February  2020, 19(2): 835-882. doi: 10.3934/cpaa.2020039

A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

3. 

School of Mathematics, Hefei University of Technology, Hefei 230009, China

4. 

Wuhan Institute of Physics and Mathematicss, Chinese Academy of Science, Wuhan 430071, China

5. 

Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China

* Corresponding author

Received  December 2018 Revised  June 2019 Published  October 2019

Fund Project: The work of S.-Y. Ha is supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03, the work of Qinghua Xiao was supported by grants from Youth Innovation Promotion Association and the National Natural Science Foundation of China #11871469, #11501556, and the work of Xiongtao Zhang was supported by grant the National Natural Science Foundation of China #11801194.

We present a two-dimensional coupled system for flocking particle-compressible fluid interactions, and study its global solvability for the proposed coupled system. For particle and fluid dynamics, we employ the kinetic Cucker-Smale-Fokker-Planck (CS-FP) model for flocking particle part, and the isentropic compressible Navier-Stokes (N-S) equations for the fluid part, respectively, and these separate systems are coupled through the drag force. For the global solvability of the coupled system, we present a sufficient framework for the global existence of classical solutions with large initial data which can contain vacuum using the weighted energy method. We extend an earlier global solvability result [20] in the one-dimensional setting to the two-dimensional setting.

Citation: Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao, Xiongtao Zhang. A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data. Communications on Pure and Applied Analysis, 2020, 19 (2) : 835-882. doi: 10.3934/cpaa.2020039
References:
[1]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Physics, 51 (2010), 103301. doi: 10.1063/1.3496895.

[2]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 24 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.

[3]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Differential Equations, 257 (2014), 2225-2255.  doi: 10.1016/j.jde.2014.05.035.

[4]

C. BarangerL. BoudinP.-E Jabin and S. Mancini, A modeling of biospray for the upper airways, CEMRACS 2004 Mathematics and applications to biology and medicine, ESAIM Proc., 14 (2005), 41-47. 

[5]

S. BerresR. BurgerK. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163.

[6]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.

[7]

L. Boudin, L. Desvillettes, C. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier-Stokes equations, Differ. Int. Equations, 22 (2009), 1247–1271.

[8]

L. Boudin, L. Desvillettes and R. Motte, A modelling of compressible droplets in a fluid, Commun. Math. Sci., 1 (2003), 657–669.

[9]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259–275.

[10]

J. A. CarrilloR.-J. Duan and A. Moussa, Global classical solutions close to equilibrium to the Vlasov-Euler-Fokker-Planck system, Kinetic and Related Models, 4 (2011), 227-258.  doi: 10.3934/krm.2011.4.227.

[11]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and non-existence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[12]

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.

[13]

Y.-P. Choi, S.-Y. Ha and Z.-C. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Active Particles Vol.I: Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkh$\ddot{a}$user Basel, (2017), 299–331.

[14]

R. CoifmanR. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.  doi: 10.2307/1970954.

[15]

R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.  doi: 10.2307/1998628.

[16]

R.-J. Duan and S.-Q. Liu, Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force, Kinet. Relat. Models, 6 (2013), 687-700.  doi: 10.3934/krm.2013.6.687.

[17]

H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541–544.

[18]

E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford University Press, Oxford, 2004.

[19]

K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74.  doi: 10.1007/BF03167396.

[20]

S.-Y. HaB.-K. HuangQ.-H. Xiao and X.-T. Zhang, Global classical solutions to 1D coupled system of flocking particles and compressible fluids with large initial data, Math. Models Methods Appl. Sci., 28 (2018), 1-60.  doi: 10.1142/S021820251850001X.

[21]

S.-Y. Ha, J. Jeong, S.-E. Noh, Q.-H. Xiao and X.-T. Zhang, Emergent of dynamic of infinity many Cucker-Smale particles in a random environment, J. Differential Equations, 262 (2017), 2554–2591. doi: 10.1016/j.jde.2016.11.017.

[22]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297–325.

[23]

X.-D. Huang and J. Li, Existence and blowup behavior of global strong solutions tothe two-dimensional barotrpic compressible Navier-Stokes system with vacuum and large initial data, J. Math. Pures Appl., 106 (2016), 123–154. doi: 10.1016/j.matpur.2016.02.003.

[24]

T. GoudonL. HeA. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal., 42 (2010), 2177-2202.  doi: 10.1137/090776755.

[25]

D. Gilbarg and N. Tridinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.

[26]

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.

