# American Institute of Mathematical Sciences

March  2014, 3(1): 191-206. doi: 10.3934/eect.2014.3.191

## Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains

 1 Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

Received  April 2013 Revised  August 2013 Published  February 2014

This paper is concerned with the system of nonlinear heat equations with constraints coupled with Navier-Stokes equations in two-dimensional domains. In 2012, Sobajima, Tsuzuki and Yokota proved the existence and uniqueness of solutions to the system with heat equations including the diffusion term $\Delta\theta$, where $\theta$ represents the temperature. This paper gives the existence result in which the Laplace operator $\Delta$ is replaced with the $p$-Laplace operator $\Delta\rho$, where $p>2$.
Citation: Yutaka Tsuzuki. Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains. Evolution Equations and Control Theory, 2014, 3 (1) : 191-206. doi: 10.3934/eect.2014.3.191
##### References:
 [1] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. [2] J. L. Boldrini and S. A. Lorca, The initial value problem for a generalized Boussinesq model, Nonlinear Anal., 36 (1999), 457-480. doi: 10.1016/S0362-546X(97)00635-4. [3] J. I. Díaz and G. Galiano, Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal., 11 (1998), 59-82. [4] T. Fukao and N. Kenmochi, Stefan problems with convection governed by Navier-Stokes equations, Adv. Math. Sci. Appl., 15 (2005), 29-48. [5] T. Fukao and M. Kubo, Nonlinear degenerate parabolic equations for a thermohydraulic model, in Discrete Contin. Dyn. Syst., Am. Inst. Math. Sci., (2007), 399-408. [6] T. Fukao and M. Kubo, Time-dependent double obstacle problem in thermohydraulics, in Nonlinear phenomena with energy dissipation, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 29 (2008), 73-92. [7] M. Kubo, Weak solutions of a thermohydraulics model with a general nonlinear heat flux, in Mathematical approach to nonlinear phenomena: Modelling, analysis and simulations, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 23 (2005), 163-178. [8] H. Morimoto, Nonstationary Boussinesq equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 39 (1992), 61-75. [9] N. Okazawa, An application of the perturbation theorem for $m$-accretive operators, II, Proc. Japan Acad. Ser. A Math. Sci., 60 (1984), 1-42. doi: 10.3792/pjaa.60.10. [10] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [11] M. Sobajima, Y. Tsuzuki and T. Yokota, Existence and uniqueness of solutions to nonlinear heat equations with constraints coupled with Navier-Stokes equations in 2D domains, Adv. Math. Sci. Appl., 22 (2012), 577-596. [12] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Amsterdam-New York, North-Holland, 1977.

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##### References:
 [1] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. [2] J. L. Boldrini and S. A. Lorca, The initial value problem for a generalized Boussinesq model, Nonlinear Anal., 36 (1999), 457-480. doi: 10.1016/S0362-546X(97)00635-4. [3] J. I. Díaz and G. Galiano, Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal., 11 (1998), 59-82. [4] T. Fukao and N. Kenmochi, Stefan problems with convection governed by Navier-Stokes equations, Adv. Math. Sci. Appl., 15 (2005), 29-48. [5] T. Fukao and M. Kubo, Nonlinear degenerate parabolic equations for a thermohydraulic model, in Discrete Contin. Dyn. Syst., Am. Inst. Math. Sci., (2007), 399-408. [6] T. Fukao and M. Kubo, Time-dependent double obstacle problem in thermohydraulics, in Nonlinear phenomena with energy dissipation, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 29 (2008), 73-92. [7] M. Kubo, Weak solutions of a thermohydraulics model with a general nonlinear heat flux, in Mathematical approach to nonlinear phenomena: Modelling, analysis and simulations, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 23 (2005), 163-178. [8] H. Morimoto, Nonstationary Boussinesq equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 39 (1992), 61-75. [9] N. Okazawa, An application of the perturbation theorem for $m$-accretive operators, II, Proc. Japan Acad. Ser. A Math. Sci., 60 (1984), 1-42. doi: 10.3792/pjaa.60.10. [10] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [11] M. Sobajima, Y. Tsuzuki and T. Yokota, Existence and uniqueness of solutions to nonlinear heat equations with constraints coupled with Navier-Stokes equations in 2D domains, Adv. Math. Sci. Appl., 22 (2012), 577-596. [12] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Amsterdam-New York, North-Holland, 1977.
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