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

Xiaoyun Cai; Liangwen Liao; Yongzhong Sun

doi:10.3934/dcdss.2014.7.917

Article Contents

2014, Volume 7, Issue 5: 917-923. Doi: 10.3934/dcdss.2014.7.917

This issue Previous Article Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces Next Article Flow-plate interactions: Well-posedness and long-time behavior

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

Received: January 31, 2013

Revised: May 31, 2013

Published: September 2014

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: 35Q30, 35D10.

Citation:

Related Papers

Cited by

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.