# American Institute of Mathematical Sciences

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.

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##### 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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