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

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.
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 and 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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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