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doi: 10.3934/dcdsb.2022040
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Exponential decay for 2D reduced gravity two-and-a-half layer model with quantum potential and drag force

1. 

School of Mathematics, Northwest University, Xi'an 710127, China

2. 

School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127, China

* Corresponding author: Lei Yao

Received  August 2021 Revised  January 2022 Early access March 2022

Fund Project: Yao is supported by National Natural Science Foundation of China #12171390, 11931013, and Natural Science Basic Research Plan for Distinguished Young Scholars in Shaanxi Province of China (Grant No. 2019JC-26)

In this paper, we study the global weak solutions to a reduced gravity two-and-a-half layer model with quantum potential and drag force in two-dimensional torus. Inspired by Bresch, Gisclon, Lacroix-Violet [Arch. Ration. Mech. Anal. (233):975-1025, 2019] and Bresch, Gisclon, Lacroix-Violet, Vasseur [J. Math. Fluid Mech., 24(11):16, 2022], we prove that the weak solutions decay exponentially in time to equilibrium state. In order to prove the decay property of weak solutions, we obtain the relative entropy inequality of weak solutions and equilibrium solutions by defining the relative entropy functional. Considering that the structure of reduced gravity two-and-a-half layer model is more complicated than the compressible Navier-Stokes equations due to the presence of cross terms $ h_{1}\nabla h_{2} $, $ h_{2}\nabla h_{1} $, we need to estimate the cross term in relative entropy.

Citation: Yunfei Su, Lei Yao, Mengmeng Zhu. Exponential decay for 2D reduced gravity two-and-a-half layer model with quantum potential and drag force. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022040
References:
[1]

P. AntonelliL. E. Hientzsch and P. Marcati, On the low Mach number limit for quantum Navier-Stokes equations, SIAM J. Math. Anal., 52 (2020), 6105-6139.  doi: 10.1137/19M1252958.

[2]

P. AntonelliL. E. Hientzsch and S. Spirito, Global existence of finite energy weak solutions to the quantum Navier-Stokes equations with non-trivial far-field behavior, J. Differential Equations, 290 (2021), 147-177.  doi: 10.1016/j.jde.2021.04.025.

[3]

P. Antonelli and S. Spirito, Global existence of finite energy weak solutions of quantum Navier-Stokes equations, Arch. Ration. Mech. Anal., 225 (2017), 1161-1199.  doi: 10.1007/s00205-017-1124-1.

[4]

P. Antonelli and S. Spirito, On the compactness of finite energy weak solutions to the quantum Navier-Stokes equations, J. Hyperbolic Differ. Equ., 15 (2018), 133-147.  doi: 10.1142/S0219891618500054.

[5]

D. BreschB. Desjardins and E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: Part II. Existence of global $\kappa$-entropy solutions to the compressible Navier-Stokes systems with degenerate viscosities, J. Math. Pures Appl., 104 (2015), 801-836.  doi: 10.1016/j.matpur.2015.05.004.

[6]

D. BreschM. Gisclon and I. Lacroix-Violet, On Navier-Stokes -Korteweg and Euler-Korteweg systems: Application to quantum fluids models, Arch. Ration. Mech. Anal., 233 (2019), 975-1025.  doi: 10.1007/s00205-019-01373-w.

[7]

D. Bresch, M. Gisclon, I. Lacroix-Violet and A. F. Vasseur, On the exponential decay for compressible Navier-Stokes-Korteweg equations with a drag term, J. Math. Fluid Mech., 24 (2022), Paper No. 11, 16 pp. doi: 10.1007/s00021-021-00639-2.

[8]

D. Bresch, A. F. Vasseur and C. Yu, Global existence of entropy-weak solutions to the compressible Navier-Stokes equations with non-linear density dependent viscosities, J. Eur. Math. Soc., 2021. doi: 10.4171/JEMS/1143.

[9]

S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, ZAMM Z. Angew. Math. Mech., 90 (2010), 219-230.  doi: 10.1002/zamm.200900297.

[10]

H. B. CuiL. Yao and Z.-A. Yao, Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model, Commun. Pure Appl. Anal., 14 (2015), 981-1000.  doi: 10.3934/cpaa.2015.14.981.

[11]

R. Duan and C. H. Zhou, On the compactness of the reduced-gravity two-and-a-half layer equations, J. Differential Equations, 252 (2012), 3506-3519.  doi: 10.1016/j.jde.2011.12.012.

[12]

E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.

[13]

D. K. Ferry and J.-R. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling, Phys. Rev. B, 48 (1993), 7944-7950.  doi: 10.1103/PhysRevB.48.7944.

