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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 218, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/218\hfil Space-time holomorphic time-periodic solutions]
{Space-time holomorphic time-periodic solutions of Navier-Stokes equations}

\author[E. Tsyganov \hfil EJDE-2013/218\hfilneg]
{Eugene Tsyganov} 

\address{Eugene Tsyganov \newline
 Department of Mathematics and Informatics Technology \\
 Bashkir State University, Russia}
\email{entsyganov@yahoo.com}

\thanks{Submitted September 12, 2013. Published October 4, 2013.}
\subjclass[2000]{34M05, 35Q30, 76D05}
\keywords{Navier-Stokes equations,  meromorphic solution}

\begin{abstract}
 We introduce a concept of space-time holomorphic solutions of partial
 differential equations and construct a meromorphic solution of Navier-Stokes
 equations, which can be  either space or time periodic.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

We study Navier-Stokes equations in Lagrangean coordinates
\begin{gather}
v_{t}-u_x=0,\label{1}\\
u_{t}+p_x=\big( \frac{u_x}{v}\big)_x,\label{2}
\end{gather}
with Cauchy data
\begin{equation}
v(0,x)=v_{0}(x), \quad  u(0,x)=u_{0}(x).
\end{equation}
Here \(t\in \mathbb{R}^{+}\) and  \(x\in \mathbb{R}\) are time and space
respectively, dependent variable \(v=v(x,t)\) denotes the specific volume,
\(u=u(t,x)\) - velocity, \(p=p(v)\) - pressure. We assume that \(p\)
satisfies the following conditions:
$$
p'<0, \quad \lim_{v\to 0+}p=+\infty, \quad \lim_{v\to +\infty}p=0.
$$
In addition we assume that  $p$ is holomorphic  in a neighborhood of
 \(\mathbb{R^{+}}\).

In this article, we continue our study of solutions of the Navier-Stokes 
equations having analyticity properties. The issue of analyticity was first 
addressed in Masuda \cite{m1} for Navier-Stokes equations for incompressible 
fluids and was further investigated in a number of papers 
(see Foias and Temam \cite{f1}, Constantin, Foias, Kukavica, Majda \cite{c1} etc.). 
The results show that analyticity arises naturally when solving the equations 
in their classical form, and can be used to study their properties.

 Of special interest is the study of complex solutions of the Navier-Stokes 
equations. Despite the fact that these models do not have a direct physical 
significance, they provide new information about the equations themselves 
(see Li and Sinai \cite{l1}).

 The study of analytic properties of weak solutions of the Navier-Stokes
equations for a compressible gas dynamic was initiated in Tsyganov \cite{t1} 
and was further developed for the multi-dimensional case in 
Hoff and Tsyganov \cite{h1}. We can point out that analyticity plays a critical 
role in the proof of backward uniqueness and in the derivation of exact rates 
of regularization and asymptotic behavior of weak solutions 
(see Tsyganov \cite{t2}).

 In this article, we study a special class of analytic solutions, which 
we call space-time holomorphic. The basic idea is to merge $t$ and $ x $ in 
one complex variable by the equality $z = it + x$, where $ i $ is the 
imaginary unit. After that, in order to find solutions of the original partial 
differential equation, we need to solve an ordinary differential equation in the 
complex plane. We give an example of a family of solutions, which are  
elementary meromorphic functions on the whole complex plane. These functions 
may blow-up (have a pole) in  finite time, however, they become smooth again 
once we  go through these positive values of time. We also prove that these 
solutions are space and time meromorphic.  

 It is important to point out that our results have no immediate applications 
to the pure real case of Navier-Stokes equations because of their 
strong non-linearity, but, nonetheless, they are significant for several reasons. 
Firstly, we offer a robust method of constructing explicit complex solutions 
of partial differential equations with two independent variables. 
This may be particularly interesting in studying systems where solutions are 
implied to be complex-valued from the beginning. Secondly, our approach  
gives new insight into Navier-Stokes equations, as the study of weak solutions  
far away from the real axis $t$ is a difficult task. Our findings give an idea 
of what one  can expect from the solutions of the Navier-Stokes equations with
 real initial data on the entire complex plane. Another important result is 
the construction of a solution  where singularities can be described explicitly.

