January  2022, 27(1): 277-300. doi: 10.3934/dcdsb.2021042

Strong solutions to a fluid-particle interaction model with magnetic field in $ \mathbb{R}^2 $

1. 

South China Research Center for Applied Mathematics and Interdisciplinary Studies and School of Mathematical Sciences, South China Normal University Guangzhou, 510631, China

2. 

School of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521041, China

3. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

* Corresponding author: Bingyuan Huang

Received  December 2019 Published  January 2022 Early access  February 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China (Nos. 11371152, 11771155, 11571117, 11871005), the Natural Science Foundation of Guangdong Province (Nos. 2017A030313003, 2019A1515011491), and the Science and Technology Program of Guangzhou (No. 2019050001). The second author is supported by the National Natural Science Foundation of China(Nos. 12026253, 12026244, 11971357), and the Natural Science Foundation of Guangdong Province (No. 2018A030310008)

A fluid-particle interaction model with magnetic field is studied in this paper. When the initial vacuum and the far field vacuum of the fluid and the particles are contained, the constant shear viscosity $ \mu $ and the bulk viscosity $ \lambda $ are $ \mu>0 $ $ \lambda = \rho^\beta $ for any $ \beta\geq 0 $, the strong solutions of the 2D Cauchy problem for the coupled system are established applying the method of weighted estimates in Li-Liang's paper on Navier-Stokes equations.

Citation: Shijin Ding, Bingyuan Huang, Xiaoyan Hou. Strong solutions to a fluid-particle interaction model with magnetic field in $ \mathbb{R}^2 $. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 277-300. doi: 10.3934/dcdsb.2021042
References:
[1]

J. Ballew, Low Mach number limits to the Navier-Stokes-Smoluchowski system. Hyperbolic Problems: Theory, Numerics, Applications, AIMS Series on Applied Mathematics, 8 (2014), 301-308. 

[2]

J. Ballew, Mathematical Topics in Fluid-Particle Interaction, Ph.D thesis, University of Maryland, USA, 2014.

[3]

J. Ballew and K. Trivisa, Weakly dissipative solutions and weak strong uniqueness for the Navier Stokes Smoluchowski system, Nonlinear Analysis Series A: Theory, Methods Applications, 91 (2013), 1-19.  doi: 10.1016/j.na.2013.06.002.

[4]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Commun. Partial Differ. Equ., 31 (2006), 1349-1379.  doi: 10.1080/03605300500394389.

[5]

J. A. CarrilloT. Karper and K. Trivisa, On the dynamics of a fluid-particle interaction model: The bubbling regime, Nonlinear Anal., 74 (2011), 2778-2801.  doi: 10.1016/j.na.2010.12.031.

[6]

Y. S. ChenS. J. Ding and W. J. Wang, Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Cont. Dyn. Serious A., 36 (2016), 5287-5307.  doi: 10.3934/dcds.2016032.

[7]

R. M. ChenJ. L. Hu and D. H. Wang, Global weak solutions to the magnetohydrodynamic and Vlasov equations, J. Math. Fluid Mech., 18 (2016), 343-360.  doi: 10.1007/s00021-015-0238-1.

[8]

Y. ChoH. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.

[9]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Diff. Equ., 190 (2003), 504-523.  doi: 10.1016/S0022-0396(03)00015-9.

[10]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.

[11]

S. J. DingB. Y. Huang and Q. R. Li, Global existence and decay estimates for the classical solutions to a compressible fluid-particle interaction model, Acta Mathematica Scientia, 39 (2019), 1525-1537.  doi: 10.1007/s10473-019-0605-8.

[12]

S. J. Ding, B. Y. Huang and X. L. Liu, Global classical solutions to the 2D compressible Navier-Stokes equations with vacuum, Journal of Mathematical Physics, 59 (2018), 081507, 19pp. doi: 10.1063/1.5000296.

[13]

D. Y. Fang, R. Z. Zi and T. Zhang, Global classical large solutions to a 1D fluid-particle interaction model: The bubbling regime, J. Math. Phys., 53 (2012), 033706, 21pp. doi: 10.1063/1.3693979.

[14]

L. Fang and Z. H. Guo, Global well-posedness of strong solutions to the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, Z. Angew. Math. Phys., 67 (2016), Art. 22, 27 pp. doi: 10.1007/s00033-016-0619-1.