[14]

M. Gisclon and I. Lacroix-Violet, About the barotropic compressible quantum Navier-Stokes equations, Nonlinear Anal., 128 (2015), 106-121.  doi: 10.1016/j.na.2015.07.006.

[15]

J. Grant, Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations, J. Phys. A: Math., Nucl. Gen., 6 (1973), 151-153.  doi: 10.1088/0305-4470/6/11/001.

[16]

Z. H. Guo, Z. L. Li and L. Yao, Existence of global weak solution for a reduced gravity two and a half layer model, J. Math. Phys., 54 (2013), 121503, 19 pp. doi: 10.1063/1.4836775.

[17]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.

[18]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581.  doi: 10.1007/PL00005543.

[19]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.  doi: 10.1137/090776068.

[20]

A. Jüngel, Dissipative quantum fluid models, Riv. Math. Univ. Parma (N.S.), 3 (2012), 217-290. 

[21]

A. Jüngel and J.-P. Milišić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution, Kinet. Relat. Models, 4 (2011), 785-807.  doi: 10.3934/krm.2011.4.785.

[22]

I. Lacroix Violet and A. F. Vasseur, Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit, J. Math. Pures Appl., 114 (2018), 191-210.  doi: 10.1016/j.matpur.2017.12.002.

[23]

J. Li and Z. P. Xin, Global existence of weak solutions to the barotropic compressible Navier-Stokes flows with degenerate viscosities, arXiv: 1504.06826v2, 2015.

[24]

P. -L. Lions, Mathematical Topics in Fluid Mechanics, Vol. II. Compressible Models, The Clarendon Press, Oxford University Press, New York, 1998.

[25]

M. I. Loffredo and L. M. Morato, On the creation of quantized vortex lines in rotating He II, Il Nuovo Cimento B, 108 (1993), 205-215.  doi: 10.1007/BF02874411.

[26]

Y. F. Su, Z. L. Li and L. Yao, Existence of global weak solutions to 2D reduced gravity two-and-a-half layer model, J. Math. Phys., 54 (2013), 121503, 19 pp. doi: 10.1063/1.4836775.

[27]

W. J. SunS. Jiang and Z. H. Guo, Helically symmetric solutions to the 3-D Navier-Stokes equations for compressible isentropic fluids, J. Differential Equations, 222 (2006), 263-296.  doi: 10.1016/j.jde.2005.06.005.

[28]

G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 2006.

[29]

A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974.  doi: 10.1007/s00222-016-0666-4.

[30]

A. F. Vasseur and C. Yu, Global weak solutions to the compressible quantum Navier-Stokes equations with damping, SIAM J. Math. Anal., 48 (2016), 1489-1511.  doi: 10.1137/15M1013730.

[31]

R. E. Wyatt, Quantum Dynamics with Trajectories, Springer-Verlag, New York, 2005.

[32]

L. YaoZ. L. Li and W. J. Wang, Existence of spherically symmetric solutions for a reduced gravity two-and-a-half layer system, J. Differential Equations, 261 (2016), 1637-1668.  doi: 10.1016/j.jde.2016.04.012.

show all references

References:
[1]

P. AntonelliL. E. Hientzsch and P. Marcati, On the low Mach number limit for quantum Navier-Stokes equations, SIAM J. Math. Anal., 52 (2020), 6105-6139.  doi: 10.1137/19M1252958.

[2]

P. AntonelliL. E. Hientzsch and S. Spirito, Global existence of finite energy weak solutions to the quantum Navier-Stokes equations with non-trivial far-field behavior, J. Differential Equations, 290 (2021), 147-177.  doi: 10.1016/j.jde.2021.04.025.

[3]

P. Antonelli and S. Spirito, Global existence of finite energy weak solutions of quantum Navier-Stokes equations, Arch. Ration. Mech. Anal., 225 (2017), 1161-1199.  doi: 10.1007/s00205-017-1124-1.

[4]

P. Antonelli and S. Spirito, On the compactness of finite energy weak solutions to the quantum Navier-Stokes equations, J. Hyperbolic Differ. Equ., 15 (2018), 133-147.  doi: 10.1142/S0219891618500054.

[5]

D. BreschB. Desjardins and E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: Part II. Existence of global $\kappa$-entropy solutions to the compressible Navier-Stokes systems with degenerate viscosities, J. Math. Pures Appl., 104 (2015), 801-836.  doi: 10.1016/j.matpur.2015.05.004.

[6]

D. BreschM. Gisclon and I. Lacroix-Violet, On Navier-Stokes -Korteweg and Euler-Korteweg systems: Application to quantum fluids models, Arch. Ration. Mech. Anal., 233 (2019), 975-1025.  doi: 10.1007/s00205-019-01373-w.