 This article is structured as follows: in Section 2  we introduce the 
notion of space-time holomorphic solutions of partial differential equations. 
Then, in Section 3, we give an example of such a solution of the Navier-Stokes 
equations and study its properties.

\section{Space-time holomorphic solutions}

In this section we introduce a concept of space-time holomorphic solutions.

\begin{definition} \label{def2.1} \rm
 Consider a partial differential equation
\begin{equation}
F\Big(u,\frac{\partial u}{\partial t},\dots ,\frac{\partial^{n} u}{\partial t^{n}},
\frac{\partial u}{\partial x},\dots,\frac{\partial^{m} u}{\partial x^{m}}\Big)=0,
\label{4}
\end{equation}
where $t\in \mathbb{R}, x\in \mathbb{R}, u\in \mathbb{C}^{l}$, and
$F\in \mathbb{R}^{k}$ is a holomorphic function of its arguments.

We say that solution $u$ of equation \eqref{4} is space-time holomorphic, 
if, in addition, it satisfies the equation
\begin{equation}
\frac{\partial u}{\partial t}=i\frac{\partial u}{\partial x},\label{5}
\end{equation}
where $i$ is the imaginary unit.
\end{definition}

Then it follows from  \eqref{5} that function $u$ satisfies 
the Cauchy-Riemann condition. So, if we set $z=x+it$, then $u$ becomes 
a holomorphic function of $z$ and the following equalities hold:
\begin{equation}
\frac{\partial u}{\partial t}=i\frac{d u}{d z}, \quad
\frac{\partial u}{\partial x}=\frac{d u}{d z}.
\end{equation}
 Now we can give another definition of space-time holomorphy.

\begin{definition} \label{def2.2} \rm
 We say that a function $u$ is a space-time holomorphic solution of 
equation \eqref{4}, if it satisfies an ordinary differential equation
\begin{equation}
F\Big(u,i\frac{d u}{d z},\dots,i^{n}\frac{d^{n} u}{d z^{n}},
\frac{d u}{d z},\dots,\frac{d^{m} u}{d z^{m}}\Big)=0\label{6}
\end{equation}
in some domain of the complex plane.
\end{definition}

\begin{remark} \label{rmk1}\rm
Instead of condition \eqref{5} we can set
\begin{equation}
i\frac{\partial u}{\partial t}=\frac{\partial u}{\partial x}  \label{13}.
\end{equation}
If we now let $z=t+ix$, then we will come to a different equation for
the space-time holomorphy:
\begin{equation}
F\Big(u,\frac{d u}{d z},\dots,\frac{d^{n} u}{d z^{n}},
i\frac{d u}{d z},\dots,i^{m}\frac{d^{m} u}{d z^{m}}\Big)=0 \label{e7}.
\end{equation}
\end{remark}

In the rest of the article, we will consider only equation \eqref{6}.

\begin{remark} \label{rmk2}\rm
Let $u=u(z)$ satisfy  \eqref{6} in some domain of the complex plane.
 If we set $x=\operatorname{Re}(z)$, $t=\operatorname{Im}(z)$, then
 $u=u(x,t)$ satisfies partial differential equation \eqref{4} in 
some domain in $\mathbb{R}^{2}$.
\end{remark}

