Solvability of generalized nonlinear heat equations  with constraints  coupled with  Navier--Stokes equations in 2D domains

Yutaka Tsuzuki

doi:10.3934/proc.2015.1079

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2015, Volume 2015: 1079-1088. Doi: 10.3934/proc.2015.1079 open access

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Solvability of generalized nonlinear heat 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

Received: July 2014

Revised: February 2015

Published: October 2015

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Abstract

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 $x$.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 47H05.

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References

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