[7]

D. Bresch, M. Gisclon, I. Lacroix-Violet and A. F. Vasseur, On the exponential decay for compressible Navier-Stokes-Korteweg equations with a drag term, J. Math. Fluid Mech., 24 (2022), Paper No. 11, 16 pp. doi: 10.1007/s00021-021-00639-2.

[8]

D. Bresch, A. F. Vasseur and C. Yu, Global existence of entropy-weak solutions to the compressible Navier-Stokes equations with non-linear density dependent viscosities, J. Eur. Math. Soc., 2021. doi: 10.4171/JEMS/1143.

[9]

S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, ZAMM Z. Angew. Math. Mech., 90 (2010), 219-230.  doi: 10.1002/zamm.200900297.

[10]

H. B. CuiL. Yao and Z.-A. Yao, Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model, Commun. Pure Appl. Anal., 14 (2015), 981-1000.  doi: 10.3934/cpaa.2015.14.981.

[11]

R. Duan and C. H. Zhou, On the compactness of the reduced-gravity two-and-a-half layer equations, J. Differential Equations, 252 (2012), 3506-3519.  doi: 10.1016/j.jde.2011.12.012.

[12]

E. FeireislA. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.  doi: 10.1007/PL00000976.

[13]

D. K. Ferry and J.-R. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling, Phys. Rev. B, 48 (1993), 7944-7950.  doi: 10.1103/PhysRevB.48.7944.

[14]

M. Gisclon and I. Lacroix-Violet, About the barotropic compressible quantum Navier-Stokes equations, Nonlinear Anal., 128 (2015), 106-121.  doi: 10.1016/j.na.2015.07.006.

[15]

J. Grant, Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations, J. Phys. A: Math., Nucl. Gen., 6 (1973), 151-153.  doi: 10.1088/0305-4470/6/11/001.

[16]

Z. H. Guo, Z. L. Li and L. Yao, Existence of global weak solution for a reduced gravity two and a half layer model, J. Math. Phys., 54 (2013), 121503, 19 pp. doi: 10.1063/1.4836775.

[17]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254.  doi: 10.1006/jdeq.1995.1111.

[18]

S. Jiang and P. Zhang, On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 215 (2001), 559-581.  doi: 10.1007/PL00005543.

[19]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.  doi: 10.1137/090776068.

[20]

A. Jüngel, Dissipative quantum fluid models, Riv. Math. Univ. Parma (N.S.), 3 (2012), 217-290. 

[21]

A. Jüngel and J.-P. Milišić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution, Kinet. Relat. Models, 4 (2011), 785-807.  doi: 10.3934/krm.2011.4.785.

[22]

I. Lacroix Violet and A. F. Vasseur, Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit, J. Math. Pures Appl., 114 (2018), 191-210.  doi: 10.1016/j.matpur.2017.12.002.

[23]

J. Li and Z. P. Xin, Global existence of weak solutions to the barotropic compressible Navier-Stokes flows with degenerate viscosities, arXiv: 1504.06826v2, 2015.

[24]

P. -L. Lions, Mathematical Topics in Fluid Mechanics, Vol. II. Compressible Models, The Clarendon Press, Oxford University Press, New York, 1998.

[25]

M. I. Loffredo and L. M. Morato, On the creation of quantized vortex lines in rotating He II, Il Nuovo Cimento B, 108 (1993), 205-215.  doi: 10.1007/BF02874411.

[26]

Y. F. Su, Z. L. Li and L. Yao, Existence of global weak solutions to 2D reduced gravity two-and-a-half layer model, J. Math. Phys., 54 (2013), 121503, 19 pp. doi: 10.1063/1.4836775.

[27]

W. J. SunS. Jiang and Z. H. Guo, Helically symmetric solutions to the 3-D Navier-Stokes equations for compressible isentropic fluids, J. Differential Equations, 222 (2006), 263-296.  doi: 10.1016/j.jde.2005.06.005.

[28]

G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics, Cambridge University Press, 2006.

[29]

A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974.  doi: 10.1007/s00222-016-0666-4.

[30]

A. F. Vasseur and C. Yu, Global weak solutions to the compressible quantum Navier-Stokes equations with damping, SIAM J. Math. Anal., 48 (2016), 1489-1511.  doi: 10.1137/15M1013730.

[31]

R. E. Wyatt, Quantum Dynamics with Trajectories, Springer-Verlag, New York, 2005.

[32]

L. YaoZ. L. Li and W. J. Wang, Existence of spherically symmetric solutions for a reduced gravity two-and-a-half layer system, J. Differential Equations, 261 (2016), 1637-1668.  doi: 10.1016/j.jde.2016.04.012.

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