\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 82, pp. 1--21.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/82\hfil Existence and uniqueness] {Existence and uniqueness of classical solutions to second-order quasilinear elliptic equations} \author[D. Denny\hfil EJDE-2010/82\hfilneg] {Diane Denny} \address{Diane Denny \newline Department of Mathematics and Statistics\\ Texas A\&M University - Corpus Christi \\ Corpus Christi, TX 78412, USA} \email{diane.denny@tamucc.edu} \thanks{Submitted April 13, 2010. Published June 18, 2010.} \subjclass[2000]{35A05} \keywords{Existence; uniqueness; quasilinear; elliptic} \begin{abstract} This article studies the existence of solutions to the second-order quasilinear elliptic equation $$-\nabla \cdot(a(u) \nabla u) +\mathbf{v}\cdot \nabla u=f$$ with the condition $u(\mathbf{x}_0)=u_0$ at a certain point in the domain, which is the 2 or the 3 dimensional torus. We prove that if the functions $a$, $f$, $\mathbf{v}$ satisfy certain conditions, then there exists a unique classical solution. Applications of our results include stationary heat/diffusion problems with convection and with a source/sink, when the value of the solution is known at a certain location. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} In this article, we consider the following quasilinear elliptic equation for $u(\mathbf{x})$ under periodic boundary conditions: \begin{gather} -\nabla \cdot(a(u) \nabla u)+\mathbf{v}\cdot \nabla u =f, \label{1.1} \\ u(\mathbf{x}_0)=u_0, \label{1.2} \end{gather} where $u_0$ is a given constant and $\mathbf{x}_0$ a given point in the domain $\Omega$. Here, $f(\mathbf{x})$ and $\mathbf{v}(\mathbf{x})$ are given smooth functions for $\mathbf{x}\in \Omega$, where the domain $\Omega =\mathbb{T}^N$, the $N$-dimensional torus, with $N=2,3$. We assume that $a(u)$ is a smooth, positive function of $u$ for $u \in \bar{G}$, where $G\subset \mathbb{R}$ is a bounded interval. The purpose of this article is to prove the existence of a unique classical solution $u(\mathbf{x})$ to \eqref{1.1}-\eqref{1.2}. What is new in this paper is the requirement that condition \eqref{1.2} holds for a quasilinear elliptic equation of the form \eqref{1.1} which includes a convection term $\mathbf{v}\cdot \nabla u$. The proof of the existence theorem uses the method of successive approximations in which an iteration scheme, based on solving a linearized version of the equation, will be defined and then convergence of the sequence of approximating solutions to a unique solution satisfying the quasilinear equation will be proven. It will be shown that there exist positive constants $\delta_0$, $\delta_1$, and $\delta_2$ such that if $\big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2 \| f\|_{s-1}^2 \leq \delta_0$, and $|\nabla \cdot \mathbf{v}|_{L^\infty} \leq \delta_1$, and $\max\{1,|\mathbf{v}|_{L^\infty}^2\}\| f\|_{s-1}^2 \leq \delta_2$, and $\| D \mathbf{v}\|_{s} \leq \frac 12$, where $s> \frac N2 +1$, and where $G_1\subset G$, then there exists a classical solution $u(\mathbf{x})$ to \eqref{1.1}-\eqref{1.2}. Here we define $|\frac{da}{du} |_{s,\bar{G}_1}=\max\{\big|\frac{d^{j+1} a}{d u^{j+1}} (u_{*}) \big|: u_{*} \in \bar{G}_1, 0 \leq j \leq s \}$. And $u(\mathbf{x}) \in \bar{G_1}$ for all $\mathbf{x}\in \Omega$. The solution $u(\mathbf{x}) \in \bar{G_1}$ will be unique if $a''(u_{*})\leq \frac{1}{a(u_{*})}(a'(u_{*}))^2$ for all $u_{*} \in \bar{G_1}$. The key to the proof lies in obtaining a priori estimates for $u$. Applications of the existence of a unique solution to \eqref{1.1}-\eqref{1.2} include stationary heat/diffusion problems with convection and with a source/sink. Solutions could be obtained for problems in which, for example, the temperature or the concentration of a substance in a fluid is monitored at a given spatial location $\mathbf{x}_0 \in \Omega$ . This article is organized as follows. First, the main result is presented and proved as Theorem \ref{T3.1} in the next section. Then lemmas supporting the proof of the theorem are proven in Appendix A (which proves the existence of a solution to the linearized equation used in the iteration scheme) and in Appendix B (which presents proofs of the a priori estimates used in the proof of the theorem). \section{Existence theorem} We use the Sobolev space $H^s(\Omega )$ (where $s$ is a non-negative integer) of real-valued functions in $L^2(\Omega )$ whose distribution derivatives up to order $s$ are in $L^2(\Omega )$, with norm given by $\|g\|_s^2=\sum_{0 \leq |\alpha |\leq s}\int_\Omega |D^\alpha g|^2d \mathbf{x}$ and inner product $(g,h)_s=\sum_{0 \leq |\alpha |\leq s}\int_\Omega (D^\alpha g)\cdot (D^\alpha h)d \mathbf{x}$. We use the notation $\|g\|_s^2=\sum_{0 \leq r \leq s} \int_\Omega |D^r g|^2 d \mathbf{x}$, where $D^r g$ is the set of all space derivatives $D^{\alpha}g$ with $|\alpha|=r$, and $|D^r g|^2= \sum_{|\alpha|=r}|D^{\alpha}g|^2$, where $r \geq 0$ is an integer. Also, $C(\Omega )$ is the space of real-valued, continuous functions with domain $\Omega$. Here, we are using the standard multi-index notation. Also, we let both $\nabla g$ and $Dg$ denote the gradient of $g$. \begin{theorem} \label{T3.1} Let $f(\mathbf{x}) \in C(\Omega)\cap H^{s-1}(\Omega)$, $\mathbf{v}(\mathbf{x}) \in C^2(\Omega)\cap H^{s+1}(\Omega)$, and let $a(u)$ be a smooth, positive function of $u$ for $u \in \bar{G}$, where $G \subset \mathbb{R}$ is a bounded interval. We require that the given data $u(\mathbf{x}_0)$ satisfy $u(\mathbf{x}_0)\in G$, where $\mathbf{x}_0\in\Omega$ and where $\Omega =\mathbb{T}^N$, the $N$-dimensional torus, with $N=2,3$. There exist positive constants $\delta_0$, $\delta_1$, and $\delta_2$, and an interval $G_1\subset G$, such that if $\big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2 \| f\|_{s-1}^2 \leq \delta_0$, and $|\nabla \cdot \mathbf{v}|_{L^\infty} \leq \delta_1$, and $\max\{1,|\mathbf{v}|_{L^\infty}^2\}\| f\|_{s-1}^2 \leq \delta_2$, and $\| D\mathbf{v}\|_{s} \leq 1/2$, then there exists a classical solution $u(\mathbf{x})$ to \eqref{1.1}-\eqref{1.2}. And $u(\mathbf{x}) \in \bar{G_1}$ for all $\mathbf{x}\in \Omega$. Here, we define $|\frac{da}{du} |_{s,\bar{G}_1}=\max\{\big|\frac{d^{j+1} a}{d u^{j+1}} (u_{*}) \big|: u_{*} \in \bar{G}_1, 0 \leq j \leq s \}$, where $s>\frac N2+1$. The solution $u(\mathbf{x}) \in \bar{G_1}$ will be unique if $a''(u_{*})\leq \frac{1}{a(u_{*})}(a'(u_{*}))^2$ for all $u_{*} \in \bar{G_1}$. The regularity of the solution is $u \in C^2(\Omega)\cap H^{s+1}(\Omega)$. \end{theorem} \begin{proof} We will construct the solution of the problem for \eqref{1.1}-\eqref{1.2} through an iteration scheme. To define the iteration scheme, we will let the sequence of approximate solutions be $\{u_k\}_{k=1}^{\infty}$. Set $u_0=u(\mathbf{x}_0)$. For $k=0,1,2,\dots$, construct $u_{k+1}$ from the previous iterate $u _k$ by solving \begin{gather} -\nabla \cdot( a(u_k)\nabla u _{k+1})+\mathbf{v}\cdot \nabla u^{k+1} = f, \label{3.1} \\ u_{k+1}(\mathbf{x}_0)=u(\mathbf{x}_0), \label{3.2} \end{gather} Existence of a sufficiently smooth solution to \eqref{3.1}, \eqref{3.2} for fixed $k$ is proven in Appendix A. The a priori estimates used in the proof are proven in Appendix B. We proceed now to prove convergence of the iterates as $k\to \infty$ to a unique classical solution of \eqref{1.1}-\eqref{1.2}. We fix an interval $G_1 \subset G$ by defining $G_1 =\{u_{*} \in G : |u_{*} -u _0|_{L^\infty } < R \}$, where $R = \mathop{\rm dist}(u_0,\partial{G})$. We fix a positive constant $c_1$ such that $a(u_{*} )> c_1$ for all $u_{*}\in \bar{G}_1$. Using a proof by induction on $k$, we assume that $u_{k}(\mathbf{x})\in \bar{G_1}$ for all $\mathbf{x}\in \Omega$, and then later we will show that $u_{k+1}(\mathbf{x})\in \bar{G_1}$ for all $\mathbf{x}\in \Omega$. \end{proof} \begin{proposition} \label{P3.1} Assume that the hypotheses of Theorem \ref{T3.1} hold. Assume that $|\nabla \cdot \mathbf{v}|_{L^\infty} \leq \frac{c_1}{C_*}$, where $C_*$ is the constant from Poincar\'{e}'s inequality $\|\bar{u} \|_0^2\leq C_*\|\nabla u \|_0^2$, and where $\bar{u}(\mathbf{x})=u(\mathbf{x})-\frac{1}{|\Omega|} \int_{\Omega} u(\mathbf{x}) d\mathbf{x}$. There exist constants $C_4$, $C_5$, $C_1$, $L$ such that if $\big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2\| f\|_{s-1}^2 \leq \frac{1}{C_4}$, and if $\max\{1,|\mathbf{v}|_{L^\infty}^2\}\| f\|_{s-1}^2 \leq \frac {R^2}{C_5^2}$, and if $\| D\mathbf{v}\|_{s} \leq \frac 12$, where $s> \frac N2 +1$, then the following hold for $k=1,2,3\dots$ \begin{gather} \|\nabla u_k\|_{s}^2 \leq 2 C_1\| f\|_{s-1}^2, \label{e3.3}\\ |u_{k} - u_0|_{L^{\infty}} \leq R ,\label{e3.5} \\ \|u_k\|_{s+1}^2 \leq L, \label{e3.4} \\ \sum_{k=0}^{\infty}| |u _{k+1}-u _k\|_{s+1}^2 < \infty \label{e3.6} \end{gather} Here, $R = \mathop{\rm dist}(u_0,\partial{G})$ and $C_1$ is the constant in \eqref{B.27} from Lemma \ref{LB.6} in Appendix B . \end{proposition} \begin{proof} The proof is done by induction on $k$. We show only the inductive step. We will derive estimates for $u _{k+1}$, and then use these estimates to show that if $u _k$ satisfies the estimates \eqref{e3.3}, \eqref{e3.5}, \eqref{e3.4} then $u _{k+1}$ also satisfies the same estimates. We will prescribe $L$ a priori, independent of $k$ so that \eqref{e3.4} holds for all $k\geq 1$. We assume by the induction hypothesis that $u_k(\mathbf{x}) \in \bar{G_1}$, and then we will show that $u_{k+1}(\mathbf{x}) \in \bar{G_1}$, for all $\mathbf{x}\in \mathbb{T}^N$. In the estimates below, we use $C$ to denote a generic constant whose value may change from one relation to the next. Recall that we let both $\nabla g$ and $Dg$ denote the gradient of $g$. \smallskip \noindent\textbf{Estimate for $\|\nabla u _{k+1}\|_s^2$:} We begin by applying estimate \eqref{B.27} from Lemma \ref{LB.6} in Appendix B to equation \eqref{3.1}, which yields $$\|\nabla u _{k+1}\|_{s}^2 \leq C_1 \Big[\sum_{j=0}^{s}(\max\{\|D(a(u _k))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\})^{j}\Big] \|f\|_{s-1}^2 \label{e3.7}$$ where $s_1=\max\{s-1,s_0\}$, and $s_0=[ \frac N2]+1=2$, and $s>\frac N2+1$, for $N=2,3$, so $s\geq 3$ and $s_1=s-1$. We consider two cases: when $\max\{\|D(a(u _k))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D(a(u _k))\|_{s_1}^2$, and when $\max\{\|D(a(u _k))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D\mathbf{v}\|_{s_1}$. \textbf{Case 1}: Suppose that $\max\{\|D(a(u _k))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D(a(u _k))\|_{s_1}^2$ in \eqref{e3.7}. To estimate the term $\|D(a(u _k))\|_{s_1}^2$, we apply the Sobolev space inequality \eqref{3.3} from Lemma \ref{LB.1} in Appendix B, which yields the following: \begin{align} \|D(a(u _k))\|_{s_1}^2&= \sum_{0 \leq r\leq s_1}\|D^r(D(a(u_k)))\|_0^2 = \sum_{0 \leq r\leq s_1}\|D^{r+1}(a(u_k))\|_0^2 \nonumber \\ &\leq \sum_{0 \leq r\leq s_1} \Big [C \big|\frac{d a}{d u}\big|_{r,\bar{G}_1}^2(1+|u_k|_{L^\infty})^{2r}\|\nabla u_k\|_{r}^2\Big] \nonumber \\ &\leq C \big|\frac{d a}{d u}\big|_{s_1,\bar{G}_1}^2(1+|u_k|_{L^\infty})^{2s_1}\|\nabla u_k\|_{s_1}^2 \nonumber \\ &\leq C \big|\frac{d a}{d u} \big|_{s_1,\bar{G}_1}^2(1+|u_k-u_0|_{L^\infty}+|u_0|_{L^\infty})^{2s_1}\|\nabla u_k\|_{s_1}^2 \nonumber \\ &\leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+|u(\mathbf{x}_0)|)^{2s}\|\nabla u_k\|_{s_1}^2 \label{e3.8} \\ &\leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+(1+|\Omega|^{1/2}) |u(\mathbf{x}_0)|)^{2s}\|\nabla u_k\|_{s_1}^2 \nonumber \\ &= C_2 \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2\|\nabla u_k\|_{s_1}^2 \nonumber \\ &\leq C_3 \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2\|\nabla u_k\|_{s_1}^2 \nonumber \end{align} where $|\frac{da}{du} |_{s,\bar{G}_1}=\max\{\big|\frac{d^{j+1} a}{d u^{j+1}} (u_{*}) \Big|: u_{*} \in \bar{G}_1, 0 \leq j \leq s \}$, from \eqref{3.3} in Lemma \ref{LB.1}. Here $C_2= C(1+R+(1+|\Omega|^{1/2})|u(\mathbf{x}_0)|)^{2s}$, and we define $C_3=M C_2$, where $M$ is a constant to be defined later and $M\geq 1$. We can assume that $C_2 \geq 1$, so that $C_3 \geq 1$. And we used the fact that $\big|\frac{d a}{d u} \big|_{r,\bar{G}_1} \leq \big|\frac{d a}{d u} \big|_{s_1,\bar{G}_1} \leq \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}$ for $r\leq s_1$ and $s_1 \leq s$. We also used the fact that $|u_k-u_0|_{L^\infty}\leq R$ holds by \eqref{e3.5}, since $u_k(\mathbf{x}) \in \bar{G_1}$ for all $\mathbf{x}\in \mathbb{T}^N$ by the induction hypothesis. We now define the constant $C_4$ to be $C_4= 4 C_3^2 C_1^2$, where $C_1$ is the constant in \eqref{e3.3} and in estimate \eqref{B.27} from Lemma \ref{LB.6} in Appendix B, and where we may assume that $C_1 \geq 1$. We assume that $\Big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2 \| f\|_{s-1}^2 \leq \frac{1}{C_4}$. Substituting \eqref{e3.8} into \eqref{e3.7}, and using estimate \eqref{e3.3}, namely $\|\nabla u_k\|_{s}^2 \leq 2 C_1\| f\|_{s-1}^2$, which holds by the induction hypothesis for $u_k$, and using the fact that $s_1 \leq s$, yields \begin{aligned} \|\nabla u _{k+1}\|_{s}^2 &\leq C_1 \Big[\sum_{j=0}^{s} C_3^j \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^{2j}\|\nabla u_k\|_{s_1}^{2j}\Big] \|f\|_{s-1}^2 \\ &\leq C_1\Big[\sum_{j=0}^{s}C_3^j (2C_1)^{j} \big|\frac{d a}{d u}\big|_{s,\bar{G}_1}^{2j}\|f\|_{s-1}^{2j}\Big] \|f\|_{s-1}^2 \\ &\leq C_1\Big[\sum_{j=0}^{s}C_3^j(2C_1)^j \Big(\frac{1}{C_4} \Big)^j\Big] \|f\|_{s-1}^2 \\ &\leq C_1\Big[\sum_{j=0}^{s} \big(\frac 12\big)^j\Big] \|f\|_{s-1}^2 \\ &\leq 2 C_1 \|f\|_{s-1}^2 \end{aligned}\label{e3.9} where we used the fact that $|\frac{d a}{d u}|_{s,\bar{G}_1}^{2}\| f\|_{s-1}^2 \leq \frac{1}{C_4}$. And we used the fact that $\frac{ 2 C_3 C_1}{C_4} \leq \frac 12$ and $\frac{1}{C_3 C_1}\leq 1$, since $C_4= 4 C_3^2 C_1^2$ and $C_3 C_1 \geq 1$. Therefore \eqref{e3.3} holds for $\|\nabla u _{k+1}\|_s^2$ when $\max\{\|D(a(u _k))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D(a(u _k))\|_{s_1}^2$. \textbf{Case 2}: Suppose that $\max\{\|D(a(u _k))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D \mathbf{v}\|_{s_1}$ in \eqref{e3.7}. From \eqref{e3.7}, we obtain $$\|\nabla u _{k+1}\|_{s}^2 \leq C_1 \Big[\sum_{j=0}^{s}\|D\mathbf{v}\|_{s_1}^{j}\Big] \|f\|_{s-1}^2 \leq C_1 \Big[\sum_{j=0}^{s} \big(\frac 12 \big)^{j}\Big] \|f\|_{s-1}^2 \leq 2 C_1\|f\|_{s-1}^2 \label{e3.10}$$ where we used the fact that $\|D\mathbf{v}\|_{s_1} \leq \|D\mathbf{v}\|_{s} \leq 1/2$. This is the same result as inequality \eqref{e3.9}, and therefore \eqref{e3.3} holds for $\|\nabla u _{k+1}\|_s^2$. \smallskip \noindent\textbf{Estimate for $|u _{k+1} -u _{0}|_{L^{\infty}}$:} To obtain an estimate for $|u _{k+1} -u _{0}|_{L^{\infty}}$, we will use Sobolev's inequality $|h|_{L^{\infty}}^2 \leq C\|h\|_{s_0}^2$ (see, e.g., \cite{a1}, \cite{e1}), where $s_0=[ \frac N2]+1=2$. We will also apply inequality \eqref{B.55} from Lemma \ref{LB.1} in Appendix B, which yields the estimate $\| u _{k+1}-u _0\|_{0}^2 \leq C \|\nabla( u _{k+1}-u _0)\|_{2}^2$. And we will use the estimate \eqref{e3.9}, \eqref{e3.10} just proven for $\|\nabla u_{k+1}\|_{s}^2$ . We then obtain the following inequality: \begin{aligned} |u _{k+1}-u _0|_{L^\infty}^2 &\leq C \| u _{k+1}-u _0\|_{s_0}^2 \leq C \| u _{k+1}-u _0\|_{s+1}^2 \\ &= C\|u _{k+1}-u_0\|_0^2+C \sum_{1 \leq |\alpha|\leq s+1}\|D^{\alpha}( u _{k+1}-u_0)\|_{0 }^2 \\ &\leq C\| u _{k+1}-u _0\|_{0}^2+C\| \nabla (u _{k+1}-u _0)\|_{s}^2 \\ &\leq C\|\nabla(u_{k+1}-u_0)\|_{2}^2 + C\|\nabla(u_{k+1}-u_0)\|_{s}^2 \\ &\leq C\|\nabla(u_{k+1}-u_0)\|_{s}^2 \\ &= C\|\nabla u_{k+1}\|_{s}^2 \\ &\leq 2 C C_1 \|f\|_{s-1}^2 \end{aligned} \label{e3.19} Therefore, from \eqref{e3.19} we have $|u _{k+1}-u _0|_{L^\infty} \leq C_5 \|f\|_{s-1}$, where we define $C_5 = ( 2 C C_1)^{1/2}$ from \eqref{e3.19}. We will assume that $\max\{1,|\mathbf{v}|_{L^\infty}^2\}\|f\|_{s-1}^2$ $\leq \frac{ R^2}{C_5^2}$. It follows that $\|f\|_{s-1} \leq \frac{ R}{C_5}$, and therefore $|u _{k+1}-u _0|_{L^{\infty}}\leq R$. And so \eqref{e3.5} holds for $u_{k+1}$, and $u_{k+1}(\mathbf{x}) \in \bar{G_1}$ for all $\mathbf{x}\in \mathbb{T}^N$. \smallskip \noindent\textbf{Estimate for $\|u _{k+1}\|_{0}^2$ and $\|u _{k+1}\|_{s+1}^2$:} To obtain an $L^2$ estimate for $u _{k+1}$ we apply inequality \eqref{B.56} from Lemma \ref{LB.1} in Appendix B, which yields \begin{aligned} \|u _{k+1}\|_0^2 &\leq C\|u _0\|_0^2+C\|\nabla u _0\|_{2 }^2+C\|\nabla u _{k+1}\|_{2}^2 \\ &\leq C\|u _0\|_0^2+C\|\nabla u _{k+1}\|_{s}^2 \\ &\leq C |\Omega||u (\mathbf{x}_0)|^2+2 C C_1 \|f\|_{s-1}^2 \end{aligned} \label{e3.11} Here we used the estimate for $\| \nabla u _{k+1}\|_s^2$ from the result just proven in \eqref{e3.9}, \eqref{e3.10}. And we used the fact that $u_0$ is a constant. From the estimates \eqref{e3.9}, \eqref{e3.10}, \eqref{e3.11} and using the fact that $\|f\|_{s-1}^2 \leq \frac{ R^2}{C_5^2}$ where $C_5^2 = 2 C C_1$, yields \begin{aligned} \| u _{k+1}\|_{s+1}^2 &= \|u _{k+1}\|_0^2+\sum_{1 \leq |\alpha|\leq s+1}\|D^{\alpha} u_{k+1}\|_{0 }^2 \\ &\leq \|u _{k+1}\|_0^2+C\|\nabla u _{k+1}\|_{s }^2 \\ &\leq C |\Omega||u (\mathbf{x}_0)|^2+2 C C_1 \|f\|_{s-1}^2 \\ &\leq C |\Omega||u (\mathbf{x}_0)|^2+C R^2 \end{aligned} \label{e3.12} We now define the constant $L$ to be $L = C |\Omega||u (\mathbf{x}_0)|^2+C R^2$ from \eqref{e3.12}. Then we have $\|u _{k+1}\|_{s+1}^2\leq L$, and so \eqref{e3.4} holds for $\|u _{k+1}\|_{s+1}^2$. Therefore \eqref{e3.3}, \eqref{e3.5}, \eqref{e3.4} hold for all $k\geq 1$, and $u_k(\mathbf{x}) \in \bar{G_1}$ for all $\mathbf{x}\in \mathbb{T}^N$ and for all $k\geq 1$. \smallskip \noindent\textbf{Estimate for $\|u _{k+1} -u _{k}\|_{s+1}^2$:} Subtracting the equation \eqref{3.1} for $u_k$ from the equation \eqref{3.1} for $u_{k+1}$ yields the following equation $$-\nabla \cdot(a(u_k)\nabla (u _{k+1}-u _k))+\mathbf{v}\cdot \nabla (u^{k+1}-u^k) =\nabla \cdot(( a(u_k)-a(u_{k-1}))\nabla u_k) \label{e3.20}$$ We consider two cases: when $\max\{\|D(a(u _k))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D(a(u _k))\|_{s_1}^2$, and when $\max\{\|D(a(u _k))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D\mathbf{v}\|_{s_1}$. \textbf{Case 1}: Suppose that $\max\{\|D(a(u _k))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D(a(u _k))\|_{s_1}^2$. Applying estimate \eqref{B.27} from Lemma \ref{LB.6} in Appendix B to equation \eqref{e3.20}, and using estimate \eqref{e3.8} for $\|D(a(u _k))\|_{s_1}^2$, and using estimate \eqref{e3.3} for $\|\nabla u_k\|_{s}^2$, yields the following: \begin{aligned} &\|\nabla (u _{k+1}-u _k)\|_{s}^2 \\ & \leq C_1\Big[\sum_{j=0}^s\|D(a(u _k))\|_{s_1}^{2j}\Big] \|\nabla \cdot((a(u _k)-a(u _{k-1}))\nabla u _k)\|_{s-1}^2 \\ &\leq C C_1\Big[\sum_{j=0}^sC_3^j \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^{2j}\|\nabla u_k\|_{s_1}^{2j}\Big] \|(a(u _k)-a(u _{k-1}))\nabla u _k\|_{s}^2 \\ &\leq C C_1\Big[\sum_{j=0}^s C_3^j \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^{2j}\|\nabla u_k\|_{s}^{2j}\Big] \|a(u _k)-a(u _{k-1})\|_s^2\|\nabla u _k\|_{s}^2 \\ &\leq C C_1\Big[\sum_{j=0}^s C_3^j(2 C_1)^{j} \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^{2j}\|f\|_{s-1}^{2j}\Big] \|a(u _k)-a(u _{k-1})\|_s^2 (2 C_1)\|f\|_{s-1}^2 \\ &\leq C C_1\Big[\sum_{j=0}^s C_3^j( 2 C_1)^j \Big(\frac{1}{C_4} \Big)^j \Big] \|a(u _k)-a(u _{k-1})\|_s^2 (2 C_1)\|f\|_{s-1}^2 \\ &\leq C C_1\Big[\sum_{j=0}^s \big(\frac 12 \big)^{j}\Big] \|a(u _k)-a(u _{k-1})\|_s^2(2 C_1)\|f\|_{s-1}^2 \\ &\leq C (2 C_1)^2 \|a(u _k)-a(u _{k-1})\|_s^2\|f\|_{s-1}^2 \end{aligned}\label{e3.13} where we used the Sobolev calculus inequality $\|g h \|_r^2 \leq C\| g\|_r^2\|h\|_r^2$ for $r>\frac N2$, where $C$ is a constant which depends on $r$ (see, e.g., \cite{e1}, \cite{m1}), and we let $r=s$ where $s>\frac N2 +1$. We also used the fact that $|\frac{d a}{d u}|_{s,\bar{G}_1}^{2}\| f\|_{s-1}^2 \leq \frac{1}{C_4}$. And we used the fact that $\frac{2 C_3 C_1}{C_4} \leq \frac 12$ and $\frac{1}{C_3 C_1}\leq 1$, since $C_4= 4 C_3^2 C_1^2$ and $C_3 C_1 \geq 1$. To estimate the term $\|a(u _k)-a(u _{k-1})\|_s^2$, we will apply the Sobolev space inquality \eqref{3.98} from Lemma \ref{LB.1} in Appendix B, which yields \begin{aligned} &\|a(u _k)-a(u _{k-1})\|_s^2\\ & \leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+|u_k|_{L^\infty}+|u_{k-1}|_{L^\infty})^2(\|u_k\|_s+\|u_{k-1}\|_s)^2 \|u_k-u_{k-1}\|_{s}^2 \\ &\leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+|u_k-u_0|_{L^\infty} +|u_{k-1}-u_0|_{L^\infty}+2|u_{0}|_{L^\infty})^2\\ &\quad\times (\|u_k\|_s^2+\|u_{k-1}\|_s^2) \|u_k-u_{k-1}\|_{s}^2 \\ &\leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(2+2R+2|u_{0}|_{L^\infty})^2 (\|u_k\|_{s+1}^2+\|u_{k-1}\|_{s+1}^2) \|u_k-u_{k-1}\|_{s}^2 \\ &\leq C L \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+(1+|\Omega|^{1/2})|u(\mathbf{x}_{0})|)^{2s} \|u_k-u_{k-1}\|_{s}^2 \\ &\leq C L C_2 \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2 \|u_k-u_{k-1}\|_{s}^2 \\ &\leq C L C_3 \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2 \|u_k-u_{k-1}\|_{s}^2 \end{aligned}\label{e3.90} where $C$ depends on $s$, and we used \eqref{e3.4} to estimate $\|u_k\|_{s+1}^2\leq L$ and $\|u_{k-1}\|_{s+1}^2\leq L$. We also used Cauchy's inequality $gh \leq \frac 12 g^2 + \frac 12 h^2$. Here, $C_2$, $C_3$ are the same constants as in \eqref{e3.8}. Then from \eqref{e3.13} and \eqref{e3.90} we obtain \begin{aligned} \|\nabla (u _{k+1}-u _k)\|_{s}^2 &\leq C L C_3(2 C_1)^2 \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2 \|f\|_{s-1}^2 \|u_k-u_{k-1}\|_{s}^2 \\ &\leq C L C_3(2 C_1)^2\Big(\frac{1}{C_4}\Big) \|u _k-u _{k-1}\|_s^2 \\ &= \frac{C L }{C_3} \|u _k-u _{k-1}\|_{s+1}^2 \end{aligned} \label{e3.14} Here we used the fact that $C_4 =4 C_3^2 C_1^2$, and that $\big|\frac{d a}{d u}\big|_{s,\bar{G}_1}^2\| f\|_{s-1}^2 \leq \frac{1}{C_4}$. \textbf{Case 2}: Suppose that $\max\{\|D(a(u _k))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D \mathbf{v}\|_{s_1}$. Applying estimate \eqref{B.27} from Lemma \ref{LB.6} in Appendix B to equation \eqref{e3.20}, and using \eqref{e3.3}, \eqref{e3.90}, and using the proof of \eqref{e3.14}, yields the inequality \begin{aligned} &\|\nabla (u _{k+1}-u _k)\|_{s}^2 \\ & \leq C_1\Big[\sum_{j=0}^s\|D\mathbf{v}\|_{s_1}^{j}\Big] \|\nabla \cdot((a(u _k)-a(u _{k-1}))\nabla u _k)\|_{s-1}^2 \\ &\leq C C_1\Big[\sum_{j=0}^s \big(\frac 12\big)^j \Big] \|a(u _k)-a(u _{k-1})\|_s^2\|\nabla u _k\|_{s}^2 \\ &\leq C ( 2 C_1) \|a(u _k)-a(u _{k-1})\|_s^2 (2 C_1)\|f\|_{s-1}^2 \\ &\leq C L C_3(2 C_1)^2 \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2 \|f\|_{s-1}^2 \|u_k-u_{k-1}\|_{s}^2 \\ &\leq C L C_3(2 C_1)^2 \Big(\frac{1}{C_4}\Big)\|u_k-u_{k-1}\|_{s}^2 \\ &\leq \frac{C L }{C_3} \|u _k-u _{k-1}\|_{s+1}^2 \end{aligned}\label{e3.91} which is the same result as \eqref {e3.14}. Here, we used the facts that $\big|\frac{d a}{d u}\big|_{s,\bar{G}_1}^2\| f\|_{s-1}^2 \leq \frac{1}{C_4}$ where $C_4= 4 C_3^2 C_1^2$, and that $\|D\mathbf{v}\|_{s_1}\leq \|D\mathbf{v}\|_{s} \leq \frac 12$. To obtain an $L^2$ estimate for $u _{k+1}-u_k$, we apply inequality \eqref{B.55} from Lemma \ref{LB.1} in Appendix B, which yields $$\|u _{k+1}-u _k\|_0^2 \leq C\|\nabla (u_{k+1}-u _k)\|_2^2 \leq C\|\nabla (u_{k+1}-u _k)\|_s^2 \label{e3.15}$$ From \eqref{e3.14}--\eqref{e3.15}, we obtain \begin{aligned} \|u _{k+1}-u _k\|_{s+1}^2 &= \|u _{k+1}-u_k\|_0^2+\sum_{1 \leq |\alpha|\leq s+1}\|D^{\alpha}( u _{k+1}-u_k)\|_{0 }^2 \\ &\leq \|u _{k+1}-u _k\|_0^2+C\|\nabla (u _{k+1}-u _k)\|_s^2 \\ &\leq C\|\nabla(u _{k+1}-u _{k})\|_s^2 \\ &\leq \frac{C L }{C_3} \|u _k-u _{k-1}\|_{s+1}^2 \end{aligned} \label{e3.16} where $L = C |\Omega||u (\mathbf{x}_0)|^2+C R^2$ from \eqref{e3.12} , and $C_2= C(1+R+(1+|\Omega|^{1/2})|u(\mathbf{x}_0)|)^{2s}$ and $C_3=MC_2$ from \eqref{e3.8}. It follows that $\frac{C L }{ C_3}\leq \frac{C}{M}$, where $C$ depends on $s$ and $s\geq 3$. We now define the constant $M$ to be large enough so that $\frac{C}{M}<1$. Then from \eqref{e3.16}, we have $$\sum_{k=0}^{\infty}\|u _{k+1}-u _k\|_{s+1}^2< \infty \label{e3.18}$$ which is the inequality \eqref{e3.6} to be proven. This completes the proof of Proposition \ref{P3.1}. \end{proof} We now complete the proof of Theorem \ref{T3.1}. From Lemma \ref{LA.4} in Appendix A, we know that $u_k \in C^2(\Omega)\cap H^{s+1}(\Omega)$ for each $k\geq 1$, where $s> \frac N2 +1$. From \eqref{e3.4} in Proposition \ref{P3.1} and from Sobolev's inequality $|h|_{L^{\infty}}^2 \leq C\|h\|_{s_0}^2$ (see, e.g., \cite{a1}, \cite{e1}), where $s_0=[ \frac N2]+1=2$, we know that $\{u_k\}_{k=1}^{\infty}$ is bounded in $C^2(\Omega)\cap H^{s+1}(\Omega)$. And from \eqref{e3.6} in Proposition \ref{P3.1}, it follows that $\|u _{k+1} -u _{k}\|_{s+1} \rightarrow 0$ as $k\to \infty$. We conclude that there exists $u \in C^2(\Omega)\cap H^{s+1}(\Omega)$ such that $\|u _{k}-u \|_{s+1}\rightarrow 0$ as $k\to \infty$. From Lemma \ref{LA.4} in Appendix A, we know that $u_{k+1}$ is a solution of the linear equation \eqref{3.1} for each $k\geq 0$, and $u_{k+1}(\mathbf{x}_0)=u(\mathbf{x}_0)$ for each $k\geq 0$, and so it follows that $u$ is a solution of the quasilinear equation \eqref{1.1}, and $u$ satisfies \eqref{1.2}. To prove uniqueness of the solution, let us assume that $u_1(\mathbf{x})$, $u_2(\mathbf{x})$ are two solutions of \eqref{1.1}-\eqref{1.2}, and $u_1(\mathbf{x})\in \bar{G_1}$ and $u_2(\mathbf{x})\in \bar{G_1}$ for all $\mathbf{x}\in \mathbb{T}^N$. We will show that there exists a constant $C_7$, such that if $\Big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2\| f\|_{s-1}^2 \leq \frac{1}{C_7}$, and if $\|D\mathbf{v}\|_{s_1}\leq \frac 12$, and if $a''(u_{*})\leq \frac{1}{a(u_{*})}(a'(u_{*}))^2$ for all $u_{*} \in \bar{G_1}$, and if $\max\{1,|\mathbf{v}|_{L^\infty}^2\}\|f\|_{s-1}^2$ $\leq \frac{R^2}{C_5^2}$, and if $|\nabla \cdot \mathbf{v}|_{L^\infty} \leq \frac{c_1}{C_*}$, where $C_5$, $c_1$, $C_*$ are the constants from the proof of Proposition \ref{P3.1}, then $u_1=u_2$. Note that since $u_1(\mathbf{x})\in \bar{G_1}$ and $u_2(\mathbf{x})\in \bar{G_1}$, it follows that $|u_1-u_0|_{L^\infty}\leq R$ and $|u_2-u_0|_{L^\infty} \leq R$, and $a(u_1) > c_1$ and $a(u_2) > c_1$, and $a''(u_{1})\leq \frac{1}{a(u_{1})}(a'(u_{1}))^2$ and $a''(u_{2})\leq \frac{1}{a(u_{2})}(a'(u_{2}))^2$. By Lemma \ref{LB.8} from Appendix B applied to equation \eqref{1.1} for $u_1$ and $u_2$, there exist constants $C_7$, $C_8$, such that if $\Big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2\| f\|_{s-1}^2 \leq \frac{1}{C_7}$, then $u_1$, $u_2$ satisfy $$\begin{gathered} \|\nabla u _1\|_{s}^2 \leq 2 C_8\|f\|_{s-1}^2, \\ \|\nabla u _2\|_{s}^2 \leq 2 C_8\|f\|_{s-1}^2 \end{gathered}\label{e3.82}$$ From Lemma \ref{LB.8} in Appendix B, the constant $C_7= 4 C_0^2 C_3^2 C_1^2 K_1^2$, and the constant $C_8=C_0 C_1 K_1$ so that we have $C_7=4 C_3^2 C_8^2$, and $C_0$ is a constant which depends on $s$, $c_1$, and the constant $K_1=\max\{1,|\mathbf{v}|_{L^\infty}^2 \}$. We may assume that $C_0\geq 1$, so that $C_1 \leq C_8$. And we have $\| u _1\|_{0}^2 \leq |\Omega| |u_1|_{L^\infty}^2$ $\leq 2|\Omega|( |u_1-u_0|_{L^\infty}^2+ |u_0|_{L^\infty}^2)$ $\leq 2|\Omega|(R^2+ |u(\mathbf{x}_0)|^2)$. So $\| u _1\|_{s+1}^2 \leq \| u _1\|_{0}^2 +C\|\nabla u _1\|_{s}^2$ $\leq 2|\Omega|(R^2+ |u(\mathbf{x}_0)|^2)+ 2 C C_8\|f\|_{s-1}^2$. It follows that $u_1 \in C^2(\Omega)\cap H^{s+1}(\Omega)$. Similarly, $u_2 \in C^2(\Omega)\cap H^{s+1}(\Omega)$. Here, we used Sobolev's inequality $|h|_{L^{\infty}}^2 \leq C\|h\|_{s_0}^2$, where $s_0=[ \frac N2]+1=2$. Subtracting \eqref{1.1} for $u_1$ from \eqref{1.1} for $u_{2}$ yields the equation $$-\nabla \cdot(a(u_1)\nabla (u _{2}-u_1))+\mathbf{v}\cdot \nabla (u_{2}-u_{1}) =\nabla \cdot(( a(u_2)-a(u_{1}))\nabla u_2) \label{e3.83}$$ To obtain an estimate for $\|u_{2}-u_1\|_{s+1}^2$, we repeat the proof of the estimate for $\|u_{k+1}-u_k\|_{s+1}^2$ from \eqref{e3.13}-\eqref{e3.16}, and apply this proof to \eqref{e3.83}. We use inequality \eqref{B.55} from Lemma \ref{LB.1} in Appendix B, which yields $\|u_2-u _1\|_0^2 \leq C \|\nabla (u_2-u _1)\|_2^2$, and we use inequality \eqref{B.27} from Lemma \ref{LB.6} in Appendix B to estimate $\|\nabla(u_2-u_1)\|_s^2$, and we use inequality \eqref{e3.8} to estimate $\|D(a(u _1))\|_{s_1}^2$ and we use inequality \eqref{e3.82} to estimate $\|\nabla u_1\|_{s}^2$ and $\|\nabla u_2\|_{s}^2$. We also use the inequality $\Big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2\| f\|_{s-1}^2 \leq \frac{1}{C_7}$ and the inequality $\|D\mathbf{v}\|_{s_1}\leq \frac 12$. We consider two cases: when $\max\{\|D(a(u _1))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D(a(u _1))\|_{s_1}^2$, and when $\max\{\|D(a(u _1))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D\mathbf{v}\|_{s_1}$. \textbf{Case 1}: Suppose that $\max\{\|D(a(u _1))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D(a(u _1))\|_{s_1}^2$. We obtain \begin{aligned} &\|u_{2}-u_1\|_{s+1}^2\\ &= \|u _{2}-u_1\|_0^2+\sum_{1 \leq |\alpha|\leq s+1}\|D^{\alpha}( u_2-u_1)\|_{0 }^2 \\ &\leq \|u_2-u _1\|_0^2+C\|\nabla (u_2-u_1)\|_s^2 \\ &\leq C\|\nabla(u_2-u_1)\|_2^2+ C\|\nabla(u_2-u_1)\|_s^2 \\ &\leq C\|\nabla(u_2-u_1)\|_s^2 \\ &\leq C C_1\Big[\sum_{j=0}^s (\max\{\|D(a(u_1))\|_{s_1}^2,\|D\mathbf{v}\|_{s_1}\})^{j}\Big] \|\nabla \cdot((a(u_2)-a(u_1))\nabla u_2)\|_{s-1}^2 \\ &\leq C C_8\Big[\sum_{j=0}^s \|D(a(u_1))\|_{s_1}^{2j}\Big] \|a(u_2)-a(u_1)\|_s^2\|\nabla u_2\|_{s}^2 \\ &\leq C C_8\Big[\sum_{j=0}^s C_3^j \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^{2j}\|\nabla u_1\|_{s}^{2j}\Big] \|a(u_2)-a(u_1)\|_s^2\|\nabla u_2\|_{s}^2 \\ &\leq C C_8\Big[\sum_{j=0}^s C_3^j(2 C_8)^{j} \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^{2j}\|f\|_{s-1}^{2j}\Big] \|a(u_2)-a(u_1)\|_s^2(2 C_8)\|f\|_{s-1}^2 \\ &\leq C C_8\Big[\sum_{j=0}^s C_3^j(2 C_8)^j \Big(\frac{1}{C_7} \Big)^j \Big] \|a(u _2)-a(u_1)\|_s^2 (2 C_8)\|f\|_{s-1}^2 \\ &\leq C C_8 \Big[\sum_{j=0}^s \big(\frac 12\big)^j\Big] \|a(u_2)-a(u_1)\|_s^2 (2C_8)\|f\|_{s-1}^2 \\ &\leq C (2C_8)^2 \|a(u_2)-a(u_1)\|_s^2\|f\|_{s-1}^2 \end{aligned}\label{e3.84} where we used that $C_1 \leq C_8$, and $C_7= 4 C_3^2 C_8^2$, and $\frac{ 2 C_3 C_8}{C_7}\leq \frac 12$, since $C_3 C_8 \geq 1$. From the third line in the proof of estimate \eqref{e3.90}, we have the inequality \begin{aligned} &\|a(u _2)-a(u _{1})\|_s^2\\ & \leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(2+2R+2|u(\mathbf{x}_{0})|)^2 (\|u_2\|_{s+1}^2+\|u_1\|_{s+1}^2) \|u_2-u_1\|_{s}^2 \\ &\leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+|u(\mathbf{x}_{0})|)^2(\|u_2\|_{0}^2 +C\|\nabla u_2\|_{s}^2+\|u_1\|_{0}^2\\ &\quad +C\|\nabla u_1\|_{s}^2) \|u_2-u_1\|_{s}^2 \end{aligned}\label{e3.85} By inequality \eqref{B.56} from Lemma \ref{LB.1} in Appendix B, we have the estimate $\|u _{2}\|_0^2 \leq C\|u _0\|_0^2+C\|\nabla u _0\|_{2 }^2+C\|\nabla u _{2}\|_{2}^2$ $\leq C |\Omega||u(\mathbf{x}_{0})|^2 +C\|\nabla u _{2}\|_{s}^2$. And a similar inequality holds for $\|u_1\|_{0}^2$. Substituting these $L^2$ estimates into \eqref{e3.85} yields \begin{aligned} &\|a(u _2)-a(u _{1})\|_s^2\\ &\leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+|u(\mathbf{x}_{0})|)^2(|\Omega||u(\mathbf{x}_{0})|^2 +\|\nabla u_2\|_{s}^2+\|\nabla u_1\|_{s}^2) \|u_2-u_1\|_{s}^2 \\ &\leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+|u(\mathbf{x}_{0})|)^2(|\Omega||u(\mathbf{x}_{0})|^2 +4 C_8\|f\|_{s-1}^2) \|u_2-u_1\|_{s}^2 \\ &\leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+|u(\mathbf{x}_{0})|)^2 \Big(|\Omega||u(\mathbf{x}_{0})|^2 +4 C_8\Big( \frac{R^2}{K_1 C_5^2}\Big)\Big) \|u_2-u_1\|_{s}^2 \\ &= C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+|u(\mathbf{x}_{0})|)^2 \Big(|\Omega||u(\mathbf{x}_{0})|^2 +\Big(\frac{ 4 C_0 C_1 K_1 R^2}{2 K_1 C C_1}\Big)\Big) \|u_2-u_1\|_{s}^2 \\ &\leq C \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+(1+|\Omega|^{1/2})|u(\mathbf{x}_{0})|)^{2s}(|\Omega||u(\mathbf{x}_{0})|^2 + R^2) \|u_2-u_1\|_{s}^2 \\ &\leq C C_3 L \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2 \|u_2-u_1\|_{s}^2 \end{aligned}\label{e3.86} where we used $\|f\|_{s-1}^2 \leq \frac{ R^2}{K_1 C_5^2}$, where $K_1=\max\{1,|\mathbf{v}|_{L^\infty}^2\}$ and $C_5 = (2 C C_1)^{1/2}$. And we used the fact that $C_8=C_0 C_1 K_1$, where $C_0$ depends on $s$, $c_1$. And we used inequality \eqref{e3.82} to estimate $\|\nabla u_1\|_{s}^2$ and $\|\nabla u_2\|_{s}^2$. Also, $L = C |\Omega||u (\mathbf{x}_0)|^2+C R^2$ from \eqref{e3.12}, and $C_2= C(1+R+(1+|\Omega|^{1/2})|u(\mathbf{x}_0)|)^{2s}$ and $C_3=M C_2$ from \eqref{e3.8}, where $M \geq 1$. Substituting \eqref{e3.86} into \eqref{e3.84} yields \begin{aligned} \|u_{2}-u_1\|_{s+1}^2 &\leq C C_3 L (2 C_8)^2 \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2\|f\|_{s-1}^2 \|u_2-u_1\|_{s}^2 \\ &\leq C C_3 L (2 C_8)^2 \Big(\frac {1}{C_7}\Big) \|u_2-u_1\|_{s}^2 \\ &= \Big(\frac{C L}{C_3 }\Big)\|u_2-u_1\|_{s+1}^2 \end{aligned}\label{e3.87} where we used the fact that $C_7= 4 C_3^2 C_8^2$ and $\big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2\| f\|_{s-1}^2 \leq \frac{1}{C_7}$ . \textbf{Case 2}: Suppose that $\max\{\|D(a(u _1))\|_{s_1}^2, \|D\mathbf{v}\|_{s_1}\}=\|D \mathbf{v}\|_{s_1}$. Repeating the proof of \eqref{e3.84}--\eqref{e3.87} yields the following: \begin{align*} &\|u_{2}-u_1\|_{s+1}^2\\ &\leq C C_1\Big[\sum_{j=0}^s (\max\{\|D(a(u_1))\|_{s_1}^2,\|D\mathbf{v}\|_{s_1}\})^{j}\Big] \|\nabla \cdot((a(u_2)-a(u_1))\nabla u_2)\|_{s-1}^2 \\ &\leq C C_8\Big[\sum_{j=0}^s \|D \mathbf{v}\|_{s_1}^{j}\Big] \|a(u_2)-a(u_1)\|_s^2\|\nabla u_2\|_{s}^2 \\ &\leq C C_8\Big[\sum_{j=0}^s \big(\frac 12\big)^j \Big] \|a(u_2)-a(u_1)\|_s^2 (2 C_8)\|f\|_{s-1}^2 \\ &\leq C (2 C_8)^2 \|a(u_2)-a(u_1)\|_s^2\|f\|_{s-1}^2 \\ &\leq \Big(\frac{C L}{ C_3 }\Big) \|u_2-u_1\|_{s+1}^2 \end{align*} which is the same estimate as \eqref{e3.87}. Recall that $L = C |\Omega||u (\mathbf{x}_0)|^2+C R^2$ from \eqref{e3.12} , and $C_2= C(1+R+(1+|\Omega|^{1/2})|u(\mathbf{x}_0)|)^{2s}$ and $C_3=M C_2$ from \eqref{e3.8}. It follows that $\frac{C L }{C_3}\leq \frac{C}{M}$, where $C$ depends on $s$. As in the proof of \eqref{e3.18}, we define the constant $M$ to be large enough so that $\frac{C}{M}<1$. It follows that $| |u_{2}-u_1\|_{s+1}^2=0$, and therefore $u_1=u_2$ and the solution is unique. This completes the proof of Theorem \ref{T3.1}. Note that $\delta_0= \min\{\frac{1}{C_4}, \frac{1}{C_7}\}$, and $\delta_1=\frac{c_1}{C_*}$, and $\delta_2=\frac { R^2}{C_5^2}$ in the statement of Theorem \ref{T3.1} in which we assume that $\Big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2 \| f\|_{s-1}^2 \leq \delta_0$, and $|\nabla \cdot \mathbf{v}|_{L^\infty} \leq \delta_1$ , and $\max\{1,|\mathbf{v}|_{L^\infty}^2\}\| f\|_{s-1}^2 \leq \delta_2$, and $\|D\mathbf{v}\|_s \leq \frac 12$. And $\delta_0$, $\delta_1$, $\delta_2$, $C_4$, $C_5$, $C_7$ depend on $s$, $c_1$, $R$, $|\Omega|$, and $|u (\mathbf{x}_0)|$. %\end{proof} \appendix \section{Existence for the linear equation} In this section, we present the proof of the existence of a solution to the linear problem \eqref{3.1}, \eqref{3.2}. \begin{lemma}\label{LA.4} Let $a_1 \in C^1(\Omega)\cap H^{s}(\Omega)$, $f \in C(\Omega)\cap H^{s-1}(\Omega)$, $\mathbf{v} \in C^1(\Omega)\cap H^{s}(\Omega)$ be given functions, where $a_1(\mathbf{x})> c_1$ for some positive constant $c_1$, for $\mathbf{x}\in\Omega$, $\Omega= \mathbb{T}^N$, $N=2$ or $N=3$. We assume that $|\nabla \cdot \mathbf{v}|_{L^\infty} \leq \frac{c_1}{ C_*}$, where $C_*$ is the constant from Poincar\'{e}'s inequality $\|\bar{u} \|_0^2\leq C_*\|\nabla u \|_0^2$, and where $\bar{u}(\mathbf{x})=u(\mathbf{x})-\frac{1}{|\Omega|}\int_{\Omega} u(\mathbf{x}) d\mathbf{x}$. Then there is a unique classical solution $u \in C^2(\Omega)\cap H^{s+1}(\Omega)$ of \begin{gather} -\nabla\cdot (a_1\nabla u)+\mathbf{v}\cdot \nabla u= f, \label{A.1}\\ u(\mathbf{x}_0)=u_0, \label{A.2} \end{gather} where $u_0$ is a given constant and $\mathbf{x}_0 \in \Omega$ is a given point, and where $s>\frac N2+1$. \end{lemma} \begin{proof} The operator in \eqref{A.1} is linear with $a_1(\mathbf{x})> c_1$ for $\mathbf{x}\in\Omega$. The existence of a zero-mean solution $\bar{u}(\mathbf{x})$ of equation \eqref{A.1} follows from the standard theory for elliptic equations, specifically, the Lax-Milgram Lemma (see, e.g., \cite{e2}). We then define the chosen solution $u(\mathbf{x})$ to \eqref{A.1}, \eqref{A.2} to be $u(\mathbf{x})= \bar{u}(\mathbf{x})-\bar{u}(\mathbf{x}_0)+u_0$. We remark that the condition for the Lax-Milgram Lemma that $\|\bar{u}\|_1^2 \leq C B[\bar{u},\bar{u}]$, where $B[\bar{u},\bar{u}]=(a_1\nabla \bar{u},\nabla \bar{u})+(\mathbf{v}\cdot \nabla \bar{u},\bar{u})$, and where $\bar{u}(\mathbf{x})=u(\mathbf{x})-\frac{1}{|\Omega|}\int_{\Omega} u(\mathbf{x}) d\mathbf{x}$, follows from the following inequality: \begin{align*} (c_1\nabla u,\nabla u )&\leq (a_1\nabla u,\nabla u ) \\ & = -(\mathbf{v}\cdot \nabla \bar{u},\bar{u}) +B[\bar{u},\bar{u}] \\ &= \frac 12(\nabla \cdot\mathbf{v}\cdot \bar{ u},\bar{u}) +B[\bar{u},\bar{u}] \\ & \leq \frac 12|\nabla \cdot \mathbf{v}|_{L^\infty}\|\bar{ u} \|_0^2+B[\bar{u},\bar{u}] \\ &\leq \frac 12 C_*|\nabla \cdot \mathbf{v}|_{L^\infty}\|\nabla{ u} \|_0^2+B[\bar{u},\bar{u}] \\ &\leq \frac {c_1}{2} \|\nabla{ u} \|_0^2+B[\bar{u},\bar{u}] \end{align*} % A3 where we used the fact that $|\nabla \cdot \mathbf{v}|_{L^\infty} \leq \frac{c_1}{ C_*}$. And so $\frac 12(c_1\nabla u,\nabla u ) \leq B[\bar{u},\bar{u}]$. From Poincar\'{e}'s inequality $\|\bar{u} \|_0^2\leq C_*\|\nabla u \|_0^2$, we obtain the desired inequality $\|\bar{u}\|_1^2=\|\bar{u}\|_0^2+\|\nabla u\|_0^2 \leq (C_*+1)\|\nabla u\|_0^2\leq \frac{2(C_*+1)}{c_1}B[\bar{u},\bar{u}]$. The regularity of the chosen solution $u(\mathbf{x})$ follows from the estimates \eqref{B.56} from Lemma \ref{LB.1} and \eqref{B.27} from Lemma \ref{LB.6} in Appendix B, applied to equation \eqref{A.1}, which yield: \begin{gather*} \|u \|_{0}^2 \leq C\|u(\mathbf{x}_0)\|_{0}^2+C\|\nabla(u(\mathbf{x}_0))\|_{2 }^2+C\|\nabla u \|_{2 }^2 \leq C |\Omega||u (\mathbf{x}_0)|^2+ C\|\nabla u \|_{s}^2 \\ \|\nabla u \|_{s}^2 \leq C_1\Big[\sum_{j=0}^{s}(\max\{\|Da_1\| _{s_1}^2, \|D\mathbf{v}\|_{s_1}\})^{j}\Big]\|f\|_{s-1}^2 \end{gather*} where $s_1=\max\{s-1,s_0\}=s-1$, and $s_0=[ \frac N2]+1=2$, and $s>\frac N2+1$, so $s\geq 3$. It follows that $u \in C^2(\Omega)\cap H^{s+1}(\Omega)$ by the above estimates and by Sobolev's inequality $|h|_{L^{\infty}}^2 \leq C\|h\|_{s_0}^2$ (see, e.g., \cite{a1,e1}). \end{proof} \section{A priori estimates} Recall that we will be using the Sobolev space $H^s(\Omega )$ (where $s\geq 0$ is an integer) of real-valued functions in $L^2(\Omega )$ whose distribution derivatives up to order $s$ are in $L^2(\Omega )$, with norm given by $\|g\|_s^2=\sum_{|\alpha |\leq s}\int_\Omega |D^\alpha g|^2d\mathbf{x}$ and inner product $(g,h)_s=\sum_{|\alpha |\leq s}\int_\Omega (D^\alpha g)\cdot (D^\alpha h)d\mathbf{x}$. The domain $\Omega$ is the N-dimensional torus $\mathbb{T}^N$, where $N=2$ or $N=3$. Here, we are using the standard multi-index notation. For convenience, we are going to denote derivatives by $g_\alpha =D^\alpha g$. And we will denote the $L^2$ inner product by $(g,h)=\int_\Omega g\cdot h$ $d\mathbf{x}$. We will use $C$ to denote a generic constant whose value may change from one relation to the next. Recall that we let both $\nabla g$ and $Dg$ denote the gradient of $g$. We begin by listing several standard Sobolev space inequalities. \begin{lemma}[Calculus Inequalities] \label{LB.1} \quad (a) Let $g(u)$ be a smooth function on $G$, where $u(\mathbf{x})$ is a continuous function and where $u(\mathbf{x}) \in G_1$ for $\mathbf{x} \in \Omega$ and $G_1 \subset G$ and $u \in H^{r}(\Omega )\cap L^{\infty}(\Omega)$. Then for $r \geq 1$, $$\|D^r(g(u))\|_0 \leq C \big|\frac{d g}{d u} \big|_{r-1,\bar{G}_1}(1+|u|_{L^\infty})^{r-1}\|Du\|_{r-1},\label{3.3}$$ where $|h |_{r,\bar{G}_1}=\max\{\big|\frac{d^j h}{d u^j} (u_{*}) \Big|: u_{*} \in \bar{G}_1, 0 \leq j \leq r \}$, and where $C$ depends on $r$, $\Omega$. (b) And $$\|g(u)-g(v)\|_r \leq C \big|\frac{d g}{d u} \big|_{r,\bar{G}_1}(1+|u|_{L^\infty}+|v|_{L^\infty}) (\|u\|_r+\|v\|_r)\|u-v\|_{r},\label{3.98}$$ where $C$ depends on $r$, $\Omega$. (c) If $Dg\in H^{r_1}(\Omega )$, $h\in H^{r-1}(\Omega )$, where $r_1=\max\{r-1,s_0\}$, $s_0=[ \frac N2]+1$, then for any $r\geq 1$, $g,h$ satisfy the estimate $$\|D^\alpha (gh)-gD^\alpha h\|_0\leq C\|Dg\|_{r_1}\|h\|_{r-1}, \label{B.75}$$ where $r=|\alpha |$, and the constant $C$ depends on $r$, $\Omega$. (d) Let $v$, $w$ be $C^1(\Omega)\cap H^{3}(\Omega)$ functions on a bounded, open, connected, convex domain $\Omega$. And let $v(\mathbf{x}_0)=w(\mathbf{x}_0)$ at a point $\mathbf{x}_0 \in \Omega$. Then $v-w$ and $v$ satisfy the estimates \begin{gather} \|v -w\|_{0}^2\leq C\|\nabla (v -w)\|_{2}^2, \label{B.55}\\ \|v \|_{0}^2 \leq C\|w\|_{0}^2+C\|\nabla w\|_{2 }^2+C\|\nabla v \|_{2 }^2 \label{B.56} \end{gather} Here $C$ is a constant which depends on $\Omega$. \end{lemma} Proofs of the inequalities (a), (b) may be found, for example, in \cite{k1}, \cite{m2}. Proof of inequalities (c), (d) may be found in \cite{DD1}. Inequalities (a), (b) also appear in \cite{e1}. Lemmas \ref{LB.6} and \ref{LB.8} provide the key a priori estimates used in the proof of the theorem. \begin{lemma} \label{LB.6} Let $a_1(\mathbf{x})$, $\mathbf{v}(\mathbf{x})$, and $f(\mathbf{x})$ be sufficiently smooth functions in the following equation $$-\nabla \cdot (a_1 \nabla u)+\mathbf{v}\cdot \nabla u = f, \label{B.5}$$ where $a_1(\mathbf{x})>c_1$, for some positive constant $c_1$, and for all $\mathbf{x}\in \Omega$, with $\Omega=\mathbb{T}^N$, and $N=2$ or $N=3$. We assume that $|\nabla \cdot \mathbf{v}|_{L^\infty} \leq \frac{c_1}{ C_*}$, where $C_*$ is the constant from Poincar\'{e}'s inequality $\|\bar{u} \|_0^2\leq C_*\|\nabla u \|_0^2$, and where $\bar{u}(\mathbf{x})=u(\mathbf{x})-\frac{1}{|\Omega|}\int_{\Omega} u(\mathbf{x}) d\mathbf{x}$. Then we obtain the inequalities: \begin{gather} \|\nabla u\|_0^2 \leq C\|f\|_{0}^2, \label{B.30}\\ \|\nabla u\|_{r}^2 \leq C \max\{\|Da_1\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\} \|\nabla u \|_{r-1}^2 +C\|f\|_{r-1}^2, \label{B.31} \\ \|\nabla u \|_{r}^2 \leq C_1\Big[\sum_{j=0}^{r}(\max\{\|Da_1\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\})^{j}\Big]\|f\|_{r-1}^2 \label{B.27} \end{gather} where $r \geq 1$, where $r_1=\max \{r-1,s_0\}$, and where $s_0=[ \frac N2]+1=2$. Here constant $C$ in \eqref{B.30} depends on $c_1$, and the constant $C$ in \eqref{B.31} depends on $r$, $c_1$, and the constant $C_1$ in \eqref{B.27} depends on $r$, $c_1$. \end{lemma} \begin{proof} First, we obtain an $L^2$ estimate. Integrating equation \eqref{B.5} by parts with $\bar{u}$, where $\bar{u}(\mathbf{x})=u(\mathbf{x})-\frac 1{|\Omega |}\int_\Omega u(\mathbf{x}) d \mathbf{x}$, yields \begin{aligned} (c_1\nabla u,\nabla u ) &\leq (a_1\nabla u,\nabla u ) = -(\nabla \cdot (a_1\nabla u),\bar{u} ) =-(\mathbf{v}\cdot \nabla u,\bar{u}) +(f,\bar{u}) \\ &= \frac 12(\nabla \cdot\mathbf{v}\cdot \bar{ u},\bar{u}) +(f,\bar{u}) \\ & \leq \frac 12|\nabla \cdot \mathbf{v}|_{L^\infty}\|\bar{ u} \|_0^2+\frac{1}{4\epsilon}\|f\|_0^2+\epsilon\|\bar{ u} \|_0^2 \\ &\leq \frac 12 C_*|\nabla \cdot \mathbf{v}|_{L^\infty}\|\nabla{ u} \|_0^2+\frac{1}{4\epsilon}\|f\|_0^2+\epsilon C_*\|\nabla u \|_0^2 \end{aligned}\label{B.9} where we used Cauchy's inequality with $\epsilon$, namely $gh \leq \frac{1}{4\epsilon}g^2+\epsilon h^2$, and where we used the fact that $a_1(\mathbf{x}) > c_1$. We also used Poincar\'{e}'s inequality (see, e.g., \cite{e1}, \cite{e2}) to estimate $\|\bar{u} \|_0^2\leq C_*\|\nabla u \|_0^2$, where $C_*$ is a constant. We assume that $|\nabla \cdot \mathbf{v}|_{L^\infty} \leq \frac{c_1}{ C_*}$. And we let $\epsilon = \frac{c_1}{4 C_*}$. Then from \eqref{B.9}, we obtain $$\label{B.10} \|\nabla u\|_0^2\leq C\|f\|_{0}^2$$ where $C$ depends on $c_1$. This is the desired inequality \eqref{B.30}. Next, after applying $D^\alpha$ to the equation \eqref{B.5}, we obtain the equation: $$\label{B.11} -\nabla \cdot (a_1\nabla u_\alpha)+\mathbf{v}\cdot \nabla u_{\alpha}=F_\alpha$$ where $F_\alpha =f_{\alpha}+[\nabla \cdot (a_1\nabla u)_\alpha -\nabla \cdot (a_1\nabla u_\alpha )]$ $-[(\mathbf{v}\cdot \nabla u)_{\alpha}-\mathbf{v}\cdot \nabla u_{\alpha}]$. From \eqref{B.11} we obtain \begin{aligned} c_1(\nabla u_\alpha, \nabla u _\alpha ) &\leq(a_1\nabla u_\alpha, \nabla u _\alpha ) \\ &= -(\nabla \cdot(a_1\nabla u_\alpha), u _\alpha) \\ &= -(\mathbf{v}\cdot \nabla u_\alpha,u_\alpha)+(F_\alpha ,u_\alpha) \\ &\leq \frac 12 |\nabla \cdot \mathbf{v}|_{L^\infty}\| u_\alpha \|_0^2+|(F_{\alpha} ,u_{\alpha} )| \\ &\leq C|D \mathbf{v}|_{L^\infty}\|\nabla{ u} \|_{k-1}^2 +|(F_{\alpha} ,u_{\alpha} )| \end{aligned} \label{B.12} where $|\alpha|=k$. Next, we estimate $|(F_{\alpha} ,u _{\alpha} )|$. We use integration by parts, and then apply inequality \eqref{B.75} from Lemma \ref{LB.1}, to obtain the following inequality: \begin{aligned} &|(F_{\alpha} ,u _{\alpha} )|\\ &\leq |( f_{\alpha},u_{\alpha})| +|([\nabla \cdot (a_1\nabla u)_{\alpha} -\nabla \cdot (a_1\nabla u_{\alpha} )],u _{\alpha} )|+|((\mathbf{v}\cdot \nabla u)_{\alpha}-\mathbf{v}\cdot \nabla u_{\alpha},u_{\alpha})| \\ &= |(f_{\alpha-\beta},u_{\alpha+\beta})|+|([(a_1\nabla u)_{\alpha} -a_1\nabla u_{\alpha} ],\nabla u _{\alpha} )|+|((\mathbf{v}\cdot \nabla u)_{\alpha}-\mathbf{v}\cdot \nabla u_{\alpha},u_{\alpha})| \\ &\leq \|f_{\alpha-\beta}\|_0\|u_{\alpha+\beta}\|_0+\|(a_1\nabla u)_{\alpha} -a_1\nabla u_{\alpha} \|_0 \|\nabla u _{\alpha} \|_0\\ &\quad +\|(\mathbf{v}\cdot \nabla u)_{\alpha}-\mathbf{v}\cdot \nabla u_{\alpha}\|_0\|u_{\alpha}\|_{0} \\ &\leq C\|f\|_{k-1}\|\nabla u\|_{k}+ C\|Da_1\|_{k_1}\|\nabla u \|_{k-1}\|\nabla u \|_k + C\|D\mathbf{v}\|_{k_1}\|\nabla u \|_{k-1}^2 \\ &\leq \frac{C}{4\epsilon}\|f\|_{k-1}^2+\epsilon\| \nabla u \|_{k}^2+ \frac{C}{4\epsilon}\|Da_1\| _{k_1}^2\|\nabla u \|_{k-1}^2+\epsilon\| \nabla u \|_k^2+ C\|D\mathbf{v}\|_{k_1}\|\nabla u \|_{k-1}^2 \end{aligned}\label{B.16} where $|\beta| = 1$, $k=|\alpha |$, and $k_1=\max \{k-1,s_0\}$, with $s_0=[\frac N2 ]+1$. Again, we used Cauchy's inequality with $\epsilon$. Substituting \eqref{B.16} into \eqref{B.12}, and adding \eqref{B.12} over $|\alpha |=k\leq r$, including the estimate \eqref{B.9}, we obtain for $r\geq 1$ the estimate $$\|\nabla u \|_r^2 \leq \frac{C}{4\epsilon}(\|Da_1\|_{r_1}^2+ \|D\mathbf{v}\|_{r_1}) \|\nabla u \|_{r-1}^2 +\frac{C}{4\epsilon}\|f\|_{r-1}^2 +\epsilon C\|\nabla u\|_{r}^2 \label{B.17}$$ where $r_1=\max \{r-1,s_0\}$, with $s_0=[\frac N2 ]+1$, and where $C$ depends on $r$, $c_1$. Here we used Sobolev's lemma to obtain $|D \mathbf{v}|_{L^\infty}\leq C\|D\mathbf{v}\|_{s_0}$. Choosing $\epsilon$ sufficiently small yields $$\|\nabla u \|_r^2 \leq C\max\{\|Da_1\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\} \|\nabla u \| _{r-1}^2 +C\|f\|_{r-1}^2 \label{B.18}$$ where $C$ depends on $r$, $c_1$. This is the desired inequality \eqref{B.31}. Applying the inequality \eqref{B.18} to $\|\nabla u \| _{r-1}^2$ which appears on the right-hand side of \eqref{B.18} yields \begin{aligned} \|\nabla u \|_r^2 &\leq C(\max\{\|Da_1\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\})\\ &\quad\times \Big[C(\max\{\|Da_1\|_{r_2}^2,\|D\mathbf{v}\|_{r_2}\}) \|\nabla u \|_{r-2}^2 +C\|f\|_{r-2}^2 \Big] +C\|f\|_{r-1}^2 \\ &\leq C(\max\{\|Da_1\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\})^2 \|\nabla u \|_{r-2}^2\\ &\quad +C(\max\{\|Da_1\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\}) \|f\|_{r-2}^2 +C\|f\|_{r-1}^2 \end{aligned}\label{B.23} where $r_1=\max \{r-1,s_0\}$, $r_2=\max \{r-2,s_0\}$, $r_2 \leq r_1$, with $s_0=[\frac N2 ]+1=2$ for $N=2,3$. Similarly, by applying the estimate \eqref{B.18} to $\|\nabla u \| _{r-j}^2$ for $j=2,3,\dots ,r-1$, which will appear in the term $C(\max\{\|Da_1\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\})^{j}\|\nabla u \| _{r-j}^2$ on the right-hand side of \eqref{B.23}, we obtain \begin{aligned} \|\nabla u \|_r^2 &\leq C\sum_{j=1}^{r-1}(\max\{\|Da_1\|_{r_1}^2, \|D\mathbf{v}\|_{r_1}\})^{j} \|f\|_{r-1-j}^2 \\ &\quad +C(\max\{\|Da_1\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\})^{r} \|\nabla u \|_{0}^2 +C\|f\|_{r-1}^2 \\ &\leq C\Big[\sum_{j=0}^{r-1}(\max\{\|Da_1\|_{r_1}^2, \|D\mathbf{v}\|_{r_1}\})^{j}\Big]\|f\|_{r-1}^2 \\ &\quad +C(\max\{\|Da_1\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\})^{r} \|\nabla u \|_{0}^2 \end{aligned} \label{B.24} Substituting the estimate $\|\nabla u\|_0^2 \leq C\|f\|_{0}^2$ into \eqref{B.24} yields \begin{align*} &\|\nabla u \|_r^2 \\ &\leq C \Big[\sum_{j=0}^{r-1}(\max\{\|Da_1\|_{r_1}^2, \|D\mathbf{v}\|_{r_1}\})^{j}\Big]\|f\|_{r-1}^2 +C(\max\{\|Da_1\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\})^{r}\|f\|_0^2 \\ &\leq C_1\Big[\sum_{j=0}^{r}(\max\{\|Da_1\|_{r_1}^2, \|D\mathbf{v}\|_{r_1}\})^{j}\Big] \|f\|_{r-1}^2 \end{align*} where $C_1$ depends on $r$, $c_1$. This completes the proof. \end{proof} \begin{lemma} \label{LB.8} Let $a(u)$ be a smooth function of $u$, and let $\mathbf{v}(\mathbf{x})$ and $f(\mathbf{x})$ be sufficiently smooth functions in the equation $$\label{B.99} -\nabla \cdot (a(u)\nabla u)+\mathbf{v}\cdot \nabla u =f$$ for $\mathbf{x} \in \Omega$, where $\Omega = \mathbb{T}^N$, $N=2,3$, where $a(u) > c_1$, for some positive constant $c_1$, and where $|u-u_0|_{L^\infty}\leq R$, where $u_0$, $R$ are given constants. We assume that $|\nabla \cdot \mathbf{v}|_{L^\infty} \leq \frac{c_1}{ C_*}$, where $C_*$ is the constant from Poincar\'{e}'s inequality $\|\bar{u} \|_0^2\leq C_*\|\nabla u \|_0^2$, and where $\bar{u}(\mathbf{x})=u(\mathbf{x})-\frac{1}{|\Omega|}\int_{\Omega} u(\mathbf{x}) d\mathbf{x}$. Then there exist constants $C_7$, $C_8$, such that if $\Big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2\| f\|_{s-1}^2 \leq \frac{1}{C_7}$, and if $\|D\mathbf{v}\|_{s} \leq \frac 12$ , and if $a''(u)\leq \frac{1}{a(u)}(a'(u))^2$, then $u$ satisfies the inequality $$\|\nabla u\|_{s}^2 \leq 2 C_8 \| f\|_{s-1}^2 \label{B.88}$$ We define $C_7= 4 C_3^2 C_8^2$ and $C_8=C_0 C_1 K_1$, where $C_1$ is the constant from estimate \eqref{B.27} in Lemma \ref{LB.6}, and where $C_2=C(1+R+(1+|\Omega|^{1/2})|u(\mathbf{x}_0)|)^{2s}$ and $C_3 = M C_2$ are the same constants as in \eqref{e3.8} from Proposition \ref{P3.1}, and $C_0$ is a constant which depends on $s$, $c_1$, where $s > \frac N2+ 1$. We define the constant $K_1=\max\{1,|\mathbf{v}|_{L^\infty}^2 \}$. And we define $|\frac{da}{du} |_{s,\bar{G}_1}=\max\{\big|\frac{d^{j+1} a}{d u^{j+1}} (u_{*}) \Big|: u_{*} \in \bar{G}_1, 0 \leq j \leq s \}$. \end{lemma} \begin{proof} First we obtain estimates for $\|\nabla u\|_{0}^2$, $\|\nabla u\|_{1}^2$, $\|\nabla u\|_{2}^2$, and $\|\nabla u\|_{r}^2$, where $3 \leq r \leq s$. It is necessary to have an estimate for $\|\nabla u\|_{j}^2$ in order to obtain an estimate for $\|\nabla u\|_{j+1}^2$, for $j=0,1,2,\dots ,s-1$. We will apply estimate \eqref{B.27} from Lemma \ref{LB.6} to obtain an estimate for $\|\nabla u\|_{r}^2$, when $3 \leq r \leq s$. From inequality \eqref{B.30} in Lemma \ref{LB.6} applied to equation \eqref{B.99}, we obtain $$\label{e3.24} \|\nabla u\|_0^2\leq C\|f\|_{0}^2$$ where $C$ depends on $c_1$. We now obtain an estimate for $\|\nabla u\|_{1}^2$. Applying $D^\alpha$ to equation \eqref{B.99}, with $|\alpha|=1$, yields $$-\nabla \cdot(a(u)\nabla u_\alpha) = \nabla \cdot ((a(u))_\alpha \nabla u)-(\mathbf{v}\cdot \nabla u)_\alpha+f_\alpha \label{e3.26}$$ Integrating \eqref{e3.26} by parts with $u_\alpha$, where $|\alpha|=1$, and using the fact from equation \eqref{B.99} that $\nabla a(u) \cdot \nabla u =$ $-a(u)\Delta u+\mathbf{v}\cdot \nabla u -f$, yields \begin{aligned} (a(u)\nabla u_\alpha,\nabla u_\alpha) &= -(\nabla \cdot(a(u)\nabla u_\alpha),u_\alpha) \\ &= (\nabla \cdot ((a(u))_\alpha \nabla u),u_\alpha)-((\mathbf{v}\cdot \nabla u)_\alpha, u_\alpha)+ (f_\alpha,u_\alpha) \\ &= -((a(u))_\alpha \nabla u,\nabla u_\alpha) -((\mathbf{v}\cdot \nabla u)_\alpha, u_\alpha)+ (f_\alpha,u_{\alpha}) \\ &= -\frac 12((a(u))_\alpha, (\nabla u \cdot \nabla u)_\alpha)-((\mathbf{v}\cdot \nabla u)_\alpha, u_\alpha) + (f_\alpha,u_{\alpha}) \\ &= -\frac 12(a^{\prime}(u) u_\alpha, (\nabla u \cdot \nabla u)_\alpha) -((\mathbf{v}\cdot \nabla u)_\alpha, u_\alpha)+ (f_\alpha,u_{\alpha}) \\ &= -\frac 12(u_\alpha, ( a^{\prime}(u)\nabla u \cdot \nabla u)_\alpha) +\frac 12( u_\alpha, (a^{\prime}(u))_\alpha(\nabla u \cdot \nabla u)) \\ &\quad -((\mathbf{v}\cdot \nabla u)_\alpha, u_\alpha)+ (f_\alpha,u_{\alpha}) \\ &= -\frac 12(u_\alpha, (\nabla a(u) \cdot \nabla u)_\alpha) +\frac 12( u_\alpha , a''(u) u_\alpha(\nabla u \cdot \nabla u)) \\ &\quad -((\mathbf{v}\cdot \nabla u)_\alpha, u_\alpha)+ (f_\alpha,u_{\alpha}) \\ &= \frac 12(u_\alpha, (a(u)\Delta u-\mathbf{v}\cdot \nabla u+f)_\alpha) +\frac 12((u_\alpha)^2, a''(u)(\nabla u \cdot \nabla u)) \\ &\quad -((\mathbf{v}\cdot \nabla u)_\alpha, u_\alpha) + (f_\alpha,u_{\alpha}) \\ &= -\frac 12(u_{\alpha+\alpha},a(u) \Delta u)+\frac 32((\mathbf{v}\cdot \nabla u), u_{\alpha+\alpha})-\frac 32 (f,u_{\alpha+\alpha}) \\ &\quad +\frac 12((u_\alpha)^2,a''(u) (\nabla u \cdot \nabla u)) \end{aligned}\label{e3.27} Adding \eqref{e3.27} over $|\alpha|=1$ yields \begin{aligned} \sum_{|\alpha|=1} (a(u)\nabla u_\alpha,\nabla u_\alpha) &= -\frac 12\sum_{|\alpha|=1}(u_{\alpha+\alpha},a(u) \Delta u) +\frac 32 \sum_{|\alpha|=1}((\mathbf{v}\cdot \nabla u), u_{\alpha+\alpha})\\ &\quad -\frac 32 \sum_{|\alpha|=1} (f,u_{\alpha+\alpha}) + \frac 12\sum_{|\alpha|=1} ((u_\alpha)^2, a''(u)(\nabla u \cdot \nabla u)) \\ &= -\frac 12 ( \Delta u,a(u) \Delta u) +\frac 32((\mathbf{v}\cdot \nabla u), \Delta u)-\frac 32 (f,\Delta u) \\ &\quad +\frac 12((\nabla u \cdot \nabla u),a''(u) (\nabla u \cdot \nabla u)) \end{aligned} \label{e3.28} Next, we estimate the term $\frac 12((\nabla u \cdot \nabla u),a''(u) (\nabla u \cdot \nabla u))$ in \eqref{e3.28}. We assume that $a''(u)\leq \frac{1}{a(u)}(a'(u))^2$. We then obtain the inequality \begin{align} &\frac 12((\nabla u \cdot \nabla u),a''(u) (\nabla u \cdot \nabla u)) \nonumber \\ &\leq \frac 12 (\frac{1}{a(u)}(a^{\prime}(u))^2(\nabla u \cdot \nabla u), (\nabla u \cdot \nabla u)) \nonumber\\ &= \frac 12((\nabla a(u) \cdot \nabla u), \frac{1}{a(u)}(\nabla a(u) \cdot \nabla u)) \\ &= \frac 12((a(u)\Delta u -\mathbf{v}\cdot \nabla u+f), \frac{1}{a(u)} (a(u)\Delta u-\mathbf{v}\cdot \nabla u+f)) \nonumber\\ &= \frac 12(\Delta u,a(u) \Delta u)+\frac 12(f,\frac{1}{a(u)}f)+\frac 12(\mathbf{v}\cdot \nabla u,\frac{1}{a(u)}\mathbf{v}\cdot \nabla u) \label{e3.29} \\ &\quad +(\Delta u, f)-(\mathbf{v}\cdot \nabla u, \Delta u)-(\mathbf{v}\cdot \nabla u,\frac{1}{a(u)} f) \nonumber \end{align} Substituting \eqref{e3.29} into \eqref{e3.28} yields \begin{aligned} \sum_{|\alpha|=1} (a(u)\nabla u_\alpha,\nabla u_\alpha) &= \frac 12(f,\frac{1}{a(u)}f) +\frac 12(\mathbf{v}\cdot \nabla u,\frac{1}{a(u)}\mathbf{v}\cdot \nabla u) \\ &\quad -\frac 12(\Delta u, f)+\frac 12(\mathbf{v}\cdot \nabla u, \Delta u)-(\mathbf{v}\cdot \nabla u,\frac{1}{a(u)} f) \\ & \leq \frac {1}{c_1}(f,f) + \frac{1}{c_1}|\mathbf{v}|_{L^\infty}^2\|\nabla u\|_0^2+\epsilon(\Delta u, \Delta u) \\ &\quad +\frac {1}{16\epsilon}(f, f) +\frac{1}{16\epsilon}|\mathbf{v}|_{L^\infty}^2\|\nabla u\|_0^2 +\epsilon(\Delta u, \Delta u) \end{aligned}\label{e3.30} where we used Cauchy's inequality with $\epsilon$, and we used the fact that $a(u)>c_1$. We now use the fact that $\sum_{|\alpha|=1} (\nabla u_\alpha,\nabla u_\alpha)$ $= \sum_{|\alpha|=1} ((u_{\alpha+\alpha},\Delta u)$ $=(\Delta u,\Delta u)$. We also use the fact that $a(u )>c_1$, and we define $\epsilon =\frac{c_1}{4}$. Then \eqref{e3.30} becomes \begin{aligned} &\sum_{|\alpha|=1} (c_1\nabla u_\alpha,\nabla u_\alpha)\leq \sum_{|\alpha|=1} (a(u)\nabla u_\alpha,\nabla u_\alpha) \\ & \leq \frac{c_1}{2}(\Delta u, \Delta u)+\frac {5}{ 4c_1}(f, f)+ \frac {5}{4c_1}|\mathbf{v}|_{L^\infty}^2\|\nabla u\|_0^2 \\ &= \frac{c_1}{2}\sum_{|\alpha|=1} ( \nabla u_\alpha,\nabla u_\alpha)+\frac {5}{4 c_1}(f, f)+ \frac{5}{4 c_1}|\mathbf{v}|_{L^\infty}^2\|\nabla u\|_0^2 \label{e3.31} \end{aligned} Subtracting the term $\frac {c_1}{2}\sum_{|\alpha|=1}(\nabla u_\alpha, \nabla u_\alpha)$ on both sides of \eqref{e3.31}, and using inequality \eqref{e3.24}, namely $\|\nabla u\|_0^2\leq C\|f\|_{0}^2$, yields the estimate $$\sum_{|\alpha|=1} (\nabla u_\alpha,\nabla u_\alpha) \leq C\max\{1,|\mathbf{v}|_{L^\infty}^2\}\|f\|_0^2 \label{e3.32}$$ where $C$ depends on $c_1$. Adding the inequalities \eqref{e3.32}, \eqref{e3.24} yields $$\|\nabla u\|_{1}^2=\|\nabla u\|_{0}^2 +\sum_{|\alpha|=1} (\nabla u_\alpha,\nabla u_\alpha) \leq C \max\{1, |\mathbf{v}|_{L^\infty}^2 \}\|f\|_0^2 = C K_1 \|f\|_0^2 \label{e3.33}$$ where $C$ depends on $c_1$, and where we define the constant $K_1=\max\{1, |\mathbf{v}|_{L^\infty}^2 \}$. We now obtain an estimate for $\|\nabla u\|_{2}^2$. Applying $D^\alpha$ to equation \eqref{B.99}, with $|\alpha|=2$, yields \begin{aligned} -\nabla \cdot(a(u)\nabla u_\alpha) &= \nabla \cdot ((a(u))_\alpha \nabla u)+\nabla \cdot ((a(u))_{\alpha-\beta} \nabla u_{\beta}) +\nabla \cdot ((a(u))_{\beta} \nabla u_{\alpha-\beta})\\ &\quad -\mathbf{v}\cdot \nabla u_\alpha-\mathbf{v}_\alpha\cdot \nabla u-\mathbf{v}_{\beta}\cdot \nabla u_{\alpha-\beta}-\mathbf{v}_{\alpha-\beta}\cdot \nabla u_{\beta}+f_\alpha \end{aligned} \label{e3.34} where $|\beta| =1$. Integrating by parts with $u_\alpha$, where $|\alpha| =2$ and $|\beta| =1$, and using inequality \eqref{3.3} from Lemma \ref{LB.1}, yields \begin{align} &((a(u)\nabla u_\alpha,\nabla u_\alpha) \nonumber \\ &=-( \nabla \cdot (a(u)\nabla u_\alpha),u_\alpha) \nonumber \\ &= (\nabla \cdot ((a(u))_\alpha \nabla u),u_\alpha)+(\nabla \cdot ((a(u))_{\alpha-\beta} \nabla u_{\beta}),u_\alpha) \nonumber \\ &\quad + (\nabla \cdot ((a(u))_{\beta} \nabla u_{\alpha-\beta}),u_\alpha)-(\mathbf{v}\cdot \nabla u_\alpha, u_\alpha) \nonumber \\ &\quad- (\mathbf{v}_\alpha\cdot \nabla u, u_\alpha)-(\mathbf{v}_{\beta}\cdot \nabla u_{\alpha-\beta},u_\alpha)-(\mathbf{v}_{\alpha-\beta}\cdot \nabla u_{\beta},u_{\alpha}) +(f_\alpha,u_\alpha) \nonumber \\ &= -((a(u))_\alpha \nabla u,\nabla u_\alpha) -((a(u))_{\alpha-\beta} \nabla u_{\beta},\nabla u_\alpha) \nonumber\\ &\quad- ((a(u))_{\beta} \nabla u_{\alpha-\beta},\nabla u_\alpha)+\frac 12 ((\nabla \cdot\mathbf{v}) u_\alpha, u_\alpha) -(\mathbf{v}_\alpha\cdot \nabla u, u_\alpha) \nonumber\\ &\quad- (\mathbf{v}_{\beta}\cdot \nabla u_{\alpha-\beta},u_\alpha)-(\mathbf{v}_{\alpha-\beta}\cdot \nabla u_{\beta},u_{\alpha}) -(f_{\alpha-\beta},u_{\alpha+\beta}) \nonumber\\ &\leq \|(a(u))_\alpha\|_0|\nabla u|_{L^\infty}\|\nabla u_\alpha\|_0+|(a(u))_{\alpha-\beta}|_{L^\infty}\|\nabla u_{\beta}\|_0\|\nabla u_\alpha\|_0 \nonumber\\ &\quad + |(a(u))_\beta|_{L^\infty}\|\nabla u_{\alpha-\beta}\|_0\|\nabla u_\alpha\|_0+\frac 12|\nabla \cdot \mathbf{v}|_{L^\infty}\| u_\alpha\|_0^2 +|\mathbf{v}_\alpha|_{L^\infty}\| \nabla u\|_0\| u_\alpha \|_0 \nonumber \\ &\quad + |\mathbf{v}_\beta|_{L^\infty}\|\nabla u_{\alpha-\beta}\|_0\|u_{\alpha}\|_0 +|\mathbf{v}_{\alpha-\beta}|_{L^\infty}\| \nabla u_\beta\|_0\| u_{\alpha}\|_0 + \|f_{\alpha-\beta}\|_{0}\|u_{\alpha+\beta}\|_{0} \nonumber \\ &\leq C\|D^2(a(u))\|_0\|\nabla u\|_{2}\|\nabla u_\alpha\|_0+C\|D(a(u))\|_2\|\nabla u\|_{1}\|\nabla u_\alpha\|_0 +C|\nabla \cdot \mathbf{v}|_{L^\infty}\| \nabla u\|_1^2 \nonumber\\ &\quad + C|D^2\mathbf{v}|_{L^\infty}\| \nabla u\|_0\|\nabla u\|_1 +C|D \mathbf{v}|_{L^\infty}\|\nabla u\|_1^2 +C \|\nabla f\|_{0}\|\nabla u_{\alpha}\|_{0} \nonumber \\ &\leq \frac{C}{4\epsilon}\|D^2(a(u))\|_0^2\|\nabla u\|_{2}^2+\epsilon\|\nabla u_\alpha\|_0^2+\frac{C}{4\epsilon}\|D(a(u))\|_2^2\|\nabla u\|_{1}^2+\epsilon\|\nabla u_\alpha\|_0^2 \nonumber\\ &\quad +C|D \mathbf{v}|_{L^\infty}\| \nabla u\|_1^2 + C|D^2\mathbf{v}|_{L^\infty}\| \nabla u\|_0^2+C|D^2\mathbf{v}|_{L^\infty}\|\nabla u\|_1^2 \nonumber\\ &\quad +C|D \mathbf{v}|_{L^\infty}\|\nabla u\|_1^2 + \frac{C}{4\epsilon}\|\nabla f\|_{0}^2+\epsilon\|\nabla u_{\alpha}\|_{0}^2 \nonumber \\ &\leq \frac{C}{4\epsilon}\|D^2(a(u))\|_0^2\|\nabla u\|_{2}^2 +\frac{C}{4\epsilon}\Big[\sum_{0\leq r \leq 2}\|D^{r+1}(a(u))\|_0^2 \Big]\|\nabla u\|_{1}^2 \nonumber \\ &\quad + C(|D\mathbf{v}|_{L^\infty}+|D^2\mathbf{v}|_{L^\infty})\| \nabla u\|_1^2 +C|D^2\mathbf{v}|_{L^\infty}\|\nabla u\|_0^2 +\frac{C}{4\epsilon}\|\nabla f\|_{0}^2+3\epsilon\|\nabla u_{\alpha}\|_{0}^2 \nonumber\\ &\leq \frac{C}{4\epsilon} \big|\frac{d a}{d u} \big|_{1,\bar{G}_1}^2(1+|u|_{L^\infty})^2\|\nabla u\|_1^2 \|\nabla u\|_{2}^2 \nonumber \\ &\quad + \frac{C}{4\epsilon}\Big[\sum_{0\leq r \leq 2 }\big|\frac{d a}{d u} \big|_{r,\bar{G}_1}^2(1+|u|_{L^\infty})^{2r}\|\nabla u\|_r^2 \Big]\|\nabla u\|_{1}^2 \nonumber \\ &\quad + C(|D\mathbf{v}|_{L^\infty}+|D^2\mathbf{v}|_{L^\infty})\| \nabla u\|_1^2 +C|D^2\mathbf{v}|_{L^\infty}\|\nabla u\|_0^2 \nonumber\\ &\quad + \frac{C}{4\epsilon}\|\nabla f\|_{0}^2+3\epsilon\|\nabla u_{\alpha}\|_{0}^2 \label{e3.35} \\ &\leq \frac{C}{4\epsilon} \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+|u|_{L^\infty})^{2s}\|\nabla u\|_1^2 \|\nabla u\|_{2}^2 \nonumber\\ &\quad + \frac{C}{4\epsilon}\big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+|u|_{L^\infty})^{2s}\Big[\sum_{0\leq r \leq 2 }\|\nabla u\|_r^2 \Big]\|\nabla u\|_{1}^2 \nonumber\\ &\quad + C(\|D\mathbf{v}\|_{s_0}+\|D^2\mathbf{v}\|_{s_0})\| \nabla u\|_1^2 +C\|D^2\mathbf{v}\|_{s_0}\|\nabla u\|_0^2+ \frac{C}{4\epsilon}\|\nabla f\|_{0}^2+3\epsilon\|\nabla u_{\alpha}\|_{0}^2 \nonumber \\ &\leq \frac{C}{4\epsilon} \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+|u|_{L^\infty})^{2s}\|\nabla u\|_1^2 \|\nabla u\|_{2}^2 + \frac{C}{4\epsilon}\big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+|u|_{L^\infty})^{2s}\|\nabla u\|_2^2 \|\nabla u\|_{1}^2 \nonumber \\ &\quad + C\|D\mathbf{v}\|_{s}\| \nabla u\|_1^2 +C\|D\mathbf{v}\|_{s}\|\nabla u\|_0^2 + \frac{C}{4\epsilon}\|\nabla f\|_{0}^2+3\epsilon\|\nabla u_{\alpha}\|_{0}^2 \nonumber \\ &\leq \frac{C}{2\epsilon} \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+|u-u_0|_{L^\infty}+|u_0|_{L^\infty})^{2s}\|\nabla u\|_1^2 \|\nabla u\|_{2}^2 \nonumber \\ &\quad + C\|D\mathbf{v}\|_{s}\| \nabla u\|_1^2 +C\|D\mathbf{v}\|_{s}\|\nabla u\|_0^2 + \frac{C}{4\epsilon}\|\nabla f\|_{0}^2+3\epsilon\|\nabla u_{\alpha}\|_{0}^2 \nonumber \\ &\leq \frac{C}{2\epsilon} \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+|u(\mathbf{x}_0)|)^{2s}\|\nabla u\|_1^2 \|\nabla u\|_{2}^2 \nonumber \\ &\quad + C\|D\mathbf{v}\|_{s}\| \nabla u\|_1^2 +C\|D\mathbf{v}\|_{s}\|\nabla u\|_0^2+ \frac{C}{4\epsilon}\|\nabla f\|_{0}^2+3\epsilon\|\nabla u_{\alpha}\|_{0}^2 \nonumber \end{align} where we used inequality \eqref{3.3} from Lemma \ref{LB.1}. We also used Sobolev's lemma to obtain $|D \mathbf{v}|_{L^\infty}\leq C\|D\mathbf{v}\|_{s_0}$ and $|D^2 \mathbf{v}|_{L^\infty}\leq C\|D^2\mathbf{v}\|_{s_0}$, where $s_0=[\frac N2 ]+1=2$ and $s\geq 3$. We also used Cauchy's inequality with $\epsilon$. We assume that $\Big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2\| f\|_{s-1}^2 \leq \frac{1}{C_7}$, where the constant $C_7$ will be defined later. And we assume that $\|D\mathbf{v}\|_{s} \leq \frac 12$. Substituting estimates \eqref{e3.33}, \eqref{e3.24} for $\|\nabla u\|_1^2$, $\|\nabla u\|_0^2$ into \eqref{e3.35}, and using the fact that $a(u )> c_1$, and letting $\epsilon=\frac{c_1}{6}$, yields \begin{aligned} &(c_1\nabla u_\alpha,\nabla u_\alpha)\\ &\leq ((a(u)\nabla u_\alpha,\nabla u_\alpha) \\ & \leq C K_1\big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+|u(\mathbf{x}_0)|)^{2s}\|f\|_0^2 \|\nabla u\|_{2}^2+ C K_1\|D\mathbf{v}\|_{s}\| f\|_{0}^2 \\ &\quad + C \|D\mathbf{v}\|_{s}\| f\|_{0}^2+C\|f\|_{1}^2+\frac{c_1}{2}\|\nabla u_{\alpha}\|_{0}^2 \\ & \leq C K_1\big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2(1+R+|u(\mathbf{x}_0)|)^{2s}\|f\|_{s-1}^2 \|\nabla u\|_{2}^2 + C K_1\| f\|_{1}^2+\frac{c_1}{2}\|\nabla u_{\alpha}\|_{0}^2 \\ &\leq C K_1\Big(\frac{1}{C_7}\Big)(1+R+|u(\mathbf{x}_0)|)^{2s}\|\nabla u\|_{2}^2+ C K_1\| f\|_{1}^2+\frac{c_1}{2}\|\nabla u_{\alpha}\|_{0}^2 \end{aligned} \label{e3.36} where $C$ depends on $s$, $c_1$, and where we used the facts that $\|D\mathbf{v}\|_{s} <1$, and that $K_1=\max\{1, |\mathbf{v}|_{L^\infty}^2 \}$. Adding \eqref{e3.36} over all $|\alpha|=2$ after moving the term $\frac{c_1}{2}\|\nabla u_{\alpha}\|_{0}^2$ to the left-hand side, and adding the estimate \eqref{e3.33} for $\|\nabla u\|_1^2$ yields \begin{aligned} \|\nabla u\|_{2}^2 &= \|\nabla u\|_1^2+\sum_{|\alpha|=2}(\nabla u_\alpha,\nabla u_\alpha) \\ & \leq C K_1\Big(\frac{1}{C_7}\Big)(1+R+|u(\mathbf{x}_0)|)^{2s}\|\nabla u\|_{2}^2 + CK_1\| f\|_{1}^2 \\ & \leq C K_1\Big(\frac{1}{C_7}\Big)(1+R+(1+|\Omega|^{1/2}) |u(\mathbf{x}_0)|)^{2s}\|\nabla u\|_{2}^2 + CK_1\| f\|_{1}^2 \\ & \leq \Big(\frac{C_0 C_2 K_1}{C_7}\Big)\|\nabla u\|_{2}^2 + C_0 K_1\| f\|_{1}^2 \\ & \leq \Big(\frac{C_0 C_3 K_1}{C_7}\Big)\|\nabla u\|_{2}^2 + C_0 K_1\| f\|_{1}^2 \end{aligned}\label{e3.37} where the constant $C_0$ depends on $s$, $c_1$, and $C_2=C(1+R+(1+|\Omega|^{1/2})|u(\mathbf{x}_0)|)^{2s}$ and $C_3=M C_2$ are the same constants as in \eqref{e3.8} from Proposition \ref{P3.1}, and $M\geq 1$. We now define $C_7= 4 C_0^2 C_3^2 C_1^2 K_1^2$, where $C_0$ is the constant from \eqref{e3.37}, and where $C_1$ is the constant from estimate \eqref{B.27} in Lemma \ref{LB.6}, and we may assume that $C_1 \geq 1$, $C_0 \geq 1$, and $C_3 \geq 1$. Substituting the definition of $C_7$ into \eqref{e3.37} yields $$\|\nabla u\|_{2}^2 \leq \frac{1}{4 C_0 C_1^2 C_3 K_1}\|\nabla u\|_{2}^2 + C_0 K_1\| f\|_{1}^2 \leq \frac 12 \|\nabla u\|_{2}^2+ C_0 K_1\| f\|_{1}^2 \label{e3.38}$$ where we used that $K_1 \geq 1$. We define $C_8=C_0 C_1 K_1$. It follows from \eqref{e3.38} that $$\|\nabla u\|_{2}^2 \leq 2 C_0 K_1\|f\|_{1}^2 \leq 2 C_8\|f\|_{1}^2\,. \label{e3.39}$$ Note that since $C_8=C_0 C_1 K_1$ we have $C_7= 4 C_3^2 C_8^2$. Next we estimate $\|\nabla u\|_{r}^2$, where $3 \leq r \leq s$. Using estimate \eqref{B.27} from Lemma \ref{LB.6} in Appendix B applied to equation \eqref{B.99} yields \begin{aligned} \|\nabla u\|_{r}^2 &\leq C_1 \Big[\sum_{j=0}^{r}(\max \{\|D(a(u ))\|_{r_1}^2,\|D\mathbf{v}\|_{r_1}\})^{j}\Big] \|f\|_{r-1}^2 \\ &\leq C_8\Big[\sum_{j=0}^{r}(\max \{\|D(a(u ))\|_{r-1}^2,\|D\mathbf{v}\|_{r-1}\})^{j}\Big] \|f\|_{r-1}^2, \end{aligned}\label{e3.40} where $r_1=\max\{r-1,s_0\}=r-1$, and $s_0=[ \frac N2]+1=2$. We consider two cases: when $\max\{\|D(a(u ))\|_{r-1}^2, \|D\mathbf{v}\|_{r-1}\}=\|D(a(u ))\|_{r-1}^2$, and when $\max\{\|D(a(u ))\|_{r-1}^2, \|D\mathbf{v}\|_{r-1}\}=\|D\mathbf{v}\|_{r-1}$. \textbf{Case 1}: Suppose that $\max\{\|D(a(u))\|_{r-1}^2, \|D\mathbf{v}\|_{r-1}\}=\|D(a(u ))\|_{r-1}^2$. From \eqref{e3.8} in Proposition \ref{P3.1}, we have $\|D(a(u))\|_{r-1}^2 \leq C_3 \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^2\|\nabla u\|_{r-1}^2$. Repeatedly applying estimate \eqref{e3.40}, letting $r=3,4,\dots ,s$ and using the fact that $\|\nabla u\|_{r-1}^2 \leq 2C_8\|f\|_{r-2}^2$, and using estimate \eqref{e3.8} for $\|D(a(u ))\|_{r-1}^2$, and using the fact that $r_1=\max\{r-1,s_0\}=r-1$ when $r\geq 3$, we obtain \begin{aligned} \|\nabla u \|_{r}^2 &\leq C_8\Big[\sum_{j=0}^{r}\|D(a(u ))\|_{r-1}^{2j}\Big] \|f\|_{r-1}^2 \\ &\leq C_8\Big[\sum_{j=0}^{r} C_3^j \big|\frac{d a}{d u} \big|_{s,\bar{G}_1}^{2j}\|\nabla u\|_{r-1}^{2j}\Big]\|f\|_{r-1}^2 \\ &\leq C_8 \Big[\sum_{j=0}^{r}C_3^j( 2 C_8)^{j} \big|\frac{d a}{d u}\big|_{s,\bar{G}_1}^{2j} \|f\|_{r-2}^{2j}\Big]\|f\|_{r-1}^2 \\ &\leq C_8 \Big[\sum_{j=0}^{s}C_3^j(2 C_8)^{j} \big|\frac{d a}{d u}\big|_{s,\bar{G}_1}^{2j} \|f\|_{s-1}^{2j}\Big]\|f\|_{r-1}^2 \\ &\leq C_8 \Big[\sum_{j=0}^{s}C_3^j(2 C_8)^j \Big(\frac{1}{C_7} \Big)^j\Big]\|f\|_{r-1}^2 \\ &\leq C_8\Big[\sum_{j=0}^{s} \big(\frac 12\big)^j\Big] \|f\|_{r-1}^2 \\ &\leq 2 C_8\|f\|_{r-1}^2 \end{aligned} \label{e3.80} where we used the fact that $\big| \frac{d a}{d u} \big|_{s,\bar{G}_1}^2\| f\|_{s-1}^2 \leq \frac{1}{C_7}$, and $C_7= 4 C_3^2 C_8^2$, and $C_3 C_8 \geq 1$. \textbf{Case 2}: Suppose that $\max\{\|D(a(u))\|_{r-1}^2, \|D\mathbf{v}\|_{r-1}\}=\|D \mathbf{v}\|_{r-1}$. From \eqref{e3.40}, and using the fact that $\|D\mathbf{v}\|_{s}\leq \frac 12$, we obtain the following: $\|\nabla u \|_{r}^2 \leq C_8\Big[\sum_{j=0}^{r}\|D\mathbf{v}\|_{r-1}^{j}\Big] \|f\|_{r-1}^2 \leq C_8\Big[\sum_{j=0}^{s} \big(\frac 12\big)^j\Big] \|f\|_{r-1}^2 \leq 2 C_8\|f\|_{r-1}^2$ for $3 \leq r \leq s$, which is the same estimate as \eqref{e3.80}. Therefore we have $\|\nabla u \|_{r}^2 \leq 2 C_8\|f\|_{r-1}^2$ for $3 \leq r \leq s$. It follows that $\|\nabla u\|_{s}^2 \leq 2 C_8 \|f\|_{s-1}^2$. This completes the proof. \end{proof} \begin{thebibliography}{99} \bibitem{a1} R. Adams and J. Fournier; \textit{Sobolev Spaces}, Academic Press, 2003. \bibitem{DD1} D. L. Denny; \textit{Existence and uniqueness of global solutions to a model for the flow of an incompressible, barotropic fluid with capillary effects}, Electronic Journal of Differential Equations 39 (2007), 1--23. \bibitem{e1} P. 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