Space-time decay rates of a nonconservative compressible two-phase flow model with common pressure

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, we study space-time decay rates of a nonconservative compressible two-phase flow model with common pressure in the whole space R³. Based on previous temporal decay results, we establish the space-time decay rate of the strong solution. The main analytical techniques involve delicate weighted energy estimates and interpolation.

Keywords

nonconsevative / compressible / two-phase flow model / space-time decay rate / weighted Sobolev space

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] J. Bear, Dynamics of Fluids in Porous Media, Environmental Scienc Series, Elsevier, New York, 1972 (reprinted with corrections, New York, Dover, 1988).

[2] C. E. Brennen, Fundamentals of Multiphase Flow, Cambridge University Press, New York, 2005. https://doi.org/10.1017/CBO9780511807169

[3] K. R. Rajagopal, L. Tao, Mechanics of mixtures, in Series on Advances in Mathematics for Applied Sciences, World Scientific, 1995. https://doi.org/10.1142/2197

[4] S. Evje, T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model, J. Comput. Phys., 192 (2003), 175-210. https://doi.org/10.1016/j.jcp.2003.07.001

[5] S. Evje, T. Flåtten, Weakly implicit numerical schemes for a two-fluid model, SIAM J. Sci. Comput., 26 (2005), 1449-1484. https://doi.org/10.1137/030600631

[6] D. Bresch, B. Desjardins, J. M. Ghidaglia, E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Rational Mech. Anal., 196 (2010), 599-6293. https://doi.org/10.1007/s00205-009-0261-6

[7] D. Bresch, X. D. Huang, J. Li, Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system, Commun. Math. Phys., 309 (2012), 737-755. https://doi.org/10.1007/s00220-011-1379-6

[8] H. B. Cui, W. J. Wang, L. Yao, C. J. Zhu, Decay rates of a nonconservative compressible generic two-fluid model, SIAM J. Math. Anal., 48 (2016), 470-512. https://doi.org/10.1137/15M1037792

[9] Y. Li, H. Q. Wang, G. C. Wu, Y. H. Zhang, Global existence and decay rates for a generic compressible two-fluid model, J. Math. Fluid Mech., 25 (2023), 77. https://doi.org/10.1007/s00021-023-00822-7

[10] G. C. Wu, L. Yao, Y. H. Zhang, Global well-posedness and large time behavior of classical solutions to a generic compressible two-fluid model, Math. Ann., 389 (2024), 3379-3415. https://doi.org/10.1007/s00208-023-02732-5

[11] S. Evje, W. J. Wang, H. Y. Wen, Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model, Arch. Ration. Mech. Anal., 221 (2016), 2352-2386. https://doi.org/10.1007/s00205-016-0984-0

[12] H. Q. Wang, J. Wang, G. C. Wu, Y. H. Zhang, Optimal decay rates of a nonconservative compressible two-phase fluid model, ZAMM Z. Angew. Math. Mech., 103 (2023), 36. https://doi.org/10.1002/zamm.202100359

[13] G. C. Wu, L. Yao, Y. H. Zhang, On instability of a generic compressible two-fluid model in R³ , Nonlinearity, 36 (2023), 4740-4757. https://doi.org/.1088/1361-6544/ace818

[14] G. C. Wu, L. Yao, Y. H. Zhang, Stability and instability of a generic non-conservative compressible two-fluid model in R³ , Phys. D, 467 (2024), 134249. https://doi.org/10.1016/j.physd.2024.134249

[15] S. Takahashi, A weighted equation approach to decay rates estimates for the Navier-Stokes equations, Nonlinear Anal., 37 (1999), 751-789. https://doi.org/10.1016/S0362-546X(98)00070-4

[16] I. Kukavica, Space-time decay for solutions of the Navier-Stokes equations, Indiana Univ. Math. J., 50 (2001), 205-222. https://doi.org/10.1512/iumj.2001.50.2084

[17] I. Kukavica, On the weighted decay for solutions of the Navier-Stokes system, Nonlinear Anal., 70 (2009), 2466-2470. https://doi.org/10.1016/j.na.2008.03.031

[18] I. Kukavica, J. J. Torres, Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations, Nonlinearity, 19 (2006), 293-303. https://doi.org/10.1088/0951-7715/19/2/003

[19] I. Kukavica, J. J. Torres, Weighted L^p decay for solutions of the Navier-Stokes equations , Comm. Partial Differ. Equations, 32 (2007), 819-831. https://doi.org/10.1080/03605300600781659

[20] N. Ju, Existence and uniqueness of the solution to the dissipative 2D Quasi-Geostrophic equations in the Sobolev space, Commun. Math. Phys., 251 (2004), 365-376. https://doi.org/10.1007/s00220-004-1062-2

[21] S. K. Weng, Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations, J. Funct. Anal., 70 (2016) 2168-187. https://doi.org/10.1016/j.jfa.2016.01.021

[22] L. C. Evans, Partial Differential Equations, 2nd edition, Marcel Dekker, 2010. https://www.ams.org/journals/notices/201004/rtx100400501p.pdf

[23] S. Evje, T. Flåtten, On the wave structure of two-phase flow models, SIAM J. Appl. Math., 67 (2006), 487-511. https://doi.org/10.1137/050633482

[24] H. A. Friis, S. Evje, T. Flåtten, A numerical study of characteristic slow-transient behavior of a compressible 2D gas-liquid two-fluid model, Adv. Appl. Math. Mech., 1 (2009), 166-200. https://doc.global-sci.org/uploads/Issue/AAMM/v1n2

[25] H. Y. Wen, L. Yao, C. J. Zhu, A blow-up criterion of strong solution to a 3D viscous liquid-gas two-phase flow model with vacuum, J. Math. Pures Appl., 97 (2012), 204-229. https://doi.org/10.1016/j.matpur.2011.09.005

[26] M. Ishii, Thermo-Fluid Dynamic Theory of Two-Phase Flow, Eyrolles, Paris, 1975. https://doi.org/10.1007/978-1-4419-7985-8

[27] S. Kawashima, Y. Shibata, J. Xu, The L^p energy methods and decay for the compressible Navier-Stokes equations with capillarity , J. Math. Pures Appl., 154 (2021), 146-184. https://doi.org/10.1016/j.matpur.2021.08.009

[28] C. Kenig, G. Ponce, G. L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Am. Math. Soc., 4 (1991), 323-347. https://doi.org/10.1090/S0894-0347-1991-1086966-0

[29] T. Kobayashi, Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in R³ , J. Differ. Equations, 184 (2002), 587-619. https://doi.org/10.1006/jdeq.2002.4158

[30] T. Kobayashi, Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R³ , Commun. Math. Phys., 200 (1999), 621-659. https://doi.org/10.1007/s002200050543

[31] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat conductive fluids, Proc. Japan Acad. Ser. A, 55 (1979), 337-342. https://doi.org/10.3792/pjaa.55.337

[32] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. https://doi.org/10.1215/kjm/1250522322

[33] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. https://doi.org/10.1007/978-3-642-10926-31

[34] A. Prosperetti, G. Tryggvason, Computational Methods for Multiphase Flow, Cambridge University Press, 2007. https://doi.org/10.1017/CBO9780511607486