199723500220 |
| Reference: | [7] Dong, J., Yuen, M.: Remarks on analytical solutions to compressible Navier-Stokes equations with free boundaries.Adv. Nonlinear Stud. 24 (2024), 941-951. Zbl 1556.35206, MR 4803714, 10.1515/ans-2023-0146 |
| Reference: | [8] Dong, J., Zhang, L.: Analytical solutions to the 1D compressible isothermal Navier-Stokes equations with density-dependent viscosity.J. Math. Phys. 62 (2021), Article ID 121503, 6 pages. Zbl 1498.76081, MR 4345197, 10.1063/5.0067503 |
| Reference: | [9] Fujii, M.: Global solutions to the rotating Navier-Stokes equations with large data in the critical Fourier-Besov spaces.Math. Nachr. 297 (2024), 1678-1693. Zbl 1541.35343, MR 4755730, 10.1002/mana.202300226 |
| Reference: | [10] Guo, Z., Jiang, S., Xie, F.: Global existence and asymptotic behavior of weak solutions to the 1D compressible Navier-Stokes equations with degenerate viscosity coefficient.Asymptotic Anal. 60 (2008), 101-123. Zbl 1166.35357, MR 2463800, 10.3233/ASY-2008-0 |
| Reference: | [11] Guo, Z., Jiu, Q., Xin, Z.: Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients.SIAM J. Math. Anal. 39 (2008), 1402-1427. Zbl 1151.35071, MR 2377283, 10.1137/070680333 |
| Reference: | [12] Guo, Z., Li, H.-L., Xin, Z.: Lagrange structure and dynamics for solutions to the spherically symmetric compressible Navier-Stokes equations.Commun. Math. Phys. 309 (2012), 371-412. Zbl 1233.35156, MR 2864798, 10.1007/s00220-011-1334-6 |
| Reference: | [13] Guo, Z., Xin, Z.: Analytical solutions to the compressible Navier-Stokes equations with density-dependent viscosity coefficients and free boundaries.J. Differ. Equations 253 (2012), 1-19. Zbl 1239.35109, MR 2917399, 10.1016/j.jde.2012.03.023 |
| Reference: | [14] Guo, Z., Zhu, C.: Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum.J. Differ. Equations 248 (2010), 2768-2799. Zbl 1193.35131, MR 2644149, 10.1016/j.jde.2010.03.005 |
| Reference: | [15] Guo, Z. H., Zhu, C. J.: Remarks on one-dimensional compressible Navier-Stokes equations with density-dependent viscosity and vacuum.Acta Math. Sin., Engl. Ser. 26 (2010), 2015-2030. Zbl 1202.35155, MR 2718098, 10.1007/s10114-009-7559-z |
| Reference: | [16] Jiang, S., Xin, Z., Zhang, P.: Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity.Methods Appl. Anal. 12 (2005), 239-252. Zbl 1110.35058, MR 2254008, 10.4310/MAA.2005.v12.n3.a2 |
| Reference: | [17] Li, H.-L., Li, J., Xin, Z.: Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations.Commun. Math. Phys. 281 (2008), 401-444. Zbl 1173.35099, MR 2410901, 10.1007/s00220-008-0495-4 |
| Reference: | [18] Li, H.-L., Zhang, X.: Global strong solutions to radial symmetric compressible Navier- Stokes equations with free boundary.J. Differ. Equations 261 (2016), 6341-6367. Zbl 1348.76124, MR 3552567, 10.1016/j.jde.2016.08.038 |
| Reference: | [19] Liang, Z., Shi, X.: Blowup of solutions for the compressible Navier-Stokes equations with density-dependent viscosity coefficients.Nonlinear Anal., Theory Methods Appl., Ser. A 93 (2013), 155-161. Zbl 1283.35090, MR 3117156, 10.1016/j.na.2013.07.025 |
| Reference: | [20] Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models.Oxford Lecture Series in Mathematics and its Applications 10. Clarendon Press, Oxford (1998). Zbl 0908.76004, MR 1637634 |
| Reference: | [21] Liu, T.-P., Xin, Z., Yang, T.: Vacuum states for compressible flow.Discrete Contin. Dyn. Syst. 4 (1998), 1-32. Zbl 0970.76084, MR 1485360, 10.3934/dcds.1998.4.1 |
| Reference: | [22] Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Ocean.Courant Lecture Notes in Mathematics 9. AMS, Providence (2003). Zbl 1278.76004, MR 1965452, 10.1090/cln/009 |
| Reference: | [23] Mellet, A., Vasseur, A.: On the barotropic compressible Navier-Stokes equations.Commun. Partial Differ. Equations 32 (2007), 431-452. Zbl 1149.35070, MR 2304156, 10.1080/03605300600857079 |
| Reference: | [24] Mensah, P. R.: A multi-scale limit of a randomly forced rotating 3-D compressible fluid.J. Math. Fluid Mech. 22 (2020), Article ID 30, 33 pages. Zbl 1451.35267, MR 4108618, 10.1007/s00021-020-00496-5 |
| Reference: | [25] Pedlosky, J.: Geophysical Fluid Dynamics.Springer, New York (1987). Zbl 0713.76005, 10.1007/978-1-4612-4650-3 |
| Reference: | [26] Vaigant, V. A., Kazhikhov, A. V.: On existence of global solutions to the two-dimensional Navier-Stokes equations for a compressible viscous fluid.Sib. Math. J. 36 (1995), 1108-1141. Zbl 0860.35098, MR 1375428, 10.1007/BF02106835 |
| Reference: | [27] Vasseur, A. F., Yu, C.: Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations.Invent. Math. 206 (2016), 935-974. Zbl 1354.35115, MR 3573976, 10.1007/s00222-016-0666-4 |
| Reference: | [28] Vong, S.-W., Yang, T., Zhu, C.: Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. II.J. Differ. Equations 192 (2003), 475-501. Zbl 1025.35020, MR 1990849, 10.1016/S0022-0396(03)00060-3 |
| Reference: | [29] Yang, T., Yao, Z.-A., Zhu, C.: Compressible Navier-Stokes equations with density-dependent viscosity and vacuum.Commun. Partial Differ. Equations 26 (2001), 965-981. Zbl 0982.35084, MR 1843291, 10.1081/PDE-100002385 |
| Reference: | [30] Yang, T., Zhao, H.: A vacuum problem for the one-dimensional compressible Navier- Stokes equations with density-dependent viscosity.J. Differ. Equations 184 (2002), 163-184. Zbl 1003.76073, MR 1929151, 10.1006/jdeq.2001.4140 |
| Reference: | [31] Yang, T., Zhu, C.: Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum.Commun. Math. Phys. 230 (2002), 329-363. Zbl 1045.76038, MR 1936794, 10.1007/s00220-002-0703-6 |
| Reference: | [32] Yeung, L. H., Yuen, M.: Analytical solutions to the Navier-Stokes equations with density-dependent viscosity and with pressure.J. Math. Phys. 50 (2009), Article ID 083101, 6 pages. Zbl 1223.76013, MR 2554422, 10.1063/1.3197860 |
| Reference: | [33] Yuen, M.: Analytical solutions to the Navier-Stokes equations.J. Math. Phys. 49 (2008), Article ID 113102, 10 pages. Zbl 1159.81330, MR 2468532, 10.1063/1.3013805 |
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