\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 93, pp. 1--14.\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-2010/93\hfil Solutions to Navier-Stokes equation] {Solutions to three-dimensional Navier-Stokes equations for incompressible fluids} \author[J. Jormakka\hfil EJDE-2010/93\hfilneg] {Jorma Jormakka} \address{Jorma Jormakka \newline National Defence University, P.O. Box 07 00861 Helsinki, Finland} \email{jorma.jormakka@mil.fi} \thanks{Submitted August 31, 2009. Published July 7, 2010.} \subjclass[2000]{35Q30} \keywords{Partial differential equations; Navier-Stokes equation; fluid dynamics} \begin{abstract} This article gives explicit solutions to the space-periodic Navier-Stokes problem with non-periodic pressure. These type of solutions are not unique and by using such solutions one can construct a periodic, smooth, divergence-free initial vector field allowing a space-periodic and time-bounded external force such that there exists a smooth solution to the 3-dimensional Navier-Stokes equations for incompressible fluid with those initial conditions, but the solution cannot be continued to the whole space. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Let $x=(x_1,x_2,x_3)\in \mathbb{R}^3$ denote the position, $t\ge 0$ be the time, $p(x,t)\in \mathbb{R}$ be the pressure and $u(x,t)=(u_i(x,t))_{1\le i\le 3}\in \mathbb{R}^3$ be the velocity vector. Let $f_i(x,t)$ be the external force. The Navier-Stokes equations for incompressible fluids filling all of $\mathbb{R}^3$ for $t\ge 0$ are \cite{feff} \begin{gather} \frac{\partial u_i}{\partial t}+\sum_{j=1}^3 {u_j\frac{\partial u_i}{\partial x_j}} = v \Delta u_i -\frac{\partial p}{\partial x_i}+f_i(x,t), \quad x\in\mathbb{R}^3,\; t\ge 0,\; 1\le i\le 3 \label{e1}\\ \operatorname{div}u=\sum_{i=1}^3 \frac{\partial u_i}{\partial x_i}=0, \quad x\in\mathbb{R}^3,\; t\ge 0 \label{e2} \end{gather} with initial conditions $$u(x,0)=u^0(x),\quad x\in \mathbb{R}^3.\label{e3}$$ Here $\Delta=\sum_{i=1}^3 \frac{\partial^2}{\partial x_i^2}$ is the Laplacian in the space variables, $v$ is a positive coefficient and $u^0(x)$ is $C^{\infty}(\mathbb{R}^3)$ vector field on $\mathbb{R}^3$ required to be divergence-free; i.e., satisfying $\operatorname{div}u^0=0$. The time derivative $\frac{\partial u_i }{\partial t}$ at $t=0$ in \eqref{e1} is taken to mean the limit when $t\to 0^+$. This article shows that there exists $C^{\infty}(\mathbb{R}^3)$, periodic, divergence-free initial vector fields $u^0$ defined at $\mathbb{R}^3$ such that there exists a family of smooth (here, in the class $C^{\infty}(\mathbb{R}^3\times [0,\infty))$) functions $u(x,t)$ and $p(x,t)$ satisfying \eqref{e1}, \eqref{e2} and \eqref{e3}. We also show that there exist a periodic and bounded external force $f_i(x,t)$ such that the solution cannot be continued to the whole $\mathbb{R}^3\times [0,\infty)$. \section{Theorems and Lemmas} The simple explicit case of $u(x,t)$ in Lemma \ref{lem1} satisfies the conditions given in \eqref{e3} and allows a free function $g(t)$ that only satisfies $g(0)=g'(0)=0$. The solution is then not unique. If the time derivatives of $u(x,t)$ are specified at $t=0$ then the solution in the lemma is unique. \begin{lemma} \label{lem1} Let \begin{gather*} u_1^0=2\pi \sin (2\pi x_2)+2\pi \cos(2\pi x_3),\\ u_2^0=2\pi \sin (2\pi x_3)+2\pi \cos(2\pi x_1),\\ u_3^0=2\pi \sin (2\pi x_1)+2\pi \cos(2\pi x_2) \end{gather*} be the initial vector field, and let $f_i(x,t)$ be chosen identically zero for $1\le i \le3$. Let $g:\mathbb{R}\to \mathbb{R}$ be a smooth function with $g(0)=g'(0)=0$ and $\beta=(2\pi)^2 v$. The following family of functions $u$ and $p$ satisfy \eqref{e1}-\eqref{e3}: \label{e4} \begin{gathered} u_1=e^{-\beta t}2\pi \left(\sin (2\pi (x_2+g(t)))+\cos (2\pi (x_3+g(t)))\right)-g'(t),\\ u_2=e^{-\beta t}2\pi \left(\sin (2\pi (x_3+g(t))) +\cos (2\pi (x_1+g(t)))\right) -g'(t),\\ u_3=e^{-\beta t}2\pi \left(\sin (2\pi (x_1+g(t))) +\cos (2\pi (x_2+g(t)))\right)-g'(t),\\ \begin{aligned} p&= -e^{-2\beta t}(2\pi)^2 \sin (2\pi (x_1+g(t))) \cos (2\pi (x_2+g(t)))\\ &\quad -e^{-2\beta t}(2\pi)^2 \sin (2\pi (x_2+g(t))) \cos (2\pi (x_3+g(t)))\\ &\quad -e^{-2\beta t}(2\pi)^2 \sin (2\pi (x_3+g(t))) \cos (2\pi (x_1+g(t))) +g''(t)\sum_{j=1}^3 x_j. \end{aligned} \end{gathered} \end{lemma} \begin{proof} The initial vector field is smooth, periodic, bounded and divergence-free. Let $(i,k,m)$ be any of the permutations $(1,2,3)$, $(2,3,1)$ or $(3,1,2)$. We can write all definitions in \eqref{e4} shorter as (here $g'(t)=dg/dt$): \begin{gather*} u_i=e^{-\beta t}2\pi \left( \sin (2\pi (x_k+g(t))) +\cos (2\pi (x_m+g(t)))\right)-g'(t),\\ \begin{aligned} p&=-e^{-2\beta t}(2\pi)^2 \sin (2\pi (x_i+g(t))) \cos (2\pi (x_k+g(t)))\\ &\quad -e^{-2\beta t}(2\pi)^2 \sin (2\pi (x_k+g(t))) \cos (2\pi (x_m+g(t)))\\ &\quad -e^{-2\beta t}(2\pi)^2 \sin (2\pi (x_m+g(t))) \cos (2\pi (x_i+g(t))) +g''(t)\sum_{j=1}^3 x_j. \end{aligned} \end{gather*} It is sufficient to proof the claim for these permutations. The permutations $(1,3,2)$, $(2,1,3)$ and $(3,2,1)$ only interchange the indices $k$ and $m$. The functions \eqref{e4} are smooth and $u(x,t)$ in \eqref{e4} satisfies \eqref{e2} and \eqref{e3} for the initial vector field in Lemma \ref{lem1}. We will verify \eqref{e1} by directly computing: \begin{gather*} \begin{aligned} \frac{\partial u_i}{\partial t} &=-\beta e^{-\beta t}2\pi \left( \sin(2\pi (x_k+g(t)))+\cos(2\pi (x_m+g(t)))\right) -g''(t) \\ &\quad +g'(t)e^{-\beta t}(2\pi)^2 \left(\cos(2\pi (x_k+g(t)))-\sin(2\pi (x_m+g(t)))\right), \end{aligned} \\ - v \Delta u_i = v e^{-\beta t}(2\pi)^3\left( \sin (2\pi (x_k+g(t)))+\cos (2\pi (x_m+g(t)))\right),\\ \begin{aligned} \frac{\partial p}{\partial x_i} &= -e^{-2\beta t}(2\pi)^3 \cos (2\pi (x_i+g(t))) \cos (2\pi (x_k+g(t)))\\ &\quad +e^{-2\beta t}(2\pi)^3 \sin (2\pi (x_m+g(t))) \sin (2\pi (x_i+g(t)))+g''(t). \end{aligned} \end{gather*} The functions $u_k$ and $u_m$ are \begin{gather*} u_k=e^{-\beta t}2\pi \left( \sin (2\pi (x_m+g(t))) +\cos (2\pi (x_i+g(t)))\right)-g'(t),\\ u_m=e^{-\beta t}2\pi \left( \sin (2\pi (x_i+g(t))) +\cos (2\pi (x_k+g(t)))\right)-g'(t). \end{gather*} The remaining term to be computed in \eqref{e1} is \begin{align*} \sum_{j\in \{i,k,m\}}u_j\frac{\partial u_i}{\partial x_j} &= u_i\frac{\partial u_i}{\partial x_i} + u_k\frac{\partial u_i}{\partial x_k} +u_m\frac{\partial u_i}{\partial x_m}\\ &=e^{-2\beta t}(2\pi)^3\cos (2\pi (x_i+g(t)))\cos (2\pi (x_k+g(t)))\\ &\quad -e^{-2\beta t}(2\pi)^3\sin (2\pi (x_m+g(t))) \sin (2\pi (x_i+g(t)))\\ &\quad -g'(t) e^{-\beta t}(2\pi)^2 \left( \cos (2\pi (x_k+g(t)))-\sin (2\pi (x_m+g(t)))\right). \end{align*} Inserting the parts to \eqref{e1} shows that $$\frac{\partial u_i}{\partial t} +\sum_{j=1}^3 {u_j\frac{\partial u_i}{\partial x_j}} - v \Delta u_i +\frac{\partial p}{\partial x_i}=0.$$ This completes the proof . \end{proof} \begin{theorem} \label{thm1} There exists a periodic, $C^{\infty}(\mathbb{R}^3)$, and divergence-free vector field $u^0(x)=(u_i^0(x))_{1\le i\le3}$ on $\mathbb{R}^3$ such that the following two claims hold: \begin{itemize} \item[C1:] The solution to \eqref{e1}-\eqref{e3} is not necessarily unique. In fact, there are infinitely many $C^{\infty}\left(\mathbb{R}^3\times [0,\infty)\right)$ functions $u(x,t)=(u_i(x,t))_{1\le i\le3}$ and $p(x,t)$ satisfying \eqref{e1}, \eqref{e2} and \eqref{e3}. \item[C2:] Periodic initial values do not guarantee that the solution is bounded. Indeed, there exist unbounded $u(x,t)$ and $p(x,t)$ satisfying \eqref{e1}, \eqref{e2} and the initial values \eqref{e3}. There also exist bounded solutions that are periodic as functions of $x$. \end{itemize} \end{theorem} \begin{proof} Let $f_i(x,t)$ be chosen identically zero for $1\le i \le3$, and let us select $g(t)={1\over 2}ct^2$ in Lemma \ref{lem1}. The value $c\in \mathbb{R}$ can be freely chosen. This shows C1. If $c=0$ then the solution is bounded and it is periodic as a function of $x$. If $c\not=0$ then $u_i(x,t)$ for every $i$ and $p(x,t)$ are all unbounded. In $u_i(x,t)$ the $ct=g'(t)$ term and in $p$ the term $c(x_1+x_2+x_3)=g''(t)\sum_{j=1}^3x_j$ are not bounded. This shows C2. The failure of uniqueness is caused by the fact that \eqref{e1}-\eqref{e3} do not determine the limits of the higher time derivatives of $u(x,t)$ when $t\to 0+$. These derivatives can be computed by differentiating \eqref{e4} but the function $g(t)$ is needed and it determines the higher time derivatives. As $g(t)$ can be freely chosen, the solutions are not unique. \end{proof} \begin{theorem} \label{thm2} There exists a smooth, divergence-free vector field $u^0(x)$ on $\mathbb{R}^3$ and a smooth $f(x,t)$ on $\mathbb{R}^3\times [0,\infty)$ and a number $C_{\alpha,m,K}>0$ satisfying $$u^0(x+e_j)=u^0(x), \quad f(x+e_j,t)=f(x,t), \quad 1\le j\le 3 \label{e5}$$ (here $e_j$ is the unit vector), and $$|\partial_x^{\alpha}\partial_t^{m}f(x,t)| \le C_{\alpha m K}(1+|t|)^{-K}\label{e6}$$ for any $\alpha$, $m$ and $K$, such that there exists $a>0$ and a solution $(p,u)$ of \eqref{e1}, \eqref{e2}, \eqref{e3} satisfying $$u(x,t)=u(x+e_j,t)\label{e7}$$ on $\mathbb{R}^3\times [0,a)$ for $1\le j\le 3$, and $$p,u\in C^{\infty}(\mathbb{R}^3\times [0,a))\label{e8}$$ that cannot be smoothly continued to $\mathbb{R}^3\times [0,\infty)$. \end{theorem} \begin{proof} Let us make a small modification to the solution in Lemma \ref{lem1}. In Lemma \ref{lem1}, $g:\mathbb{R}\to \mathbb{R}$ is a smooth function with $g(0)=g'(0)=0$, but we select $$g(t)={1\over 2}ct^2{1\over a-t}$$ where $c\not=0$ and $a>0$. The initial vector field $u^0(x)$ in Lemma \ref{lem1} is smooth, periodic and divergence-free. The period is scaled to one in \eqref{e4}. The $f(x,t)$ is zero and therefore is periodic in space variables with the period as one. Thus, \eqref{e5} holds. The constant $C_{\alpha,m,K}$ is selected after the numbers $\alpha,m,K$ are selected. The force $f(x,t)$ is identically zero, thus \eqref{e6} holds. The solution \eqref{e4} in Lemma \ref{lem1} is periodic in space variables with the period as one. Thus \eqref{e7} holds. The solution $u(x,t)$ in \eqref{e4} is smooth if $t0$ satisfying $$u^0(x+e_j)=u^0(x) , \quad f(x+e_j,t)=f(x,t) , \quad 1\le j\le 3.\label{e5'}$$ (here $e_j$ is the unit vector), and $$|\partial_x^{\alpha}\partial_t^{m}f(x,t)| \le C_{\alpha m K}(1+|t|)^{-K} \label{e6'}$$ for any $\alpha$, $m$ and $K$, such that there exist no solutions $(p,u)$ of \eqref{e1}, \eqref{e2}, \eqref{e3} on $\mathbb{R}^3\times [0,\infty)$ satisfying $$u(x,t)=u(x+e_j,t)\label{e7'}$$ on $\mathbb{R}^3\times [0,\infty)$ for $1\le j\le 3$, and $$p,u\in C^{\infty}(\mathbb{R}^3\times [0,\infty)).\label{e8'}$$ \end{theorem} \begin{proof} Let the solution in Theorem \ref{thm2} with the particular $g$ be denoted by $U$ and $a$ be larger than $1$. A feedback control force $f(x,t)$ is defined by using the values of the function $u(x,t')$ for $t'\le t$. In practise there is a control delay and $t't_1$ for some fixed $t_1$ satisfying \$0