# 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 & 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, (2010).  doi: 10.1007/978-1-4419-5542-5.  Google Scholar [2] J. L. Boldrini and S. A. Lorca, The initial value problem for a generalized Boussinesq model,, Nonlinear Anal., 36 (1999), 457.  doi: 10.1016/S0362-546X(97)00635-4.  Google Scholar [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.   Google Scholar [4] T. Fukao and N. Kenmochi, Stefan problems with convection governed by Navier-Stokes equations,, Adv. Math. Sci. Appl., 15 (2005), 29.   Google Scholar [5] T. Fukao and M. Kubo, Nonlinear degenerate parabolic equations for a thermohydraulic model,, in Discrete Contin. Dyn. Syst., (2007), 399.   Google Scholar [6] T. Fukao and M. Kubo, Time-dependent double obstacle problem in thermohydraulics,, in Nonlinear phenomena with energy dissipation, 29 (2008), 73.   Google Scholar [7] M. Kubo, Weak solutions of a thermohydraulics model with a general nonlinear heat flux,, in Mathematical approach to nonlinear phenomena: Modelling, 23 (2005), 163.   Google Scholar [8] H. Morimoto, Nonstationary Boussinesq equations,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 39 (1992), 61.   Google Scholar [9] N. Okazawa, An application of the perturbation theorem for $m$-accretive operators, II,, Proc. Japan Acad. Ser. A Math. Sci., 60 (1984), 1.  doi: 10.3792/pjaa.60.10.  Google Scholar [10] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar [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.   Google Scholar [12] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Amsterdam-New York, (1977).   Google Scholar

show all references

##### References:
 [1] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces,, Springer, (2010).  doi: 10.1007/978-1-4419-5542-5.  Google Scholar [2] J. L. Boldrini and S. A. Lorca, The initial value problem for a generalized Boussinesq model,, Nonlinear Anal., 36 (1999), 457.  doi: 10.1016/S0362-546X(97)00635-4.  Google Scholar [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.   Google Scholar [4] T. Fukao and N. Kenmochi, Stefan problems with convection governed by Navier-Stokes equations,, Adv. Math. Sci. Appl., 15 (2005), 29.   Google Scholar [5] T. Fukao and M. Kubo, Nonlinear degenerate parabolic equations for a thermohydraulic model,, in Discrete Contin. Dyn. Syst., (2007), 399.   Google Scholar [6] T. Fukao and M. Kubo, Time-dependent double obstacle problem in thermohydraulics,, in Nonlinear phenomena with energy dissipation, 29 (2008), 73.   Google Scholar [7] M. Kubo, Weak solutions of a thermohydraulics model with a general nonlinear heat flux,, in Mathematical approach to nonlinear phenomena: Modelling, 23 (2005), 163.   Google Scholar [8] H. Morimoto, Nonstationary Boussinesq equations,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 39 (1992), 61.   Google Scholar [9] N. Okazawa, An application of the perturbation theorem for $m$-accretive operators, II,, Proc. Japan Acad. Ser. A Math. Sci., 60 (1984), 1.  doi: 10.3792/pjaa.60.10.  Google Scholar [10] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar [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.   Google Scholar [12] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Amsterdam-New York, (1977).   Google Scholar
 [1] Yutaka Tsuzuki. Solvability of generalized nonlinear heat equations with constraints coupled with Navier--Stokes equations in 2D domains. Conference Publications, 2015, 2015 (special) : 1079-1088. doi: 10.3934/proc.2015.1079 [2] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [3] Julia García-Luengo, Pedro Marín-Rubio, José Real. Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1603-1621. doi: 10.3934/cpaa.2015.14.1603 [4] J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 [5] Songsong Lu, Hongqing Wu, Chengkui Zhong. Attractors for nonautonomous 2d Navier-Stokes equations with normal external forces. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 701-719. doi: 10.3934/dcds.2005.13.701 [6] Hakima Bessaih, Benedetta Ferrario. Statistical properties of stochastic 2D Navier-Stokes equations from linear models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2927-2947. doi: 10.3934/dcdsb.2016080 [7] Ruihong Ji, Yongfu Wang. Mass concentration phenomenon to the 2D Cauchy problem of the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1117-1133. doi: 10.3934/dcds.2019047 [8] Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3039-3057. doi: 10.3934/dcds.2015.35.3039 [9] Julia García-Luengo, Pedro Marín-Rubio, José Real. Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 181-201. doi: 10.3934/dcds.2014.34.181 [10] Adam Larios, Yuan Pei. Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020031 [11] Igor Kukavica. Interior gradient bounds for the 2D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 873-882. doi: 10.3934/dcds.2001.7.873 [12] Zilai Li, Zhenhua Guo. On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3903-3919. doi: 10.3934/dcdsb.2017201 [13] Julia García-Luengo, Pedro Marín-Rubio, José Real, James C. Robinson. Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203 [14] Julia García-Luengo, Pedro Marín-Rubio. Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2127-2146. doi: 10.3934/cpaa.2020094 [15] Henri Schurz. Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D. Conference Publications, 2013, 2013 (special) : 673-684. doi: 10.3934/proc.2013.2013.673 [16] Guangrong Wu, Ping Zhang. The zero diffusion limit of 2-D Navier-Stokes equations with $L^1$ initial vorticity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 631-638. doi: 10.3934/dcds.1999.5.631 [17] Grzegorz Łukaszewicz. Pullback attractors and statistical solutions for 2-D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 643-659. doi: 10.3934/dcdsb.2008.9.643 [18] Pedro Marín-Rubio, José Real. Pullback attractors for 2D-Navier-Stokes equations with delays in continuous and sub-linear operators. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 989-1006. doi: 10.3934/dcds.2010.26.989 [19] Shuguang Shao, Shu Wang, Wen-Qing Xu, Bin Han. Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle. Kinetic & Related Models, 2016, 9 (4) : 767-776. doi: 10.3934/krm.2016015 [20] Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004

2018 Impact Factor: 1.048