[15]

S. J. DingB. Y. Huang and H. Y. Wen, Global well-posedness of classical solutions to a fluid-particle interaction model in $R^3$, J. Differential Equations, 263 (2017), 8666-8717.  doi: 10.1016/j.jde.2017.08.048.

[16]

X. D. HuangJ. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure. Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.

[17]

B. Y. Huang, J. R. Huang and H. Y. Wen, Low Mach number limit of the compressible Navier-Stokes-Smoluchowski equations in multi-dimensions, J. Math. Phys., 60 (2019), 061501, 20pp. doi: 10.1063/1.5089229.

[18]

B. Y. HuangS. J. Ding and H. Y. Wen, Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum, Discrete Contin. Dyn. Syst S., 9 (2016), 1717-1752.  doi: 10.3934/dcdss.2016072.

[19]

B. K. HuangL. Q. Liu and L. Zhang, On the existence of global strong solutions to 2D compressible Navier-Stokes-Smoluchowski equations with large initial data, Nonlinear Analysis: Real World Applications, 49 (2019), 169-195.  doi: 10.1016/j.nonrwa.2019.03.005.

[20]

P. Jiang, Global well-posedness and large time behavior of classical solutions to the Vlasov-Fokker-Planck and magnetohydrodynamics equations, J. Differential Equations, 262 (2017), 2961-2986.  doi: 10.1016/j.jde.2016.11.020.

[21]

Q. S. JiuY. Wang and Z. P. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum, J. Math. Fluid Mech., 16 (2014), 483-521.  doi: 10.1007/s00021-014-0171-8.

[22]

Q. S. JiuY. Wang and Z. P. Xin, Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces, J. Differ. Eqs., 255 (2013), 351-404.  doi: 10.1016/j.jde.2013.04.014.

[23] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible models Oxford University Press, Oxford, 1998. 
[24]

J. Li and Z. P. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Annals of PDE, 5 (2019), Paper No. 7, 37 pp. doi: 10.1007/s40818-019-0064-5.

[25]

J. Li and Z. L. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.

[26]

Y. K. Song, H. J. Yuan, Y. Chen and Z. D. Guo, Strong solutions for a 1D fluid-particle interaction non-newtonian model: The bubbling regime, J. Math. Phys., 54 (2013), 091501, 12pp. doi: 10.1063/1.4820446.

[27]

H. Y. Wen and L. M. Zhu, Global well-posedness and decay estimates of strong solutions to a two-phase model with magnetic field, Journal of Differential Equations, 264 (2018), 2377-2406.  doi: 10.1016/j.jde.2017.10.027.

show all references

References:
[1]

J. Ballew, Low Mach number limits to the Navier-Stokes-Smoluchowski system. Hyperbolic Problems: Theory, Numerics, Applications, AIMS Series on Applied Mathematics, 8 (2014), 301-308. 

[2]

J. Ballew, Mathematical Topics in Fluid-Particle Interaction, Ph.D thesis, University of Maryland, USA, 2014.

[3]

J. Ballew and K. Trivisa, Weakly dissipative solutions and weak strong uniqueness for the Navier Stokes Smoluchowski system, Nonlinear Analysis Series A: Theory, Methods Applications, 91 (2013), 1-19.  doi: 10.1016/j.na.2013.06.002.

[4]

J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid-particle interaction model, Commun. Partial Differ. Equ., 31 (2006), 1349-1379.  doi: 10.1080/03605300500394389.

[5]

J. A. CarrilloT. Karper and K. Trivisa, On the dynamics of a fluid-particle interaction model: The bubbling regime, Nonlinear Anal., 74 (2011), 2778-2801.  doi: 10.1016/j.na.2010.12.031.

[6]

Y. S. ChenS. J. Ding and W. J. Wang, Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Cont. Dyn. Serious A., 36 (2016), 5287-5307.  doi: 10.3934/dcds.2016032.

[7]

R. M. ChenJ. L. Hu and D. H. Wang, Global weak solutions to the magnetohydrodynamic and Vlasov equations, J. Math. Fluid Mech., 18 (2016), 343-360.  doi: 10.1007/s00021-015-0238-1.