[27]

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. Differential Equations, 255 (2013), 351-404.  doi: 10.1016/j.jde.2013.04.014.

[28]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible Models, Oxford University Press, New York, 1996.

[29]

Y. Mei, Global classical solutions to the 2D compressible MHD equations with large data and vacuum, J. Differential Equations, 258 (2015), 3304-3359.  doi: 10.1016/j.jde.2014.11.023.

[30]

Y. Mei, Corrigendum to "Global classical solutions to the 2D compressible MHD equations with large data and vacuum", J. Differential Equations, 258 (2015), 3360-3362.  doi: 10.1016/j.jde.2015.02.001.

[31]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Comm. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4.

[32]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063.  doi: 10.1142/S0218202507002194.

[33]

A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Ser. Math. Appl., vol. 27 (2004) Oxford Univ. Press, Oxford.

[34]

J. Simon, Compact sets in the space $L_p(0, t;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[35]

E. M. Stein, Harmonic Analysis, Princeton University Press, 1995.

[36]

D.-H. Wang and C. Yu, Global weak solutions to the inhomogeneous Navier-Stokes-Vlasov equations, J. Differ. Equations, 259 (2015), 3976-4008.  doi: 10.1016/j.jde.2015.05.016.

[37]

F. A. Williams, Combustion Theory, Benjamin Cummings, 1985.

[38]

V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316.  doi: 10.1007/BF02106835.

[39]

C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl., 100 (2013), 275-293.  doi: 10.1016/j.matpur.2013.01.001.

show all references

References:
[1]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Physics, 51 (2010), 103301. doi: 10.1063/1.3496895.

[2]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Time-asymptotic interaction of flocking particles and an incompressible viscous fluid, Nonlinearity, 24 (2012), 1155-1177.  doi: 10.1088/0951-7715/25/4/1155.

[3]

H.-O. BaeY.-P. ChoiS.-Y. Ha and M.-J. Kang, Global existence of strong solution for the Cucker-Smale-Navier-Stokes system, J. Differential Equations, 257 (2014), 2225-2255.  doi: 10.1016/j.jde.2014.05.035.

[4]

C. BarangerL. BoudinP.-E Jabin and S. Mancini, A modeling of biospray for the upper airways, CEMRACS 2004 Mathematics and applications to biology and medicine, ESAIM Proc., 14 (2005), 41-47. 

[5]

S. BerresR. BurgerK. H. Karlsen and E. M. Tory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 64 (2003), 41-80.  doi: 10.1137/S0036139902408163.

[6]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.

[7]

L. Boudin, L. Desvillettes, C. Grandmont and A. Moussa, Global existence of solution for the coupled Vlasov and Navier-Stokes equations, Differ. Int. Equations, 22 (2009), 1247–1271.

[8]

L. Boudin, L. Desvillettes and R. Motte, A modelling of compressible droplets in a fluid, Commun. Math. Sci., 1 (2003), 657–669.

[9]

L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequality with weights, Compos. Math., 53 (1984), 259–275.

[10]

J. A. CarrilloR.-J. Duan and A. Moussa, Global classical solutions close to equilibrium to the Vlasov-Euler-Fokker-Planck system, Kinetic and Related Models, 4 (2011), 227-258.  doi: 10.3934/krm.2011.4.227.

[11]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and non-existence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.

[12]

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.

[13]

Y.-P. Choi, S.-Y. Ha and Z.-C. Li, Emergent dynamics of the Cucker-Smale flocking model and its variants, Active Particles Vol.I: Advances in Theory, Models, Applications (eds. N. Bellomo, P. Degond, and E. Tadmor), Birkh$\ddot{a}$user Basel, (2017), 299–331.

[14]

R. CoifmanR. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635.  doi: 10.2307/1970954.

[15]

R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.  doi: 10.2307/1998628.

[16]

R.-J. Duan and S.-Q. Liu, Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force, Kinet. Relat. Models, 6 (2013), 687-700.  doi: 10.3934/krm.2013.6.687.

[17]

H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541–544.

[18]

E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford University Press, Oxford, 2004.

[19]

K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74.  doi: 10.1007/BF03167396.

[20]

S.-Y. HaB.-K. HuangQ.-H. Xiao and X.-T. Zhang, Global classical solutions to 1D coupled system of flocking particles and compressible fluids with large initial data, Math. Models Methods Appl. Sci., 28 (2018), 1-60.  doi: 10.1142/S021820251850001X.

