# American Institute of Mathematical Sciences

## 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  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021042
##### References:
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Equ., 190 (2003), 504-523.  doi: 10.1016/S0022-0396(03)00015-9.  Google Scholar [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.  Google Scholar [11] S. J. Ding, B. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] S. J. Ding, B. 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.  Google Scholar [16] X. D. Huang, J. 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.  Google Scholar [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.  Google Scholar [18] B. Y. Huang, S. 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.  Google Scholar [19] B. K. Huang, L. 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.  Google Scholar [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.  Google Scholar [21] Q. S. Jiu, Y. 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.  Google Scholar [22] Q. S. Jiu, Y. 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.  Google Scholar [23] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible models Oxford University Press, Oxford, 1998.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.   Google Scholar [2] J. Ballew, Mathematical Topics in Fluid-Particle Interaction, Ph.D thesis, University of Maryland, USA, 2014.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] J. A. Carrillo, T. 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.  Google Scholar [6] Y. S. Chen, S. 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.  Google Scholar [7] R. M. Chen, J. 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.  Google Scholar [8] Y. Cho, H. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] S. J. Ding, B. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] S. J. Ding, B. 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.  Google Scholar [16] X. D. Huang, J. 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.  Google Scholar [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.  Google Scholar [18] B. Y. Huang, S. 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.  Google Scholar [19] B. K. Huang, L. 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.  Google Scholar [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.  Google Scholar [21] Q. S. Jiu, Y. 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.  Google Scholar [22] Q. S. Jiu, Y. 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.  Google Scholar [23] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible models Oxford University Press, Oxford, 1998.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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