American Institute of Mathematical Sciences

Advanced
October  2014, 7(5): 917-923. doi: 10.3934/dcdss.2014.7.917

Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model

Xiaoyun Cai 1, , Liangwen Liao 1, and  Yongzhong Sun 2,

1. 

Department of Mathematics, Nanjing University, Nanjing, Jiangsu, 210093, China, China
2. 

Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093

Received  February 2013 Revised  June 2013 Published  May 2014

We establish the global-in-time existence of strong solution to the initial-boundary value problem of a 2-D Kazhikov-Smagulov type model for incompressible nonhomogeneous fluids with mass diffusion for arbitrary size of initial data.
Keywords: well-posedness, mass diffusion, global regularity, initial-boundary value problem., Kazhikhov-Smagulov model.
Mathematics Subject Classification: 35Q30, 35D1.
Citation: Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917
References:
[1]

S. Antonsev, A. Kazhikhov and V. Monakov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Translated from the Russian, Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990.  Google Scholar

[2]

H. Beirao da Veiga, Diffusion on viscous fluids: Existence and asymptotic properties of solutions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 10 (1983), 341-355.  Google Scholar

[3]

D. Bresch, El. H. Essoufi and M. Sy, Effect of density dependent viscosities on multiphasic incompressible fluid models, J. Math. Fluid Mech., 9 (2007), 377-397. doi: 10.1007/s00021-005-0204-4.  Google Scholar

[4]

X. Cai, L. Liao and Y. Sun, Global regularity for the initial value problem of a 2-D Kazhikhov-Smagulov type model, Nonlinear Anal., 75 (2012), 5975-5983. doi: 10.1016/j.na.2012.06.011.  Google Scholar

[5]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.  Google Scholar

[6]

P. Embid, Well-posedness of the nonlinear equations for zero Mach number combustion, Comm. PDE, 12 (1987), 1227-1283. doi: 10.1080/03605308708820526.  Google Scholar

[7]

P. Embid, On the reactive and nondiffusive equations for zero Mach number flow, Comm. PDE, 14 (1989), 1249-1281. doi: 10.1080/03605308908820652.  Google Scholar

[8]

A. Kazhikhov and Sh. Smagulov, The correctness of boundary value problems in a diffusion model of an inhomogeneous fluid, Sov. Phys. Dokl., 22 (1977), 249-252. Google Scholar

[9]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tceva, Linear and Quasi-Linear Parabolic Equations, Amer. Math. Soc., Providence, RI, 1968. Google Scholar

[10]

P.-L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[11]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Dimensions, Appl. Math. Sci., 53, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[12]

P. Secchi, On the initial value problem for the equations of motion of viscous incompressible fluids in the presence of diffusion, Boll. Un. Mat. Ital., B (6), 1 (1982), 1117-1130.  Google Scholar

[13]

P. Secchi, On the motion of viscous fluids in the presence of diffusion, SIAM J. Math. Anal., 19 (1988), 22-31. doi: 10.1137/0519002.  Google Scholar

[14]

V. A. Solonnikov, $L^p$-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci., (New York), 105 (2001), 2448-2484. doi: 10.1023/A:1011321430954.  Google Scholar

[15]

Y. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Diff. Equa., 225 (2013), 1069-1085. doi: 10.1016/j.jde.2013.04.032.  Google Scholar

[16]

C. Wang and Z. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. in Math., 228 (2011), 43-62. doi: 10.1016/j.aim.2011.05.008.  Google Scholar

show all references

References:
[1]

S. Antonsev, A. Kazhikhov and V. Monakov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Translated from the Russian, Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990.  Google Scholar

[2]

H. Beirao da Veiga, Diffusion on viscous fluids: Existence and asymptotic properties of solutions, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4), 10 (1983), 341-355.  Google Scholar

[3]

D. Bresch, El. H. Essoufi and M. Sy, Effect of density dependent viscosities on multiphasic incompressible fluid models, J. Math. Fluid Mech., 9 (2007), 377-397. doi: 10.1007/s00021-005-0204-4.  Google Scholar

[4]

X. Cai, L. Liao and Y. Sun, Global regularity for the initial value problem of a 2-D Kazhikhov-Smagulov type model, Nonlinear Anal., 75 (2012), 5975-5983. doi: 10.1016/j.na.2012.06.011.  Google Scholar

[5]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988.  Google Scholar

[6]

P. Embid, Well-posedness of the nonlinear equations for zero Mach number combustion, Comm. PDE, 12 (1987), 1227-1283. doi: 10.1080/03605308708820526.  Google Scholar

[7]

P. Embid, On the reactive and nondiffusive equations for zero Mach number flow, Comm. PDE, 14 (1989), 1249-1281. doi: 10.1080/03605308908820652.  Google Scholar

[8]

A. Kazhikhov and Sh. Smagulov, The correctness of boundary value problems in a diffusion model of an inhomogeneous fluid, Sov. Phys. Dokl., 22 (1977), 249-252. Google Scholar

[9]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tceva, Linear and Quasi-Linear Parabolic Equations, Amer. Math. Soc., Providence, RI, 1968. Google Scholar

[10]

P.-L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[11]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Dimensions, Appl. Math. Sci., 53, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[12]

P. Secchi, On the initial value problem for the equations of motion of viscous incompressible fluids in the presence of diffusion, Boll. Un. Mat. Ital., B (6), 1 (1982), 1117-1130.  Google Scholar

[13]

P. Secchi, On the motion of viscous fluids in the presence of diffusion, SIAM J. Math. Anal., 19 (1988), 22-31. doi: 10.1137/0519002.  Google Scholar

[14]

V. A. Solonnikov, $L^p$-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci., (New York), 105 (2001), 2448-2484. doi: 10.1023/A:1011321430954.  Google Scholar

[15]

Y. Sun and Z. Zhang, Global regularity for the initial-boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Diff. Equa., 225 (2013), 1069-1085. doi: 10.1016/j.jde.2013.04.032.  Google Scholar

[16]

C. Wang and Z. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. in Math., 228 (2011), 43-62. doi: 10.1016/j.aim.2011.05.008.  Google Scholar

[1]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021205
[2]

Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015
[3]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[4]

Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212
[5]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[6]

Haifeng Hu, Kaijun Zhang. Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1601-1626. doi: 10.3934/dcdsb.2014.19.1601
[7]

Elena Rossi. Well-posedness of general 1D initial boundary value problems for scalar balance laws. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3577-3608. doi: 10.3934/dcds.2019147
[8]

Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917
[9]

Xiaoqiang Dai, Shaohua Chen. Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021114
[10]

Iñigo U. Erneta. Well-posedness for boundary value problems for coagulation-fragmentation equations. Kinetic & Related Models, 2020, 13 (4) : 815-835. doi: 10.3934/krm.2020028
[11]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[12]

Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292
[13]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381
[14]

Martn P. Árciga Alejandre, Elena I. Kaikina. Mixed initial-boundary value problem for Ott-Sudan-Ostrovskiy equation. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 381-409. doi: 10.3934/dcds.2012.32.381
[15]

Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148
[16]

Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135
[17]

Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297
[18]

Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087
[19]

Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737
[20]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences