American Institute of Mathematical Sciences

Advanced
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

Yutaka Tsuzuki 1,

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 $p>2$.
Keywords: subdifferential operators., Nonlinear heat equations, time-dependent constraints, $p$-Laplacian, 2D Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 47H0.
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]

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
[15]

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
[16]

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
[17]

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
[18]

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
[19]

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
[20]

Zoran Grujić. Regularity of forward-in-time self-similar solutions to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 837-843. doi: 10.3934/dcds.2006.14.837

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences