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% Asymptotic behavior of solutions to the generalized Becker-Döring
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% Poster for the HYKE-3 meeting in Rome, 13-15 April 2005.
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\title{}
\author{}
\date{}
\begin{document}
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\begin{center}
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\begin{center}
\Huge \textbf{Asymptotic behavior of solutions to the
generalized Becker-D\"oring equations for general initial
data}
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\begin{flushright}
Jos\'e Alfredo Ca\~nizo Rinc\'on\\
Dept. Matem\'atica Aplicada\\
Universidad de Granada, Spain\\
\vspace{.1cm} \small \emph{These results were obtained under the
supervision and help of St\'ephane Mischler. I wish to thank
him for his explanations and suggestions. The author has been
supported by a HYKE grant.}
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\noindent
\fcolorbox{black}{salmon}{
\begin{minipage}[t]{.96\linewidth}
\vspace{.05cm}
\begin{center}
\vspace{.1cm}
\section*{\Huge Main result}
\Large In \cite{C04} (to appear) we prove that the well-known
asymptotic behavior of solutions to the generalized
Becker-D\"oring system takes place for general initial data,
extending the previous knowledge that placed some restrictions
on it.
\end{center}
\end{minipage}
}
\section*{The coa\-gu\-la\-tion-frag\-men\-ta\-tion equations}
The coagulation-fragmentation equations describe the
evolution of a large number of clusters which can stick together or
break. Here we deal with the discrete version.
\begin{center}
\begin{tabular}{lll}
$c_j$ & $\equiv$ & number density of\\
&& clusters of size j
\end{tabular}
\vspace{.2cm}
% Thus, $c_j$ can be measured, for instance, in clusters per cubic
% meter, in units of $\unit{m^{-3}}$.
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\begin{center}
% -------------------------------------
\includegraphics[width=3cm]{frag.eps}
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\begin{tabular}{cc}
$b_{jk}$ $\equiv$ & rate of occurrence of\\
& reaction $j+k \to j,k$
\end{tabular}
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\includegraphics[width=3cm]{coag.eps}
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\begin{tabular}{cc}
$a_{jk}$ $\equiv$ & rate of occurrence of\\
& reaction $j \to j+k$
\end{tabular}
\end{center}
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\vspace{.2cm}
\noindent
\colorbox{marronrp3}{
\begin{minipage}[t]{.96\linewidth}
\Large
\begin{align*}
\frac{d}{dt} c_j
& = && \frac{1}{2} \sum_{k=1}^{j-1} a_{k,j-k} c_k c_{j-k}
& \text{Coagulation gain}\\
&& - &\sum_{k=1}^{\infty} a_{jk} c_j c_k
& \text{Coagulation loss}\\
&& + &\sum_{k=j+1}^{\infty} b_{j,k-j} c_k
& \text{Fragmentation gain}\\
&& - &\frac{1}{2} \sum_{k=1}^{j-1} b_{k,j-k} c_j
& \text{Fragmentation loss}
\end{align*}
\vspace{.02cm}
\end{minipage}
}
\vspace{.4cm}
{\textcolor{rojo}{The generalized Becker-Döring system is the special
case where $a_{jk}$ and $b_{jk}$ are zero whenever $\min\{j,k\} >
N$ for some $N$. For $N=1$ the system is the Becker-Döring
system.} }
\section*{Asymptotic Behavior}
The study of the long-time behavior of solutions to these equations is
expected to be a model of physical processes such as phase transition.
Under certain general conditions which include a detailed balance we
can ensure the existence of equilibrium states. In these conditions,
there is a critical mass $\rho_s \in ]0,\infty[$ such that any
solution that initially has mass $\rho_0 \leq \rho_s$ will converge
for large times, in a certain strong sense, to an equilibrium solution
with mass $\rho_0$. On the other hand, any solution with mass above
$\rho_s$ converges (in a weak sense) to the only equilibrium with mass
$\rho_s$; this weak convergence can then be interpreted as a phase
transition in the physical process modelled by the equation.
Convergence in this weak sense means that a fixed part of the total
mass of particles is found to be forming larger and larger clusters as
time passes and the mean size of clusters goes to infinity. The
physical interpretation of this, depending on the context, can be a
change of phase or the apparition of crystals, for example.
