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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 255, pp. 1--6.\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-2013/255\hfil Lavrent'ev problem for separated flows]
{Lavrent'ev problem for separated flows \\
with an external perturbation}
\author[D. K. Potapov, V. V. Yevstafyeva \hfil EJDE-2013/255\hfilneg]
{Dmitriy K. Potapov, Victoria V. Yevstafyeva} % in alphabetical order
\address{Dmitriy K. Potapov \newline
Saint Petersburg State University,
7-9, University emb., 199034 St. Petersburg, Russia}
\email{potapov@apmath.spbu.ru}
\address{Victoria V. Yevstafyeva \newline
Saint Petersburg State University,
7-9, University emb., 199034 St. Petersburg, Russia}
\email{vica@apmath.spbu.ru}
\thanks{Submitted July 2, 2013. Published November 20, 2013.}
\subjclass[2000]{35J25, 35J60, 35Q35}
\keywords{Lavrent'ev model; separated flows; external perturbation;
\hfill\break\indent discontinuous nonlinearity; semiregular solution;
variational method}
\begin{abstract}
We study the Lavrent'ev mathematical model for separated flows with
an external perturbation. This model consists of a differential
equation with discontinuous nonlinearity and a boundary condition.
Using a variational method, we show the existence of a semiregular
solution. As a particular case, we study the one-dimensional model.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction and statement of the problem}
The Lavrent'ev model for separated flows as a main tool for
hydrodynamics is discussed in \cite{lavr}.
Separated flows are constructed with a scheme of some ``mixed'' ideal
fluid motion that is potential outside a separation zone and has a
constant vorticity inside.
The mathematical model of the Lavrent'ev problem is given in \cite{krasn}.
The resonance case and a nonlinear perturbation in a general form
for this problem are presented in \cite{ozz}.
In the present article the Lavrent'ev model under an external continuous
perturbation is studied.
Unlike \cite{ozz}, here the coercive case is considered and the
external perturbation is given in a concrete form. Actually
the external perturbations such as a jump, an exponential, a polynomial or a
sine are simplifying and are not quite adequate to real perturbations.
So, we consider the model of the external
perturbation in the special analytical form that has not been studied for the
Lavrent'ev problem.
In a bounded domain $\Omega \subset\mathbb{R}^2$ with a boundary $\Gamma$
of class $C_{2,\alpha}$, where $0<\alpha \leq 1$, we solve the Dirichlet problem for an elliptic equation with discontinuous nonlinearity
\begin{gather} \label{uravn}
-\Delta u(x)=\mu \operatorname{sign} (u(x))+f(\|x\|), \quad x\in\Omega, \\
\label{gran}
u(x) \vert_{ \Gamma}=0.
\end{gather}
Here $\Delta$ is the Laplace operator, a parameter $\mu>0$ is the vorticity,
a function $f\in C(\overline{\Omega})$.
We study the model of the external perturbation in the form
\begin{equation}
\label{vozd}
f(t)=e^{\alpha t}\sin(\omega t+\varphi),
\end{equation}
where $\alpha$, $\omega$, $\varphi$ are real constants.
Here $\omega$ is a frequency, $\varphi$ is a phase angle that
allows us to define a deviation at $t=0$.
The external perturbation of \eqref{vozd} is considered
in \cite{vica1998,vica2004}.
Function of \eqref{vozd} describes a fading oscillatory
process at $\alpha<0$ and an accruing oscillatory process at $\alpha>0$.
In this article we study the existence of solutions for the Lavrent'ev
problem \eqref{uravn}, \eqref{gran} under \eqref{vozd}.
Let \eqref{uravn} be exposed to the nonperiodic external perturbation
of \eqref{vozd} with a decreasing amplitude at $\alpha<0$.
For example, a shock sea wave that arises as a result of explosion may be
described by the function $f(t)$ with a strongly decreasing amplitude.
On the other hand, to describe a calming down storm that is accompanied with
the fading fluctuations of waves it is possible to use the function $f(t)$
with a poorly decreasing amplitude.
Also, we notice that
\[
|f(\|x\|)|=|e^{\alpha\cdot \|x\|}\cdot\sin(\omega\cdot \|x\|+\varphi)|\leq
e^{\alpha\cdot \|x\|}\leq 1
\]
as $\alpha<0$, $\|x\|\geq 0$. So $f$ is bounded.
\section{Preliminaries}
In this section we recall some definitions and
a basic result to control problems for the distributed systems of the elliptic type with a spectral parameter and discontinuous nonlinearity under an external
perturbation (see \cite{pot2012_2}).
In a bounded domain
$\Omega \subset\mathbb{R}^n$ ($n\geq 2$) with a boundary $\Gamma$ of class
$C_{2,\alpha}$ ($0<\alpha \leq 1$), we consider the controlled
system with an external perturbation in the form
\begin{gather}
\begin{aligned}
Lu(x)&\equiv-\sum_{i,j=1}^n {(a_{ij}(x)u_{x_i})}_{x_j}+ c(x)u(x)\\
&= \lambda g(x,u(x))+Bv(x)+Dw(x), \quad x\in\Omega, \label{odin}
\end{aligned} \\
\label{dva}
Gu \big|_{ \Gamma}=0.
\end{gather}
Here $L$ is a uniformly elliptic and formally self-adjoint differential
operator with coefficients
$a_{ij} \in C_{1,\alpha}(\overline{\Omega})$ and
$c \in C_{0,\alpha}(\overline{\Omega})$;
$\lambda$ is a positive parameter;
the function $g:\Omega \times\mathbb{R}\rightarrow\mathbb{R}$ is
superpositionally measurable and for almost all $x \in \Omega$ the section
$g(x,\cdot)$ has only discontinuities of the first kind on
$\mathbb{R}$, $g(x,u) \in [g_{-}(x,u),g_{+}(x,u)]$ for all
$u \in \mathbb{R}$, where
\[
g_{-}(x,u)=\liminf_{\eta \to u}g(x,\eta),\quad
g_{+}(x,u)=\limsup_{\eta\to u}g(x,\eta),
\]
$|g(x,u)|\leq a(x)$ for all $u\in \mathbb{R}$,
$a\in L_q(\Omega)$, $q>\frac{2n}{n+2}$;
the operator $B: U\to L_q(\Omega)$ is linear and bounded, $U$ is the
Banach space of controls,
the function $v(x)$ in \eqref{odin} is viewed as a control,
the control $v\in U_{\rm ad}\subset U$, $U_{\rm ad}$ is the set of all admissible
controls for system \eqref{odin}, \eqref{dva};
the operator $D : W\to L_q(\Omega)$ is linear and bounded, $W$ is the
Banach space of perturbations,
the function $w(x)$ in \eqref{odin} describes a perturbation, the perturbation
$w\in W$. The boundary condition \eqref{dva} is either the Dirichlet condition
$u(x) \vert _{\Gamma}=0$, or the Neumann condition
$\frac{\partial u }{\partial {\mathbf{n}}_L}(x)\vert_{\Gamma}=0$
with the conormal derivative
$\frac{\partial u }{\partial {\mathbf{n}}_L}(x) \equiv \sum_{i,j=1}^n
a_{ij}(x)u_{x_i}\cos({\mathbf{n}},x_j)$, where
$\mathbf{n}$ is the outward normal to $\Gamma$ and
$\cos({\mathbf{n}},x_j)$ are the direction cosines of the normal $\mathbf{n}$,
or the Robin condition
$\frac{\partial u }{\partial {\mathbf{n}}_L}(x)+\sigma (x)u(x)\vert_{\Gamma}=0$,
where the function $\sigma \in C_{1,\alpha}(\Gamma)$ is nonnegative and
does not identically vanish on $\Gamma$.
Such eigenvalue problems for elliptic equations with
discontinuous nonlinearities but without control and perturbation
($v(x)\equiv 0$ and $w(x)\equiv 0$) was established earlier
(see \cite{pot2001}--\cite{pot2013}).
\noindent
\begin{definition} \label{def1}\rm
A strong solution of problem \eqref{odin},
\eqref{dva} at the fixed control $v$ and the fixed perturbation $w$
is a function $u\in \textbf{W}_r^2(\Omega)$, $r>1$, satisfying
\eqref{odin} for almost all $x\in\Omega$ and such that the trace $Gu(x)$ on
$\Gamma$ equals zero.
\end{definition}
\begin{definition} \label{def2} \rm
A semiregular solution of problem \eqref{odin},
\eqref{dva} at the fixed control $v$ and the fixed perturbation $w$
is a strong solution $u$ such that $u(x)$ is a
point of continuity of the function $g(x,\cdot)$ for almost all
$x\in\Omega$.
\end{definition}
\begin{definition} \label{def3} \rm
A jump discontinuity of a function
$f: \mathbb{R}\to \mathbb{R}$ is a point $u\in \mathbb{R}$ such
that $f(u-)\frac{2n}{n+2}$) is fixed;
\item[(3)] there exists a $u_0\in X$ such that
\[
\int_{\Omega}dx\int_0^{u_0(x)}g(x,s)ds>0;
\]
\item[(4)] if the solution space $N(L)$ of the problem
\begin{gather*}
Lu=0,\quad x\in\Omega,\\
Gu|_{\Gamma}=0
\end{gather*}
is nonzero (the resonance case), then it is additionally assumed that
\[
\lim_{u\in N(L), \|u\|\to +\infty}\int_{\Omega}dx\int_0^{u(x)}g(x,s)ds
=-\infty;
\]
\item[(5)] the operator $B : U\to L_q(\Omega)$ is linear and bounded,
the control space $U$ is Banach, the set of admissible controls
$U_{\rm ad}\subset U$ is nonempty;
\item[(6)] the operator $D : W\to L_q(\Omega)$ is linear and bounded, the
perturbation space $W$ is Banach.
