\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 37, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/37\hfil Reproductive solutions] {Reproductive solutions for the g-Navier-Stokes and g-Kelvin-Voight equations} \author[L. Friz, M. A. Rojas-Medar, M. D. Rojas-Medar \hfil EJDE-2016/37\hfilneg] {Luis Friz, Marko Antonio Rojas-Medar, Mar\'ia Drina Rojas-Medar} \address{Luis Friz \newline Grupo de Matem\'atica Aplicada, Dpto. de Ciencias B\'asicas, Facultad de Ciencias, Universidad del B\'io-B\'io, Campus Fernando May, Casilla 447, Chill\'an, Chile} \email{lfriz@ubiobio.cl} \address{Marko Antonio Rojas-Medar \newline Instituto de Alta Investigaci\'on, Universidad de Tarapac\'a, Casilla 7D, Arica, Chile} \email{marko.medar@gmail.com} \address{Mar\'ia Drina Rojas-Medar \newline Dpto. de Matem\'aticas, Facultad de Ciencias B\'asicas, Universidad de Antofagasta, Antofagasta, Chile} \email{maria.rojas@uantof.cl} \thanks{Submitted July 3,2015. Published January 26, 2016.} \subjclass[2010]{35Q35, 76D03} \keywords{Reproductive solution; g-Navier-Stokes system} \begin{abstract} This article presents the existence of reproductive solutions of g-Navier-Stokes and g-Kelvin-Voight equations. In this way, for weak solutions, we reach basically the same result as for classic Navier-Stokes equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} On one hand, in this work we consider the g-Navier-Stokes equation \begin{equation} \begin{gathered} { \frac{\partial\mathbf{u}}{\partial t}- \nu\Delta\mathbf{u}+(\mathbf{u}}\cdot\nabla)\mathbf{u}+ \nabla p = \mathbf{f},\quad \text{in }]0,T[\times\Omega, \\ { \frac{1}{g}(\nabla(g\mathbf{u}))= \frac{\nabla g}{g}\cdot\mathbf{u}+\nabla\cdot\mathbf{u}} = 0, \quad \text{in }]0,T[\times\Omega, \end{gathered} \label{ecua:pulenta1} \end{equation} defined on a domain $\Omega\subseteq\mathbb{R}^2$. This system is derived in \cite{roh4} from the 3-D Navier-Stokes equations \begin{gather*} { \frac{\partial\mathbf{U}}{\partial t}-\nu\Delta\mathbf{U}+ (\mathbf{U}}\cdot\nabla)\mathbf{U}+\nabla\Phi = \mathbf{f},\quad \text{in }]0,T[\times\Omega_g,\\ { \nabla\cdot\mathbf{U}} = 0, \quad \text{in }]0,T[\times\Omega_g, \end{gather*} where $\Omega_g=\{(y_1,y_2,y_3): (y_1,y_2)\in\Omega,\ 0\leq y_3\leq g(y_1,y_2)\}$, with the boundary conditions \[ { \mathbf{U}\cdot\mathbf{n}=0\quad\text{on } \partial_{\rm top} \Omega_g\cap\partial_{\rm bottom}\Omega_g} \] being, \begin{gather*} \partial_{\rm top}\Omega_g = \{(y_1,y_2,y_3)\in\Omega_g: y_3=g(y_1,y_2)\},\\ \partial_{\rm bottom}\Omega_g = \{(y_1,y_2,y_3)\in\Omega_g: y_3=0\}. \end{gather*} More precisely, the authors assume that \[ \mathbf{U}(y_1,y_2,y_3)=(\mathbf{U}_1(y_1,y_2), \mathbf{U}_2(y_1,y_2),\mathbf{U}_3(y_1,y_2,y_3)), \] and they define the following new variables and unknowns \begin{gather*} y_1=x_1,\quad y_2=x_2,\quad y_3=x_3g(x_1,x_2), \\ \mathbf{U}_1(y_1,y_2)= \mathbf{u}_1(x_1,x_2),\quad \mathbf{U}_2(y_1,y_2)=\mathbf{u}_2(x_1,x_2),\quad \mathbf{U}_3(y_1,y_2,y_3)=\mathbf{u}_3(x_1,x_2,x_3) \end{gather*} Finally, they prove that $\mathbf{u}=(\mathbf{u}_1,\mathbf{u}_2)$ is solution of the two equation of \eqref{ecua:pulenta1} and $\mathbf{u}_3=x_3\nabla g\cdot\mathbf{u}$. The interested reader can also review \cite{roh2}, \cite{roh1} and \cite{roh3}. Although the g-Navier-Stokes system is defined in two dimension domain, we will also study the tridimensional case. In this article, at first we seek a reproductive solution (or weak periodic solution) of \eqref{ecua:pulenta1}, i.e. solutions satisfying \begin{equation} \mathbf{u}(0,x)=\mathbf{u}(T,x),\quad x\in\Omega,\label{cond:repro} \end{equation} instead of a initial condition. In the case of the Navier-Stokes equation, the study of the reproductive solutions was initiated by Kaniel and Shinbrot in \cite{kaniel}, the reader can also see the classical textbook \cite{lions} by Lions. In \cite{blanca} the authors review some results concerning the existence, uniqueness and regularity of reproductive and time periodic solutions of the Navier-Stokes equations and some variants defined in bounded domains. In order to obtain a reproductive solution, they introduce a Galerkin discretization of the problem, proving existence of approximate solution to certain initial conditions. Then, a Leray-Schauder argument, by means of fixed point process, permits to obtain a reproductive Galerkin solution, which converges towards a continuous reproductive solution. To be more precise, in this work the first purpose is to solve the system \begin{equation} \begin{gathered} \frac{\partial\mathbf{u}}{\partial t}- \nu\Delta\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u}+ \nabla p = \mathbf{f}, \quad \text{in }]0,T[\times\Omega, \\ \frac{1}{g}(\nabla\cdot(g\mathbf{u}))= \frac{\nabla g}{g}\cdot\mathbf{u}+\nabla\cdot\mathbf{u} = 0, \quad \text{in }]0,T[\times\Omega,\\ \mathbf{u}(0,x) = \mathbf{u}(T,x), \quad \text{in }\Omega,\\ \mathbf{u}(t,x) = \beta(t,x), \quad \text{on }[0,T]\times\partial\Omega. \end{gathered} \label{ecua:pulenta} \end{equation} Here $\beta\in C^1(\mathbb{R},H^{1/2}(\partial\Omega)^{n})$ is $T$-periodic function and satisfies the (g-SOC) condition \begin{equation} \int_{\partial\Omega}g \beta\cdot\mathbf{n}ds=0. \label{gsoc} \end{equation} This definition is inspired by that given in \cite{morimoto} when $g\equiv1$, the so-called (SOC) condition, \begin{equation} \int_{\partial\Omega} \beta\cdot\mathbf{n}ds=0.\label{soc} \end{equation} Moreover, in a similar manner to the Navier-Stokes system, we can prove uniqueness of the solution in the bidimensional case. On the other hand, in this paper we also consider the g-Kelvin-Voight equation \begin{equation} \begin{gathered} \begin{aligned} & \frac{\partial\mathbf{u}}{\partial t} -\frac{\nu}{g}(\nabla\cdot g\nabla)\mathbf{u} +{ \frac{\nu}{g}}(\nabla g\cdot\nabla)\mathbf{u} - { \frac{\alpha}{g}}(\nabla\cdot g\nabla)\mathbf{u}_{t}\\ & +\frac{\alpha}{g}(\nabla g\cdot\nabla)\mathbf{u}_{t}+ \mathbf{u}\cdot\nabla\mathbf{u} + \nabla p=f,\quad \text{in }]0,T[\times\Omega \end{aligned} \\ { \frac{1}{g}}(\nabla\cdot(g\mathbf{u}))= { \frac{\nabla g}{g}}\cdot\mathbf{u}+ \nabla\cdot\mathbf{u} = 0,\quad \text{in }]0,T[\times\Omega \end{gathered} \label{ecua:pulenta3} \end{equation} The derivation of this system is analogous to the g-Navier-Stokes. In fact, it is deduced from the Kelvin-Voight system \begin{gather*} { \frac{\partial\mathbf{U}}{\partial t}- \nu\Delta\mathbf{U}-\alpha\Delta\mathbf{U}_{t}+(\mathbf{U}}\cdot\nabla)\mathbf{U}+ \nabla P = \mathbf{F},\quad \text{in }]0,T[\times\Omega_g, \\ { \nabla\cdot\mathbf{U}} = 0, \quad \text{in }]0,T[\times\Omega_g, \end{gather*} where $\Omega_g=\{(y_1,y_2,y_3): (y_1,y_2)\in\Omega$, $0\leq y_3\leq g(y_1,y_2)\}$. We refer interested readers to the article \cite{kaya} and the reference given there. The second purpose of this article is to solve the system \begin{equation} \begin{gathered} \begin{aligned} &{ \frac{\partial\mathbf{u}}{\partial t}}- { \frac{\nu}{g}}(\nabla\cdot g\nabla)\mathbf{u} +{ \frac{\nu}{g}}(\nabla g\cdot\nabla)\mathbf{u} - { \frac{\alpha}{g}}(\nabla\cdot g\nabla)\mathbf{u}_{t} \\ & +\frac{\alpha}{g}(\nabla g\cdot\nabla)\mathbf{u}_{t} +\mathbf{u}\cdot\nabla\mathbf{u} + \nabla p=f,\quad \text{in }]0,T[\times\Omega \end{aligned}\\ { \frac{1}{g}}(\nabla\cdot(g\mathbf{u}))= { \frac{\nabla g}{g}}\cdot\mathbf{u}+\nabla\cdot\mathbf{u} = 0,\quad \text{in }]0,T[\times\Omega\\ \mathbf{u}(0,x) = \mathbf{u}(T,x),\quad \text{in }\Omega\\ \mathbf{u}(t,x) = 0,\quad \text{in }]0,T[\times\partial\Omega \end{gathered} \label{ecua:pulenta3-1} \end{equation} in other words, we seek a reproductive solution for the g-Kelvin-Voight equation. This article is organized as follows. In section 2 the basic definitions and results are introduced. Section 3 is devoted to proving the existence of the reproductive solution of the g-Navier-Stokes system, both for the case $\beta = 0$ and the case $\beta\not= 0$. Finally, in section 4 the existence of the reproductive solution of the g-Kelvin-Voight system is proved. \section{Preliminaries} In this section, we introduce notation and spaces to be used later. Let $\Omega\subseteq\mathbb{R}^{n}$, $n=2,3$ be a bounded domain with smooth boundary $\partial\Omega$. We assume that $g\in W^{1,\infty}(\Omega)$ satisfies \begin{equation} 00$ is the first eigenvalue of the g-Stokes operator in $\Omega$ (see \cite{kaya}), i.e. the spectral problem \begin{equation} \begin{gathered} { -\frac{1}{g}}(\nabla\cdot g\nabla)\mathbf{w}^j+ \nabla p^j = \lambda_{j}\mathbf{w}^j, \quad\text{in } \Omega,\\ \nabla\cdot g\mathbf{w}^j = 0 \quad\text{in } \Omega,\\ \mathbf{w}^j = 0 \quad\text{on } \partial\Omega. \end{gathered} \label{prob-espectral} \end{equation} Problem \eqref{prob-espectral} has eigenvalues $0<\lambda_1\leq\lambda_2\leq\ldots\leq\lambda_{j}\leq\ldots$ and corresponding eigenfunctions $\mathbf{w}^1,\mathbf{w}^2, \ldots,\mathbf{w}^j,\ldots$ form an orthonormal basis in $\mathbf{H}_g$ and total basis in $\mathbf{V}_g$, where $\mathbf{H}_g$ and $\mathbf{V}_g$ are defined in the following manner: \begin{gather*} \mathcal{V} = \{\mathbf{u}\in\mathcal{D}(\Omega):\nabla\cdot(g\mathbf{u})=0\},\\ \mathbf{H}_g \text{ is the closure of }\mathcal{V}\text{ in }\mathbf{L}^2(\Omega),\\ \mathbf{V}_g \text{ is the closure of $\mathcal{V}$ in }\mathbf{H}_0^1(\Omega). \end{gather*} Where $\mathbf{H}_g$ is endowed with the scalar product \[ (\mathbf{u},\mathbf{v})_g=\int_{\Omega}(\mathbf{u} \cdot\mathbf{v})gdx\quad \text{and}\quad |\mathbf{u}|^2= (\mathbf{u},\mathbf{u})_g. \] Notice that this inner product is equivalent to the usual inner product defined in $\mathbf{L}^2(\Omega)$. Similarly, we define in $\mathbf{V}_g$ the equivalent inner product: \[ ((\mathbf{u},\mathbf{v}))_g=\int_{\Omega}g\nabla\mathbf{u} \cdot\nabla\mathbf{v}dx. \] Let us recall that $\beta$ satisfies condition \eqref{soc} if \[ \int_{\partial\Omega}\beta\cdot\mathbf{n}ds=0. \] In this case, Morimoto \cite[p. 636]{morimoto} proved the next Lemma. \begin{lemma} \label{lemajapones} Suppose $\beta\in C^1(\mathbb{R}, \mathbf{H}^{1/2}(\partial\Omega)^{n})$ is $T$-periodic and satisfies (SOC). Then for every $\varepsilon>0$, there exists a solenoidal and $T$-periodic function $\mathbf{b}\in C^1(\mathbb{R};\mathbf{H}^1(\Omega))$ such that \begin{gather*} \nabla_{x}\cdot\mathbf{b}(t,x) = 0\quad\text{a.e } x\in\Omega,\; \forall t\in\mathbb{R},\\ \mathbf{b}(t,x) = \beta(t,x),\quad x\in\partial\Omega,\; \forall t\in\mathbb{R},\\ |((\mathbf{u}\cdot\nabla)\mathbf{b},\mathbf{u})| \leq \varepsilon\|\nabla\mathbf{u}\|^2,\ \forall\mathbf{u}\in V,\forall t\in\mathbb{R}. \end{gather*} \end{lemma} Now, if $\beta\in C^1(\mathbb{R},\mathbf{H}^{1/2}(\partial\Omega)^{n})$ is $T$-periodic and satisfies the \eqref{gsoc} condition: \[ \int_{\partial\Omega}g\beta\cdot\mathbf{n}ds=0, \] we have the following proposition. \begin{proposition} \label{prop-chilensis} Suppose $\beta\in C^1(\mathbb{R},\mathbf{H}^{1/2}(\partial\Omega)^{n})$ is $T$-periodic and satisfies \eqref{gsoc}. Then for every $\varepsilon>0$ there exists a $T$-periodic function $\Psi\in C^1(\mathbb{R};\mathbf{H}^1(\Omega))$ such that: \begin{gather*} \nabla_{x}\cdot(g(x)\Psi(t,x)) = 0\quad \text{a.e } x\in\Omega,\quad \forall t\in\mathbb{R}\\ \Psi(t,x) = \beta(t,x),\quad \text{a.e. } x\in\partial\Omega,\; \forall t\in\mathbb{R},\\ |((\mathbf{v}\cdot\nabla)\Psi,\mathbf{v})_g| \leq C(\Omega,g)(\varepsilon+\|\nabla g\|_{L^{\infty}}|\nabla\Psi|) |\nabla\mathbf{v} |^2,\quad \forall t\in\mathbb{R}, \end{gather*} for all $\mathbf{v}\in V_g$. \end{proposition} \begin{proof} For $\varepsilon>0$, define $\Psi(t,x)={\frac{\mathbf{b}(t,x)}{g(x)}}$, where $\mathbf{b}(t,x)\in C^1(\mathbb{R},\mathbf{H}^1(\Omega))$ is given by Lemma \ref{lemajapones}. It is clear that $\Psi\in\mathbf{V}_g$ is $T$-periodic and $\Psi=\beta$ on $[0,T]\times\partial\Omega$. We have \begin{align*} ((\mathbf{v}\cdot\nabla)\Psi,\mathbf{v})_g & = \sum_{i,j=1}^{3}\int_{\Omega}\mathbf{v}_{i} \frac{\partial\Psi_{j}}{\partial x_{i}}\mathbf{v}_{j}gdx\\ & = \int_{\Omega}\sum_{i,j=1}^{3}\big(\frac{1}{g^2}\big) g\mathbf{v}_{i}\frac{\partial(g\Psi_{j})}{\partial x_{i}} g\mathbf{v}_{j}dx-\int_{\Omega}\sum_{i,j=1}^{3} \mathbf{v}_{i}\frac{\partial g}{\partial x_{i}}\Psi_{j}\mathbf{v}_{j}dx \end{align*} Now, from Lemma $\ref{lemajapones}$ \begin{align*} \big|\int_{\Omega}\sum_{i,j=1}^{3}\big(\frac{1}{g^2}\big) g\mathbf{v}_{i}\frac{\partial(g\Psi_{j})}{\partial x_{i}}g\mathbf{v}_{j}dx\big| & \leq \frac{1}{m_0^2}|((g\mathbf{v}\cdot\nabla)(g\Psi),g\mathbf{v})|\\ & \leq \frac{\varepsilon}{m_0^2}|\nabla(g\mathbf{v})|^2 \\ &\leq\varepsilon C(\Omega,g)|\nabla\mathbf{v}|^2 \end{align*} moreover, \begin{align*} \big|\int_{\Omega}\sum_{i,j=1}^{3}\mathbf{v}_{i} \frac{\partial g}{\partial x_{i}}\Psi_{j}\mathbf{v}_{j}\big| & \leq \|\nabla g\|_{L^{\infty}}|\mathbf{v}|_{L^{3}}|\Psi|_{L^{6}}|\mathbf{v}|\\ & \leq C(\Omega,g)\|\nabla g\|_{L^{\infty}}|\nabla\Psi||\nabla\mathbf{v}|^2 \end{align*} Therefore, \[ |((\mathbf{v}\cdot\nabla)\Psi,\mathbf{v})_g |\leq C(\Omega,g)(\varepsilon+\|\nabla g\|_{L^{\infty}}|\nabla\Psi|)|\nabla\mathbf{v}|^2. \] \end{proof} \begin{remark} \label{obs-bg} \rm Similarly to the case of the Navier-Stokes equation, we can define the trilinear form $b_g:\mathbf{V}_g\times\mathbf{V}_g\times \mathbf{V}_g\to\mathbb{R}$ by \[ b_g(\mathbf{u},\mathbf{v},\mathbf{w})=\sum_{i,j=1}^{n}\int_{\Omega} \mathbf{u}_{i}\frac{\partial\mathbf{v}_{j}}{\partial x_{i}}\mathbf{w}_{j}gdx \] for every $\mathbf{u},\mathbf{v},\mathbf{w}\in\mathbf{V}_g$. It is not difficult (see \cite{temam}) to prove that \[ b_g(\mathbf{u},\mathbf{v},\mathbf{v}) = 0, \] for each $\mathbf{u},\mathbf{v}\in\mathbf{V}_g$, moreover (see \cite{kaya}), if we further assume that $\Delta g=0$ we have \[ b_g\big(\frac{\nabla g}{g},\mathbf{v},\mathbf{v}\big) = 0, \] for all $\mathbf{v}\in\mathbf{V}_g$. \end{remark} Define the g-Laplacian operator as \[ -\Delta_g\mathbf{u}=-\frac{1}{g}(\nabla\cdot g\nabla)\mathbf{u}=-\Delta\mathbf{u}-\frac{1}{g}\nabla g\cdot\nabla\mathbf{u}. \] Now, we can rewrite the first equation of \eqref{ecua:pulenta} as follows: \[ { \frac{\partial\mathbf{u}}{\partial t}- \nu\Delta_g\mathbf{u}+\nu\frac{\nabla g}{g}\cdot\nabla\mathbf{u} +(\mathbf{u}\cdot\nabla) \mathbf{u}+\nabla p}=\mathbf{f}. \] \section{Existence of reproductive and periodic solutions for the g-Navier-Stokes system} The variational formulation of \eqref{ecua:pulenta} is the following: given $\mathbf{f}\in L^2(0,T;\mathbf{V}'_g)$ and $\mathbf{u}_0\in\mathbf{V}_g$ to find $\mathbf{u}-\Psi\in L^{\infty}(0,T;\mathbf{H}_g)\cap L^2(0,T;\mathbf{V}_g)$ such that \begin{equation} \begin{gathered} \begin{aligned} & \frac{d}{dt}(\mathbf{u}-\Psi, \mathbf{v})+\nu((\mathbf{u}-\Psi,\mathbf{v}))_g+b_g (\mathbf{u}-\Psi,\mathbf{u}-\Psi,\mathbf{v})\\ & +b_g(\Psi,\mathbf{u}-\Psi,\mathbf{v})+ b_g(\mathbf{u}-\Psi,\Psi,\mathbf{v}) +\nu b_g\big({ \frac{\nabla g}{g}, \mathbf{u}-\Psi,\mathbf{v}}\big)\\ &= \langle f,\mathbf{v}\rangle -L(\Psi,\mathbf{v}) \end{aligned} \\ \mathbf{u}(0)=\mathbf{u}_0+\Psi(0) \end{gathered} \label{form-variacional} \end{equation} for all $\mathbf{v}\in\mathbf{V}_g$. Here $\Psi$ is given in Proposition \ref{prop-chilensis}, $b_g$ is the trilinear form given in Remark \ref{obs-bg} and \[ L(\Psi,\mathbf{v})= \big({ \frac{d\Psi}{dt},\mathbf{v}}\big)+ \nu((\Psi,\mathbf{v}))_g+b_g(\Psi,\Psi,\mathbf{v})+ \nu b_g\big({ \frac{\nabla g}{g},\Psi,\mathbf{v}}\big). \] \begin{definition} \label{definicion-debil-g-nav} \rm Let $\mathbf{u}_0\in\mathbf{H}_g$ and $\mathbf{f}\in L^2(0,T;\mathbf{V}'_g)$. A function $\mathbf{u}\in L^{\infty}(0,T;\mathbf{H}_g)\cap L^2(0,T;\mathbf{V}_g$) is a weak solution of the problem \eqref{ecua:pulenta1} with initial data $\mathbf{u}(0)=\mathbf{u}_0$ and boundary data $\mathbf{u}=\beta$ on $[0,T]\times\partial\Omega$, if $\mathbf{u}$ verifies \eqref{form-variacional} for all $\mathbf{v}\in\mathbf{V}_g$. \end{definition} In the case $\beta\equiv0$, we have the following theorem. \begin{theorem}[{\cite[thm 6.1]{roh2}}] \label{teo-koreano} Assume $\mathbf{f}\in L^2(0,T;\mathbf{V}'_g)$ and $\mathbf{u}_0\in\mathbf{H}_g$. Then there exists at least a weak solution of the problem \eqref{ecua:pulenta1}, in the sense of the Definition \ref{definicion-debil-g-nav}. Moreover, $\mathbf{u}$ is weakly continuous from $[0,T]$ into $\mathbf{H}_g$. \end{theorem} \begin{proposition} \label{unicidad-2-d} If $\Omega\subseteq\mathbb{R}^2$, under the assumptions of Theorem \ref{teo-koreano}, the weak solution of \eqref{ecua:pulenta1} with initial data $\mathbf{u}(0)=\mathbf{u}_0$ is unique. \end{proposition} \begin{proof} Let $\mathbf{u}_1$ and $\mathbf{u}_2$ be two solutions of the problem $\eqref{form-variacional}$ with initial data $\mathbf{u}_0$. If we