\noindent
\begin{center}
\noindent
\colorbox{marronrp3}{
\begin{minipage}[t]{.96\linewidth}
\begin{align*}
& \text{\Large Below critical mass}
&\to \quad
&\begin{cases}
\text{ \Large Trend to equilibrium }\\
\text{ \Large Strong convergence }
\end{cases}
&
\\
&\text{\Large Over critical mass }
&\to \quad
&\begin{cases}
\text{ \Large Large clusters created}\\
\text{\Large Weak convergence }
\end{cases}
&
\end{align*}
\end{minipage}
}
\end{center}
\begin{center}
\vspace{.5cm}
\Large
\begin{tabular}[t]{c|c}
\multicolumn{2}{c}{\huge \textbf{Previous results}}
\vspace{.3cm}
\\
Becker-Döring& Ball, Carr, Penrose\\
system & \cite{BCP86,BC88} (1986-88)\\
\hline
Generalized Becker-Döring &Carr, da Costa\\
(rapidly decaying initial data) &\cite{CdC94} (1994)\\
\hline
Generalized Becker-Döring & da Costa \\
(small initial data) & \cite{dC98} (1998)
\end{tabular}
\end{center}
% ---------------------------------------------------------------------------
\section*{Sketch of the proof}
Our proof is a generalization of a method used in unpublished notes by
Ph. Lauren\c{c}ot and S. Mischler \cite{LM}, inspired by the proof of
uniqueness of solutions to the Becker-D\"oring equation in
\cite{LM02e}.
It is known that, under common assumptions, \emph{there is always} at
least weak convergence to a certain equilibrium state;
\textcolor{rojo}{the problem reduces to show that for an initial
density under the critical one solutions converge \emph{strongly} to
the equilibrium \emph{with the same density}}. To prove this, it is
enough to show that the tails of the solutions are small enough, so
that strong convergence holds. The following estimate, roughly stated
here, is the key of our proof:
\noindent
\colorbox{marronrp3}{
\begin{minipage}[t]{.96\linewidth}
\vspace{.2cm}
\centerline{\huge \textbf{Main estimate}}
\vspace{.05cm}
\Large
If $c = \{c_j\}_{j \geq 1}$ is a solution to the generalized
Becker-Döring equations with density below the critical one, then
there is some sequence $r_i$ (which tends to zero as $i \to
\infty$) such that the tails of the solution have mass below
$r_i$; this is,
\begin{equation*}
\sum_{k=i}^\infty k c_k(t) \leq r_i
\end{equation*}
for all times $t$ after some time $t_0$.
\\\hspace{.05cm}
\end{minipage}
}
\vspace{.5cm}
The proof of this consists mainly of an estimate obtained by
differentiating the quantity $H_i := (G_i-r_i)_+$ (the positive part
of $G_i - r_i$), proving with a differential inequality that it must
remain zero for all times starting from a certain $t_0$.
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\small
\begin{thebibliography}{}
\bibitem{BCP86} J. M. Ball, J. Carr, O. Penrose, \emph{The
Becker-D\"oring cluster equations: basic properties and asymptotic
behaviour of solutions}, Comm. Math. Phys. 104, 657--692 (1986)
\bibitem{BC88} J. M. Ball, J. Carr, \emph{Asymptotic behaviour of
solutions to the Becker-D\"oring equations for arbitrary initial
data}, Proc. Roy. Soc. Edinburgh Sect. A, 108, 109-116 (1988)
\bibitem{C04} J. A. Cañizo, \emph{Asymptotic behavior of solutions to
the generalized Becker-Döring equations for general initial data},
preprint.
\bibitem{CdC94} J. Carr, F. P. da Costa, \emph{Asymptotic behaviour of
solutions to the coagulation-fragmentation equations. II. Weak
fragmentation}, J. Stat. Phys. 77, 89--123 (1994)
\bibitem{dC98} F. P. da Costa, \emph{Asymptotic behaviour of low
density solutions to the generalized Becker-D\"oring equations},
NoDEA Nonlinear Differential Equations Appl. 5, 23--37, (1998)
\bibitem{LM} Ph. Lauren\c{c}ot, S. Mischler, \emph{Notes on the
Becker-D\"oring equation}, personal communication.
\bibitem{LM02e} Ph. Lauren{\c{c}}ot, S. Mischler, \emph{From the
{B}ecker-{D}\"oring to the {L}ifshitz-{S}lyozov-{W}agner
equations}, J. Statist. Phys. 106, 5-6, pages 957--991 (2002).
\end{thebibliography}
\end{multicols}
\end{document}