\end{itemize}
Then for any $v\in U_{\rm ad}$ and $w\in W$ there exists a semiregular solution
of problem \eqref{odin}, \eqref{dva}.
\end{theorem}
Under conditions (1)--(4) of the above theorem,
when control and perturbation are absent, and
using a variational method, existence results were obtained
in \cite[Theorems 3 and 4]{pot2001}, and \cite[Theorem~3]{pot2013}.
\section{Solution of the problem}
Let us verify that all the conditions of Theorem \ref{thm1} are
fulfilled for the Lavrent'ev problem \eqref{uravn}, \eqref{gran} under
\eqref{vozd}.
We have
\[
J_1(u)=\frac{1}{2}\sum_{i=1}^2\int_{\Omega}u_{x_i}^2dx=
\frac{1}{2}\int_{\Omega}(u_{x_1}^2+u_{x_2}^2)dx=
\frac{1}{2}{\|u\|}^2\geq 0 \quad
\forall u\in H_{\circ}^1(\Omega).
\]
Condition (1) is satisfied.
For almost all $x\in\Omega$ the function $\operatorname{sign}(\cdot)$ has only
jump discontinuity $u=0$ as
$-1=\operatorname{sign}(0-)<\operatorname{sign}(0+)=1$;
$\operatorname{sign}(0)=0$ and
$|\operatorname{sign}(u)|\leq 1$ for all $u\in \mathbb{R}$,
$1\in L_q(\Omega)$, $q>\frac{2 \cdot 2}{2+2}=1$ are valid.
Therefore condition (2) of Theorem \ref{thm1} is fulfilled.
As in \cite{pot2010}, it can be shown that there exists a
$u_0\in H_{\circ}^1(\Omega)$ such that
\[
\int_{\Omega}dx\int_0^{u_0(x)}\operatorname{sign}(s)ds>0.
\]
Condition (3) of Theorem \ref{thm1} holds.
Since the space $N(-\Delta)$ of solutions for the problem
\[
-\Delta u=0,\\
u \big|_{ \Gamma}=0
\]
is zero, it follows that no additional assumption in
condition (4) of Theorem \ref{thm1} is needed.
Clearly, condition (5) of Theorem \ref{thm1} is not required
as the control in \eqref{uravn} is absent.
We see that at the perturbation $f$ in \eqref{uravn} there is the
identical operator $I$, i.e., $If=f$.
The operator $I$ is linear and bounded.
The space $C(\overline{\Omega})$ of
the perturbations is a Banach space.
Condition (6) of Theorem \ref{thm1} is satisfied.
Thus all the conditions of Theorem \ref{thm1} for the
Lavrent'ev problem \eqref{uravn}, \eqref{gran} under \eqref{vozd} are fulfilled.
This implies that the Lavrent'ev problem has a semiregular solution.
In the present paper we show the existence of the semiregular solution of the
Dirichlet problem for the elliptic equation with the discontinuous nonlinearity
by the variational method unlike in \cite{krasn}.
If, in addition, the variational functional corresponding to problem
\eqref{uravn}, \eqref{gran} has no more than a countable number of points of a
global minimum, then, according to \cite{lep,pot2012_3},
there is the regular solution of problem \eqref{uravn}, \eqref{gran}; i.e.,
the semiregular solution with the property of correctness.
Earlier (see \cite{lavr}--\cite{ozz}) the regular solutions for the Lavrent'ev
problem were not investigated.
We note that other theoretical results for the Lavrent'ev problem
are received similarly to results for the Gol'dshtik mathematical model for
se\-parated flows of incompressible fluid~\cite{gold}, which are analyzed
in \cite{pot2010,pot2004,pot2012_1}.
\section{One-dimensional model}
Further we consider the one-dimensional analog of model
\eqref{uravn}, \eqref{gran}.
We have
\begin{gather}
\label{uravn1}
-u''(x)=\mu \operatorname{sign} (u(x))+f(x), \quad x\in [0,1],\\
\label{gran1}
u(0)=u(1)=0.
\end{gather}
A system of ordinary differential equations that contains a hysteresis
nonlinearity such as a relay and the external perturbation of \eqref{vozd} is
studied in \cite{vica1998,vica2004}. By replacement of variables, this
system can be reduced to model \eqref{uravn1}, \eqref{gran1}.
Solvability for this problem was established earlier.
Arguing as above, we see that other results for problem
\eqref{uravn1}, \eqref{gran1} can be obtained as well as for the
one-dimensional analog of Gol'dshtik's model that is considered
in \cite{pot2004,pot2011}.
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\end{document}