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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, (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., 10 (1983), 341.   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.  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.  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, (1988).   Google Scholar

[6]

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

[9]

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

[10]

P.-L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models,, Oxford Lecture Series in Mathematics and its Applications, (1996).   Google Scholar

[11]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Dimensions,, Appl. Math. Sci., (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., 1 (1982), 1117.   Google Scholar

[13]

P. Secchi, On the motion of viscous fluids in the presence of diffusion,, SIAM J. Math. Anal., 19 (1988), 22.  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., 105 (2001), 2448.  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.  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.  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, (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., 10 (1983), 341.   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.  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.  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, (1988).   Google Scholar

[6]

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

[9]

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

[10]

P.-L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models,, Oxford Lecture Series in Mathematics and its Applications, (1996).   Google Scholar

[11]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Dimensions,, Appl. Math. Sci., (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., 1 (1982), 1117.   Google Scholar

[13]

P. Secchi, On the motion of viscous fluids in the presence of diffusion,, SIAM J. Math. Anal., 19 (1988), 22.  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., 105 (2001), 2448.  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.  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.  doi: 10.1016/j.aim.2011.05.008.  Google Scholar

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