# American Institute of Mathematical Sciences

2015, 2015(special): 1079-1088. doi: 10.3934/proc.2015.1079

## Solvability of generalized nonlinear heat 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

Received  July 2014 Revised  February 2015 Published  November 2015

This paper is concerned with a system of nonlinear heat equations with constraints coupled with Navier--Stokes equations in two-dimensional domains. In 2012, Sobajima, the author and Yokota proved existence and uniqueness of solutions to the system with heat equations with the linear diffusion term $\Delta\theta$ and the nonlinear term $|\theta|^{q-1}\theta$. Recently, the author generalized the result for the equation with the $p$-Laplace operator $\Delta p$ and the logistic nonlinear term $|\theta|^{q-1}\theta - \alpha\theta$. This paper gives an existence result for the equation with $\Delta p$ and the more general nonlinear term $h(x,\theta)-\alpha\theta$ depending on the spacial variable $x$.
Citation: 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
##### References:
 [1] T. Fukao and M. Kubo, Time-dependent double obstacle problem in thermohydraulics,, in Nonlinear phenomena with energy dissipation, Vol.29 (2008), 73.   Google Scholar [2] N. Okazawa, An application of the perturbation theorem for $m$-accretive operators. II,, Proc. Japan Acad. Ser. A Math. Sci., 60 (1984), 10.   Google Scholar [3] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65.   Google Scholar [4] 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 [5] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Amsterdam-New York, (1977).   Google Scholar [6] Y. Tsuzuki, Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains,, Evol. Equ. Control Theory, 3 (2014), 191.   Google Scholar

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##### References:
 [1] T. Fukao and M. Kubo, Time-dependent double obstacle problem in thermohydraulics,, in Nonlinear phenomena with energy dissipation, Vol.29 (2008), 73.   Google Scholar [2] N. Okazawa, An application of the perturbation theorem for $m$-accretive operators. II,, Proc. Japan Acad. Ser. A Math. Sci., 60 (1984), 10.   Google Scholar [3] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65.   Google Scholar [4] 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 [5] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Amsterdam-New York, (1977).   Google Scholar [6] Y. Tsuzuki, Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains,, Evol. Equ. Control Theory, 3 (2014), 191.   Google Scholar
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