[21]

S.-Y. Ha, J. Jeong, S.-E. Noh, Q.-H. Xiao and X.-T. Zhang, Emergent of dynamic of infinity many Cucker-Smale particles in a random environment, J. Differential Equations, 262 (2017), 2554–2591. doi: 10.1016/j.jde.2016.11.017.

[22]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297–325.

[23]

X.-D. Huang and J. Li, Existence and blowup behavior of global strong solutions tothe two-dimensional barotrpic compressible Navier-Stokes system with vacuum and large initial data, J. Math. Pures Appl., 106 (2016), 123–154. doi: 10.1016/j.matpur.2016.02.003.

[24]

T. GoudonL. HeA. Moussa and P. Zhang, The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium, SIAM J. Math. Anal., 42 (2010), 2177-2202.  doi: 10.1137/090776755.

[25]

D. Gilbarg and N. Tridinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.

[26]

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.

[27]

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. Differential Equations, 255 (2013), 351-404.  doi: 10.1016/j.jde.2013.04.014.

[28]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible Models, Oxford University Press, New York, 1996.

[29]

Y. Mei, Global classical solutions to the 2D compressible MHD equations with large data and vacuum, J. Differential Equations, 258 (2015), 3304-3359.  doi: 10.1016/j.jde.2014.11.023.

[30]

Y. Mei, Corrigendum to "Global classical solutions to the 2D compressible MHD equations with large data and vacuum", J. Differential Equations, 258 (2015), 3360-3362.  doi: 10.1016/j.jde.2015.02.001.

[31]

A. Mellet and A. Vasseur, Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations, Comm. Math. Phys., 281 (2008), 573-596.  doi: 10.1007/s00220-008-0523-4.

[32]

A. Mellet and A. Vasseur, Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations, Math. Models Methods Appl. Sci., 17 (2007), 1039-1063.  doi: 10.1142/S0218202507002194.

[33]

A. Novotny and I. Straskraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Ser. Math. Appl., vol. 27 (2004) Oxford Univ. Press, Oxford.

[34]

J. Simon, Compact sets in the space $L_p(0, t;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[35]

E. M. Stein, Harmonic Analysis, Princeton University Press, 1995.

[36]

D.-H. Wang and C. Yu, Global weak solutions to the inhomogeneous Navier-Stokes-Vlasov equations, J. Differ. Equations, 259 (2015), 3976-4008.  doi: 10.1016/j.jde.2015.05.016.

[37]

F. A. Williams, Combustion Theory, Benjamin Cummings, 1985.

[38]

V. A. Vaigant and A. V. Kazhikhov, On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., 36 (1995), 1283-1316.  doi: 10.1007/BF02106835.

[39]

C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl., 100 (2013), 275-293.  doi: 10.1016/j.matpur.2013.01.001.

[1]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

[2]

Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic and Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016

[3]

Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052

[4]

Peter Constantin, Gregory Seregin. Global regularity of solutions of coupled Navier-Stokes equations and nonlinear Fokker Planck equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1185-1196. doi: 10.3934/dcds.2010.26.1185

[5]

Bingyuan Huang, Shijin Ding, Huanyao Wen. Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1717-1752. doi: 10.3934/dcdss.2016072

[6]

Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic and Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041

[7]

Xiaoping Zhai, Zhaoyang Yin. Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2829-2859. doi: 10.3934/dcds.2017122

[8]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371

[9]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure and Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[10]

Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263

[11]

Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234

[12]

Fei Jiang, Song Jiang, Junpin Yin. Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 567-587. doi: 10.3934/dcds.2014.34.567

[13]

Yaobin Ou, Pan Shi. Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 537-567. doi: 10.3934/dcdsb.2017026

[14]

Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure and Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353

[15]

Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67

[16]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[17]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041

[18]

Ping Chen, Ting Zhang. A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients. Communications on Pure and Applied Analysis, 2008, 7 (4) : 987-1016. doi: 10.3934/cpaa.2008.7.987

[19]

Ben Duan, Zhen Luo. Dynamics of vacuum states for one-dimensional full compressible Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2543-2564. doi: 10.3934/cpaa.2013.12.2543

[20]

Yuming Qin, Lan Huang, Shuxian Deng, Zhiyong Ma, Xiaoke Su, Xinguang Yang. Interior regularity of the compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 163-192. doi: 10.3934/dcdss.2009.2.163

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (323)
  • HTML views (90)
  • Cited by (0)

[Back to Top]