\section{Meromorphic solutions of Navier-Stokes equations}

The equations of the space-time holomorphy \eqref{6} for Navier-Stokes 
equations \eqref{1}, \eqref{2} are the following:
\begin{gather}
iv_z-u_z=0;\\
iu_z+p_z=\big(\frac{u_z}{v}\big)_z.
\end{gather}
We express $v_z$ in terms of $u_z$ and then substitute it into the 
second equation:
\begin{equation}
-v_z+p_z=\big(\frac{iv_z}{v}\big)_z.
\end{equation}
Then we integrate the above equality to obtain
\begin{equation}
-v+p=i\frac{v_z}{v}+C,\label{14}
\end{equation}
where $C$ is an arbitrary constant of integration.
 We rewrite the equation and integrate it once again:
\begin{equation}
z+C_{1}=i\int \frac{dv}{-v^{2}+vp+C_2v}
\end{equation}
We will primarily be interested in those functions $ p $, for which the equation
is integrable by quadratures.
Setting $p=\dfrac{1}{v}$, $C_2=0$ and $C_{1}=C$, we can obtained
the answer in closed form:
$$
z+C=i\frac{1}{2}(\ln(v+1)-\ln(1-v)),
$$
from which we obtain $v$:
\begin{equation}
v=\frac{e^{-2i(z+C)}-1}{e^{-2i(z+C)}+1}.
\end{equation}
Now we can find $u$:
\begin{equation}
u=i\frac{e^{-2i(z+C)}-1}{e^{-2i(z+C)}+1}+C_2
\end{equation}
for a new constant $C_2$.
We point out that the functions $v$ and $u$ are meromorphic
 on the whole complex plane and have the following properties:
\begin{equation}
\lim_{z\to \infty,\,\operatorname{Im}(z)>0} v=1, \quad
\lim_{z\to \infty,\,\operatorname{Im}(z)>0} u=i+C_2.
\end{equation}

\begin{remark} \label{rmk3}\rm
We can see that the corresponding solution $v=v(x,t)$, $u=u(x,t)$ of \eqref{4} 
is periodic with respect to the space variable $x$.
\end{remark}

\begin{remark} \label{rmk4}\rm
If we use condition \eqref{13} instead of \eqref{5}, then space-time 
holomorphic solutions for Navier-Stokes equations with $p=1/v$ will be
\begin{equation}
v=\frac{e^{2(z+C)}-1}{e^{2(z+C)}+1}, \quad
u=i\frac{e^{2(z+C)}-1}{e^{2(z+C)}+1}+C_2.
\end{equation}
The corresponding functions $v=v(x,t)$, $u=u(x,t)$ are periodic with
respect to the time variable $t$.
\end{remark}

 We are interested primarily in the zeros and poles of $v$:
\[
\text{zeroes: } z=\pi k-C, \quad \text{poles: } 
z=\frac{\pi}{2}+\pi k -C, \quad k\in \mathbb{Z}.
\]
 Depending on the value of $ C $ we can  have the following situation in 
the half-plane $ \operatorname{Im}(z) \geq 0 $:

(1) $\operatorname{Im}(C)>0$. Then the functions $ v $ and $ u $ are 
holomorphic in $ \operatorname{Im}(z)> 0 $ and continuous on 
$ \operatorname{Im}(z) \geq 0 $.
In this case, the pair of functions $ v = v (x, t) $, $ u = u (x, t) $ 
is a smooth solution of the Navier-Stokes equations \eqref {1}, \eqref {2} 
in the half-plane $ \mathbb {R} \times \mathbb {R} ^ {+} $ with smooth 
initial data; 

(2) $\operatorname{Im}(C)=0$. Then $ (v, u) $ is a smooth solution 
of \eqref {1},\eqref {2} in the half-plane $t> 0$ with meromorphic initial data;

(3) $\operatorname{Im}(C)<0$. In this case, $ (v, u) $ is a meromorphic 
solution for $ t> 0 $ with smooth initial data at $ t = 0 $.

\begin{remark} \label{rmk5}\rm
Equation \eqref{2} does not hold at the points where $v=0$. 
These singularities are, however, removable, since $(v,u)$ exist and 
is smooth at  such points.
\end{remark}

\begin{remark} \label{rmk6}\rm
 Functions $ v = v (t, x)$, $u = u (t, x) $ are meromorphic in  $ t $ 
for any fixed $ x \in \mathbb{R} $.  To prove this we use the following 
identities:
\begin{equation}
v(t,x):= v(t+x), \quad
 u(t,x):= u(t+x), \quad t\in \mathbb{C}, \; x\in \mathbb{R}.
\end{equation}
The solution is also meromorphic in $ x $ for any fixed $ t \in \mathbb{R} $.
This follows from the obvious definitions
\begin{equation}
v(t,x):= v(it+x), \quad u(t,x):= u(it+x), \quad x\in \mathbb{C}, \;
 t\in \mathbb{R}.
\end{equation}
\end{remark}

\begin{thebibliography}{0}

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\bibitem{h1} D. Hoff, E. Tsyganov;
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\bibitem{t2} E. Tsyganov;
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\end{thebibliography}

\end{document}