define $\mathbf{w=u}_1-\mathbf{u}_2$, then it satisfies the variational formulation \[ \frac{1}{2}\frac{d}{dt}(\mathbf{w},\mathbf{v})_g +\nu((\mathbf{w},\mathbf{v}))_g+ \nu\Big(\big(\frac{\nabla g}{g}\cdot\nabla\big)\mathbf{w}, \mathbf{v}\Big)_g=-b_g(\mathbf{u}_1,\mathbf{u}_1, \mathbf{v})+b_g(\mathbf{u}_2,\mathbf{u}_2,\mathbf{v}) \] By replacing $\mathbf{v}=\mathbf{w}$ we get \begin{align*} \frac{d}{dt}|\mathbf{w}|^2+2\nu\|\mathbf{w}\|^2 & = -2\nu\Big(\big(\frac{\nabla g}{g}\cdot\nabla\big)\mathbf{w}, \mathbf{w}\Big)_g-2b_g(\mathbf{u}_1,\mathbf{u}_1, \mathbf{w})+2b_g(\mathbf{u}_2,\mathbf{u}_2,\mathbf{w})\\ & = -2\nu\Big(\big(\frac{\nabla g}{g}\cdot\nabla\big)\mathbf{w}, \mathbf{w}\Big)_g-2b_g(\mathbf{w},\mathbf{u}_1,\mathbf{w}); \end{align*} therefore, since \[ -2\nu\Big(\big(\frac{\nabla g}{g}\cdot\nabla\big)\mathbf{w}, \mathbf{w}\Big)_g\leq2\nu\frac{\|\nabla g\|_{\infty}} {m_0\lambda_1}\|\mathbf{w}\|^2, \] by \cite[Lemma 2.1]{anh}, we also have \begin{align*} 2b_g(\mathbf{w},\mathbf{u}_1,\mathbf{w}) & \leq C\|\mathbf{u}_1\||\mathbf{w}|\|\mathbf{w}\|\\ & \leq \varepsilon\|\mathbf{w}\|^2+C_{\varepsilon}\|\mathbf{u}_1 \|^2|\mathbf{w}|^2. \end{align*} Now, for $\varepsilon$ small enough we can obtain \[ \frac{d}{dt}|\mathbf{w}|^2\leq C_{\varepsilon}\|\mathbf{u}_1\|^2|\mathbf{w}|^2; \] then by using Gronwall's inequality, we conclude that $\mathbf{w}=\mathbf{0}$. \end{proof} \begin{remark} \rm After some tedious calculations, it is possible to see that Theorem \ref{teo-koreano} and Proposition \ref{unicidad-2-d} remain valid even if the $\beta$ is not null. \end{remark} Our main result is the following. \begin{theorem} \label{teo-bakan} For any $\mathbf{f}\in L^2(0,T;\mathbf{V}_g')$ and $\|\nabla g\|_{\infty}$ small enough there exists a weak solution of \eqref{ecua:pulenta} i.e. the weak solution $\mathbf{u}\in L^{\infty}(0,T;\mathbf{H}_g)\cap L^2(0,T;\mathbf{V}_g)$ has the so-called reproductive property, i.e. a solution of the variational problem \eqref{form-variacional} which satisfies $\mathbf{u}(0,x)=\mathbf{u}(t,x)$. \end{theorem} \begin{remark} \rm Note that if $n=2$ and the external force $\mathbf{f}\in L^2(\mathbb{R};\mathbf{V}'_g)$ is a $T$-periodic in time function, the above Theorem \ref{teo-bakan} furnishes a $T$-periodic weak solution for \eqref{ecua:pulenta}. In fact, it is a strong solution and actually very regular. This is so because we can prove that $\mathbf{u}\in C^\infty (\Omega) $ for $t>0$, where $\mathbf{u}$ is solution of the problem \eqref{ecua:pulenta1} with initial condition $\mathbf{u}_0\in \mathbf{H}_g$. Thus $\mathbf{u}_p\in C^\infty (\Omega)$ for $t\in [T,2T]$ and, by the $T$-periodicity, we conclude that $\mathbf{u}_p(t)=\mathbf{u}_p(t+T)\in C^\infty (\Omega) $, here $\mathbf{u}_p$ is the reproductive solution. In particular, $\mathbf{u}_p(0)\in C^\infty (\Omega) $. \end{remark} \subsection{Proof of Theorem \ref{teo-bakan} when $\beta\equiv 0$} Let $\{\mathbf{w}^{i}\}_{i=1}^{\infty}$ be orthonormal bases in $\mathbf{H}_g$ and total bases in $\mathbf{V}_g$ obtained in spectral problem $\eqref{prob-espectral}$. As k$^{th}$-approximated solution of equation \eqref{form-variacional} we choose \begin{equation} \mathbf{u}^{k}(t,x)={ \sum_{i=1}^{k}c_{i}^{k}(t)\mathbf{w}^{i}(x)\,} \label{galerkin} \end{equation} satisfying for all $i=1,\dots,k$, and for all $t\in(0,T)$ the system of equations \begin{equation} \begin{gathered} { \frac{d}{dt}}(\mathbf{u}^{k},\mathbf{v})_g+ \nu((\mathbf{u}^{k},\mathbf{v}))_g+b_g(\mathbf{u}^{k}, \mathbf{u}^{k},\mathbf{v}) +\nu b_g\big({ \frac{\nabla g}{g},\mathbf{u}^{k},\mathbf{v}}\big) = \langle\mathbf{f},\mathbf{v}\rangle\\ \mathbf{u}^{k}(0) = P_{k}\mathbf{u}_0 \end{gathered} \label{ecua1} \end{equation} for all $\mathbf{v}\in\mathbf{V}^{k}=\langle\{\mathbf{w}^1, \mathbf{w}^2,\ldots,\mathbf{w}^{k}\}\rangle$. Taking $\mathbf{v}=\mathbf{u}^{k}$, we have \[ \frac{d}{dt}|\mathbf{u}^{k}|^2+2\nu|\nabla\mathbf{u}^{k}|^2 = \langle f,\mathbf{u}^{k}\rangle-2\nu\Big(\big(\frac{\nabla g} {g}\cdot\nabla\big)\mathbf{u}^{k},\mathbf{u}^{k}\Big)_g \] Therefore, by using the Poincar\'e inequality, \[ { |\mathbf{v}|^2\leq\frac{1}{\lambda_1}| \nabla\mathbf{v}|^2\quad \forall\,\mathbf{v}\in \mathbf{H}_0^1(\Omega),} \] we have \begin{equation} \frac{d}{dt}|\mathbf{u}^{k}|^2+2\nu|\nabla\mathbf{u}^{k}|^2 \leq{ \frac{1}{\nu}}\|\mathbf{f}\|_{V^{*}}^2+\nu| \nabla\mathbf{u}^{k}|^2+2\nu\frac{\|\nabla g\|_{\infty}}{m_0 \lambda_1^{1/2}}|\nabla\mathbf{u}^{k}|^2\,. \label{ecua8} \end{equation} Finally, we obtain \[ \frac{d}{dt}|\mathbf{u}^{k}|^2+\nu \lambda_1\gamma_0|\mathbf{u}^{k}|^2 \leq\frac{1}{\nu}\|\mathbf{f}\|_{V^{*}}^2 \,, \] where $\gamma_0=1-{ \frac{2\|\nabla g\|_{\infty}}{m_0\lambda_1^{1/2}}>0}$ for $\|\nabla g\|_{\infty}$ small. The above inequality implies \[ \frac{d}{dt}(e^{\nu\lambda_1\gamma_0t}| \mathbf{u}^{k}|^2)\leq\frac{e^{\nu\lambda_1\gamma_0t}}{\nu}\| \mathbf{f}\|_{V^{*}}^2\,. \] Integrating from $0$ to $T$ we have \begin{equation} e^{\nu\lambda_1\gamma_0T}|\mathbf{u}^{k}(T)|^2 \leq|\mathbf{u}^{k}(0)|^2+\frac{1}{\nu}\int_0^T e^{\nu\lambda_1\gamma_0t}\|\mathbf{f}(t)\|_{V^{*}}^2. \label{integraldesigualdad} \end{equation} Next, we show that $\mathbf{u}^{k}$ is nothing but one fixed point of the operator $\Phi^{k}$ defined in what follows. Let $L^{k}: [0,T]\to\mathbb{R}^{k}$ the mapping defined by \[ L^{k}(t)=\mathbf{y}(t)= (c_1^{k}(t),\dots,c_{k}^{k}(t)), \] where the time dependent functions $\{c_{i}^{k}(t)\}_{i=1}^{k}$ are the coefficients of the expansion of $\mathbf{u}^{k}$, as done in \eqref{galerkin}. Since we have chosen the basis $\{\mathbf{w}^{i}(x)\}_{i=1}^{\infty}$ orthonormal in $\mathbf{H}_g$, we have \begin{equation} \|\mathbf{y}(t)\|_{\mathbb{R}^{k}}= |\mathbf{u}^{k}(t)|\quad\forall t\in\,[0,T]\,.\label{igualdadnormas} \end{equation} Next, we define the operator $\Phi^{k}: \mathbb{R}^{k} \to \mathbb{R}^{k}$ as \[ \Phi^{k}(\mathbf{x})=\mathbf{y}(T) \] where $\mathbf{x}=(x_1,x_2,\ldots,x_{k})$ and $\mathbf{y}(T)=L^{k}(T)$ is the vector-coefficients at time $T$ of the solution of \eqref{ecua1} with initial condition \[ \mathbf{u}_0^{k}(x)={ \sum_{i=1}^{m}x_{i}\;\mathbf{w}^{i}(x)}, \] It is not difficult to see that $\Phi^{k}$ is continuous and we claim that $\Phi^{k}$ has at least one fixed point. It will be a consequence of Leray-Schauder's Homotopy Theorem. To prove this, it is enough to show that for any $\lambda\in[0,1]$, a solution of the equation \begin{equation} \lambda\Phi^{k}(\mathbf{x}(\lambda))=\mathbf{x}(\lambda)\label{igualdad2} \end{equation} has a bound independent of $\lambda$. Since $\mathbf{x}(0)=0$, we restrict the proof to $\lambda\in(0,1]$. In such case \eqref{igualdad2} may be rewritten as \[ \Phi^{k}(\mathbf{x}(\lambda)) = \frac{1}{\lambda}\mathbf{x}(\lambda)\,. \] By the definition of $\Phi^{k}$ and \eqref{igualdadnormas}, we deduce from \eqref{integraldesigualdad}, that \[ { e^{\nu\lambda_1\gamma_0T} \| \frac{1}{\lambda}\mathbf{x}(\lambda)\| _{\mathbb{R}^{k}}^2 \leq\| \mathbf{x}(\lambda)\|_{\mathbb{R}^{k}}^2+\int_0^T e^{\nu\lambda_1\gamma_0T}\|\mathbf{f}(t)\|_{\mathbf{V}^{\ast}}^2dt} \] Since we impose $\mathbf{u}^{k}(0)=\mathbf{u}^{k}(T)$, we obtain \begin{equation} \|\mathbf{x}(\lambda)\|_{\mathbb{R}^{k}}^2 \leq\frac{1}{e^{\nu\lambda_1\gamma_0T}-1}\,\int_0^T e^{\nu\lambda_1\gamma_0T}\|\mathbf{f}(t) \|_{V^{\ast}}dt\equiv M(T,\mathbf{f}),\label{inequality1} \end{equation} for all $\lambda\in(0,1]$. Obviously, this upper bound do not depends on $\lambda\in[0,1]$ and so we have stated that the operator $\Phi^{k}$ has at least one fixed point, denoted by $\mathbf{x}(1)$ and then there exists a reproductive Galerkin solution $\mathbf{u}^{k}$, namely it satisfies $\mathbf{u}^{k}(0)=\mathbf{u}^{k}(T)$. Note that, from $\eqref{inequality1}$, we have that $\mathbf{u}^{k}\in L^{\infty}(0,T;\mathbf{H}_g)$, for every $k\in\mathbb{N}$ and it is uniformly bounded. From \eqref{ecua8} and by definition of $\gamma_0$ we can obtain the inequality \[ { \frac{d}{dt}|\mathbf{u}^{k}|^2+ \nu\gamma_0|\nabla\mathbf{u}^{k}|^2\leq\frac{1}{\nu}\|\mathbf{f}\|^2}\,. \] Since $\mathbf{u}^{k}$ is a Galerkin reproductive solution and by integrating from $0$ to $T$ we have \begin{equation} \int_0^T|\nabla\mathbf{u}^{k}|^2dt\leq\frac{1} {\gamma_0\nu^2}\int_0^T\|\mathbf{f}\|^2dt= \widetilde{M}(T,\mathbf{f}).\label{cota-grad} \end{equation} In other words, $\mathbf{u}^{k}\in L^2(0,T;\mathbf{V}_g) \cap L^{\infty}(0,T;\mathbf{H}_g)$, for each $k\in\mathbb{N}$ and it is uniformly bounded. It is not difficult to prove that ${ \frac{d}{dt}\mathbf{u}^{k}\in L^2(0,T;\mathbf{V}'_g)}$ and it is uniformly bounded. By using compactness results (see \cite{simon}) with the triplets $\mathbf{H}_g\hookrightarrow\mathbf{V}'_g \hookrightarrow\mathbf{V}'_g$ and $\mathbf{V}_g\hookrightarrow\mathbf{H}_g \hookrightarrow\mathbf{V}'_g$, we have that $(\mathbf{u}^{k})$ is relatively compact in $L^2(0,T;\mathbf{H}_g)\cap C([0,T];\mathbf{V}'_g)$. Thus, since $\mathbf{u}^{k}(0)=\mathbf{u}^{k}(T)$ and $\mathbf{u}^{k}(0)\to\mathbf{u}(0)$, we get that $\mathbf{u}(0)=\mathbf{u}(T)$ in $\mathbf{V}'_g$, but we also have that $\mathbf{u}\in C([0,T];\mathbf{H}_g)$, because $\mathbf{u}\in L^2(0,T;\mathbf{H}_g)$ and ${ \frac{d}{dt}\mathbf{u}\in L^2(0,T;\mathbf{V}'_g)}$ (see \cite{temam} ), therefore $\mathbf{u}(0)=\mathbf{u}(T)$ in $\mathbf{H}_g$. \subsection{Proof of Theorem \ref{teo-bakan}, general case} Let us define $\hat{\mathbf{u}}=\mathbf{u}-\Psi$, where $\Psi$ is given in Proposition \ref{prop-chilensis}, which satisfies \begin{equation} \begin{gathered} \begin{aligned} & \frac{\partial\hat{\mathbf{u}}}{\partial t}- \nu\Delta\hat{\mathbf{u}}+(\hat{\mathbf{u}}\cdot\nabla) \hat{\mathbf{u}}+(\hat{\mathbf{u}}\cdot\nabla)\Psi + (\Psi\cdot\nabla)\hat{\mathbf{u}}+\nabla p \\ &=f-{ \frac{\partial\Psi}{\partial t}} + \nu\Delta\Psi-(\Psi\cdot\nabla)\Psi\quad\text{in } ]0,T[\times\Omega\,, \end{aligned} \\ { \frac{1}{g}(\nabla(g\hat{\mathbf{u}}))= { \frac{\nabla g}{g}\cdot\hat{\mathbf{u}}+ \nabla\cdot\hat{\mathbf{u}}}} = 0\quad\text{in } ]0,T[\times\Omega\,,\\ \hat{\mathbf{u}}(0,x) = \hat{\mathbf{u}}_0(x)\quad\text{in } ]0,T[\times\Omega, \\ \hat{\mathbf{u}}(t,x) = 0 \quad\text{on } [0,T]\times\partial\Omega\,. \end{gathered} \label{ecua:pulenta2} \end{equation} Since $\Psi$ is a $T$-periodic function it is only necessary to prove that there exists a reproductive solution of the problem \eqref{ecua:pulenta2}. The variational formulation is as follows: Find $\hat{\mathbf{u}}\in L^{\infty}(0,T;\mathbf{H}_g)\cap L^2(0,T; \mathbf{V}_g)$ such that for all $\mathbf{v}\in\mathbf{V}_g$ we have \begin{equation} \begin{aligned} & \frac{d}{dt}(\hat{\mathbf{u}},\mathbf{v})_g+ \nu((\hat{\mathbf{u}},\mathbf{v}))_g+b_g(\hat{\mathbf{u}}, \hat{\mathbf{u}},\mathbf{v}) + b_g(\hat{\mathbf{u}},\Psi,\mathbf{v})\\ &+b_g(\Psi,\hat{\mathbf{u}},\mathbf{v})+ \nu b_g\big({ \frac{\nabla g}{g},}\hat{\mathbf{u}},\mathbf{v}\big) \\ &= \langle\mathbf{f},\mathbf{v}\rangle-L(\Psi,\mathbf{v})\,, \end{aligned} \label{fvariautilde} \end{equation} where \[ L(\Psi,\mathbf{v})=\Big({\frac{d\Psi}{dt},\mathbf{v}}\Big)_g +\nu((\Psi,\mathbf{v}))_g+b_g(\Psi,\Psi,\mathbf{v}) +\nu b_g\big({\frac{\nabla g}{g},}\Psi,\mathbf{v}\big). \] After some calculations, we can write \begin{align*} |L(\Psi,\mathbf{v})| & \leq \Big(|{ \frac{d\Psi}{dt}}| +\frac{\nu\|\nabla g\|_{\infty}} {m_0}|\nabla\Psi|\Big)|\mathbf{v}|+(\nu|\nabla\Psi| +|\nabla\Psi|^2)|\nabla\mathbf{v}|\\ & \leq \frac{1}{2\varepsilon_1}\Big(| { \frac{d\Psi}{dt}}| +\frac{\nu\|\nabla g\|_{\infty}} {m_0}|\nabla\Psi|\Big)^2 +\frac{\varepsilon_1}{2}| \mathbf{v}|^2\\ &\quad +\frac{1}{2\varepsilon_1}(|\nabla\Psi|^2 + \nu|\nabla\Psi|)^2+ \frac{\varepsilon_1}{2}|\nabla\mathbf{v}|^2\,. \end{align*} Let us put \[ F=\frac{1}{2\varepsilon_1} \Big(| { \frac{d\Psi}{dt}}| +\frac{\nu\|\nabla g\|_{\infty}}{m_0}|\nabla\Psi|\Big)^2 +\frac{1}{2\varepsilon_1}(|\nabla\Psi|^2+ \nu|\nabla\Psi|)^2 +\frac{1}{2\varepsilon_1}\|\mathbf{f}\|_{V_g^{*}}\,. \] By replacing $\mathbf{v}$ by $\hat{\mathbf{u}}$ in \eqref{fvariautilde} we obtain \begin{align*} \frac{d}{dt}|\hat{\mathbf{u}}|^2+2\nu|\nabla\hat{\mathbf{u}}|^2 &\leq \frac{\varepsilon_1}{2}|\hat{\mathbf{u}}|^2+F +\Big(\varepsilon_1C(\Omega,g) + \varepsilon_1\\ &\quad + C(\Omega,g)\|\nabla g\|_{\infty}|\nabla\Psi|+ 2\nu\frac{\|\nabla g\|_{\infty}}{m_0\lambda^{1/2}} \Big)|\nabla\hat{\mathbf{u}}|^2\,. \end{align*} By choosing $\varepsilon_1$ and $\|\nabla g\|_{\infty}$ small enough, we obtain \begin{equation} \frac{d}{dt}|\hat{\mathbf{u}}(t)|^2+C|\hat{\mathbf{u}}(t)|^2 \leq F(t), \label{j-p2} \end{equation} where $C>0$, we can obtain a reproductive solution by following the same argument as in the proof of the case $\beta\equiv0$. \section{Existence of reproductive solutions for the g-Kelvin-Voight system} The variational formulation of problem \eqref{ecua:pulenta3} is: Given $\mathbf{f}\in L^2(0,T;\mathbf{V}'_g)$ and $\mathbf{u}_0\in\mathbf{H}_g$, find $\mathbf{u}\in\mathbf{V}_g$ such that \begin{equation} \begin{gathered} \begin{aligned} &\frac{d}{dt}(\mathbf{u},\mathbf{v})_g +\nu((\mathbf{u},\mathbf{v})) + \alpha((\mathbf{u}_{t},\mathbf{v})) +\nu b_g\big({ \frac{\nabla g}{g}}, \mathbf{u},\mathbf{v}\big)\\ &+\alpha b_g\big({ \frac{\nabla g}{g}},\mathbf{u}_{t}, \mathbf{v}\big) + b_g(\mathbf{u},\mathbf{u},\mathbf{v}) =\langle \mathbf{f},\mathbf{v}\rangle \end{aligned}\\ \mathbf{u}(0) = \mathbf{u}_0\,, \end{gathered} \label{for-celebi-variacional} \end{equation} for all $\mathbf{v}\in\mathbf{V}_g$. \begin{definition} \rm Let $\mathbf{u}_0\in\mathbf{H}_g$ and $\mathbf{f}\in L^2(0,T;\mathbf{V}'_g)$. A function $\mathbf{u}\in L^{\infty}(0,T;\mathbf{H}_g)\cap L^2(0,T;\mathbf{V}_g)$ is a weak solution of the problem \eqref{ecua:pulenta3} with initial condition $\mathbf{u}(0)=\mathbf{u}_0$ if $\mathbf{u}$ verifies $\eqref{for-celebi-variacional}$ for all $\mathbf{v}\in\mathbf{V}_g$. \end{definition} \begin{theorem}[\cite{kaya}] If $\mathbf{f}\in\mathbf{L}^2(\Omega)$, $\Omega\subseteq\mathbb{R}^2$ $\mathbf{u}_0\in\mathbf{V}_g$ and $g$ satisfying \eqref{condicion-g} and $\Delta g=0$, then there exists a unique weak solution of \eqref{ecua:pulenta3}. \end{theorem} \begin{remark} \rm It is possible to prove that the hypothesis that $\mathbf{f}$ does not depend on time $t$ can be removed and replaced by $\mathbf{f}\in L^2(0,T;\mathbf{V}'_g)$, and the theorem is still valid. \end{remark} The main result of this section is the following. \begin{theorem} \label{teo-bakan-2} For $\|\mathbf{f}\|_{L^2(0,T;\mathbf{V}_g')}$ and $\|\nabla g\|_{\infty}$ small enough there exists a weak solution of \eqref{ecua:pulenta3-1} i.e. the weak solution $\mathbf{u}\in L^{\infty}(0,T;\mathbf{H}_g)\cap L^2(0,T;\mathbf{V}_g)$ has the so-called reproductive property, i.e. a solution of the variational problem \eqref{for-celebi-variacional} which satisfies $\mathbf{u}(0,x)=\mathbf{u}(T,x)$. \end{theorem} \subsection{Proof of Theorem \ref{teo-bakan-2}} In the same manner as in the proof of Theorem \ref{teo-bakan}, we define \begin{equation} \mathbf{u}^{k}(t,x)={ \sum_{i=1}^{k}c_{i}^{k}(t) \mathbf{w}^{i}(x)}\label{galerkin2} \end{equation} as the solution of the variational problem \begin{align*} & \frac{d}{dt}(\mathbf{u}^{k},\mathbf{v})_g+ \nu((\mathbf{u}^{k},\mathbf{v})) + \alpha((\mathbf{u}_{t}^{k}, \mathbf{v}))+\nu b_g\big({ \frac{\nabla g}{g}}, \mathbf{u}^{k},\mathbf{v}\big)\\ &+\alpha b_g\big({ \frac{\nabla g}{g}},\mathbf{u}_{t}^{k}, \mathbf{v}\big) + b_g(\mathbf{u}^{k},\mathbf{u}^{k}, \mathbf{v})=\langle \mathbf{f},\mathbf{v}\rangle \end{align*} for all $\mathbf{v}\in\mathbf{V}^{k} =\langle \{\mathbf{w}^1,\ldots,\mathbf{w}^{k}\}\rangle$. The proof of the following lemma can be found in \cite[pp 499-501]{kaya}. For simplicity, we denote \[ y(t) = \|\mathbf{u}^{k}(t)\|_g^2+(\alpha+\nu) \|\nabla\mathbf{u}^{k}(t)\|_g^2. \] \begin{lemma} \label{lematrigli} For $\|\nabla g\|_{\infty}$ small enough there exist positive constants $\beta$ and $\delta$ such that the function $y(t)$ satisfies \[ \frac{dy}{dt}+\beta y\leq\delta y^2+C\|\mathbf{f}(t)\|_g^2. \] \end{lemma} \begin{proposition}\label{prop-olas} Let $M_1>0$ be such that \[ \delta s<\frac{\beta}{2},\quad\forall s\in]0,M_1]. \] Let us suppose that $\delta$ satisfies $\|\mathbf{f}\|_{L^{\infty}(0,T;\mathbf{V}'_g)}^2 \leq{ \frac{\beta}{2}M_1}$. If $y(0)\leq M_1$, then $y(t)\leq M_1$, for all $t\in[0,T]$. \end{proposition} \begin{proof} From Lemma \ref{lematrigli}, $y$ satisfies the differential inequality \begin{equation} y'+(\beta-\delta y)y\leq\|\mathbf{f}(t)\|_g^2.\label{desigualdad-trigli0} \end{equation} By hypothesis, there exists $\sigma>0$ such that \begin{equation} \delta s\leq\frac{\beta}{2},\quad \forall s\in[M_1,M_1+\sigma]. \label{desigualdad-trigli} \end{equation} At first, we will prove that \[ y(t)