[8]

Y. ChoH. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.  doi: 10.1016/j.matpur.2003.11.004.

[9]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Diff. Equ., 190 (2003), 504-523.  doi: 10.1016/S0022-0396(03)00015-9.

[10]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, manuscripta Math., 120 (2006), 91-129.  doi: 10.1007/s00229-006-0637-y.

[11]

S. J. DingB. Y. Huang and Q. R. Li, Global existence and decay estimates for the classical solutions to a compressible fluid-particle interaction model, Acta Mathematica Scientia, 39 (2019), 1525-1537.  doi: 10.1007/s10473-019-0605-8.

[12]

S. J. Ding, B. Y. Huang and X. L. Liu, Global classical solutions to the 2D compressible Navier-Stokes equations with vacuum, Journal of Mathematical Physics, 59 (2018), 081507, 19pp. doi: 10.1063/1.5000296.

[13]

D. Y. Fang, R. Z. Zi and T. Zhang, Global classical large solutions to a 1D fluid-particle interaction model: The bubbling regime, J. Math. Phys., 53 (2012), 033706, 21pp. doi: 10.1063/1.3693979.

[14]

L. Fang and Z. H. Guo, Global well-posedness of strong solutions to the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, Z. Angew. Math. Phys., 67 (2016), Art. 22, 27 pp. doi: 10.1007/s00033-016-0619-1.

[15]

S. J. DingB. Y. Huang and H. Y. Wen, Global well-posedness of classical solutions to a fluid-particle interaction model in $R^3$, J. Differential Equations, 263 (2017), 8666-8717.  doi: 10.1016/j.jde.2017.08.048.

[16]

X. D. HuangJ. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure. Appl. Math., 65 (2012), 549-585.  doi: 10.1002/cpa.21382.

[17]

B. Y. Huang, J. R. Huang and H. Y. Wen, Low Mach number limit of the compressible Navier-Stokes-Smoluchowski equations in multi-dimensions, J. Math. Phys., 60 (2019), 061501, 20pp. doi: 10.1063/1.5089229.

[18]

B. Y. HuangS. J. Ding and H. Y. Wen, Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum, Discrete Contin. Dyn. Syst S., 9 (2016), 1717-1752.  doi: 10.3934/dcdss.2016072.

[19]

B. K. HuangL. Q. Liu and L. Zhang, On the existence of global strong solutions to 2D compressible Navier-Stokes-Smoluchowski equations with large initial data, Nonlinear Analysis: Real World Applications, 49 (2019), 169-195.  doi: 10.1016/j.nonrwa.2019.03.005.

[20]

P. Jiang, Global well-posedness and large time behavior of classical solutions to the Vlasov-Fokker-Planck and magnetohydrodynamics equations, J. Differential Equations, 262 (2017), 2961-2986.  doi: 10.1016/j.jde.2016.11.020.

[21]

Q. S. JiuY. Wang and Z. P. Xin, Global well-posedness of 2D compressible Navier-Stokes equations with large data and vacuum, J. Math. Fluid Mech., 16 (2014), 483-521.  doi: 10.1007/s00021-014-0171-8.

[22]

Q. S. JiuY. Wang and Z. P. Xin, Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces, J. Differ. Eqs., 255 (2013), 351-404.  doi: 10.1016/j.jde.2013.04.014.

[23] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible models Oxford University Press, Oxford, 1998. 
[24]

J. Li and Z. P. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Annals of PDE, 5 (2019), Paper No. 7, 37 pp. doi: 10.1007/s40818-019-0064-5.

[25]

J. Li and Z. L. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl., 102 (2014), 640-671.  doi: 10.1016/j.matpur.2014.02.001.

[26]

Y. K. Song, H. J. Yuan, Y. Chen and Z. D. Guo, Strong solutions for a 1D fluid-particle interaction non-newtonian model: The bubbling regime, J. Math. Phys., 54 (2013), 091501, 12pp. doi: 10.1063/1.4820446.

[27]

H. Y. Wen and L. M. Zhu, Global well-posedness and decay estimates of strong solutions to a two-phase model with magnetic field, Journal of Differential Equations, 264 (2018), 2377-2406.  doi: 10.1016/j.jde.2017.10.027.

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