\documentclass[reqno]{amsart} \usepackage{amssymb} \AtBeginDocument{{\noindent\small 2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela.\newline {\em Electronic Journal of Differential Equations}, Conference 13, 2005, pp. 21--27.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{21} \begin{document} \title[\hfilneg EJDE/Conf/13 \hfil The relativistic Enskog equation] {The relativistic Enskog equation near the vacuum} \author[R. Galeano, B. Orozco, O. Vasquez\hfil EJDE/Conf/13 \hfilneg] {Rafael Galeano Andrades, Bernardo Orozco Herrera,\\ Maria Ofelia Vasquez Avila} % in alphabetical order \address{Rafael Galeano Andrades \hfill\break Program de Matematicas, Universidad de Cartagena, Cartagena, Colombia} \email{ecuacionesdif@yahoo.com} \address{Bernardo Orozco Herrera \hfill\break Program de Matematicas, Universidad de Cartagena, Cartagena, Colombia} \email{orozcoberna@hotmail.com} \address{Maria Ofelia Vasquez Avila\hfill\break Program de Matematicas, Universidad de Cartagena, Cartagena, Colombia} \email{movasquez@epm.net.co} \date{} \thanks{Published May 30, 2005.} \subjclass[2000]{35Q75, 82-02} \keywords{Relativistic; Enskog equation; near the vacuum} \begin{abstract} We prove an existence and uniqueness theorem for the solution with data near the vacuum in the Hard sphere. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} The relativistic Boltzmann equation is written as \[ V\cdot\nabla_{x}F =-C(F,F),\] where the dot represents the Lorentz inner product $(+---)$ of 4-vectors $v=(v_{1},v_{2},v_{3})$, $V=(v_{0},v_{1},v_{2},v_{3})$, $X=(x_{0},x_{1},x_{2},x_{3})$, $x=(x_{1},x_{2},x_{3})$, $x_{0}= -t$ and $C(F,F)$ is the collision integral. Normalizing the speed of light $c=1$ and the particle mass $m=1$, we have $V\cdot V=1$ or $v_{0} =\sqrt{1+|v|^{2}}$. For convenience, we separate the time and space variables, and then divide by $v_{0}$ the relativistic Bolttzman equation to obtain, \begin{equation} \partial_{t}F + \hat{v}\cdot\nabla_{x}F =Q(F,F) \end{equation} where \begin{gather*} Q(F,F) = v_{0}^{-1}C(F,F)\,\text{y}\, \hat{v} =\frac{v}{v_{0}}= \frac{v}{\sqrt{1+|v|^{2}}}\,,\\ \begin{aligned} Q(F,F)(v)=&\frac{1}{2v_{0}}\int\int\int \delta(U^{2} -1)\delta (U'^{2} -1)\delta (V'^{2} -1)s \sigma(s,\theta)\delta^{4}\\ &\times (U+V-U'-V')[F(u')F(v')-F(u)F(v)]d^{4}Ud^{4}U'd^{4}V \end{aligned} \end{gather*} where $U^{2}=U\cdot U=u_{0}^{2}-|u|^{2}$, $|u|^{2} =u_{1}^{2}+u_{2}^{2} +u_{3}^{2}$, $\delta$ is the delta function in one variable, $\delta^{4}$ is the delta function in four variables, and all of the $F$ are evaluated at the same space-time point $(t,x)$. Furthermore $\sigma(s,\theta)$ is called the \emph{differential cross section or the scattering kernel}; it is a function of variables $s$ and $\theta$ which will be defined below. The delta functions express the conservation of momentum and energy: \begin{gather*} u'+v' = u+v\,. \\ \sqrt{1+|u'|^{2}}+\sqrt{1+|v'|^{2}}=\sqrt{1+|u|^{2}}+\sqrt{1+|v|^{2}} \end{gather*} Let us begin by defining the remaining variables in the collision integral. We define \begin{align*} S=(U+V)^{2} &= (u_{0}+v_{0})^{2}-|u+v|^{2}\\ &=2u_{0}v_{0} - 2u\cdot v + u_{0}^{2}-|u|^{2}+v_{0}^{2}-|v|^{2}\\ &=2(\sqrt{1+|u|^{2}}\sqrt{1+|v|^{2}}-2u\cdot v +1)\,. \end{align*} Now \begin{align*} 4g^{2} &=-(U-V)^{2}\\ &=-(u_{0}- v_{0})^{2}+|u-v|^{2}\\ &=2u_{0}v_{0} - 2u\cdot v - u_{0}^{2}+|u|^{2}-v_{0}^{2}+|v|^{2}\\ &=2(\sqrt{1+|u|^{2}}\sqrt{1+|v|^{2}}-u\cdot v +1)\\ &=s-4 \end{align*} and \[ \cos\theta=\frac{(V-U)\cdot(V'-U')}{(V-U)^{2}}. \] Furthermore, we define the Moller velocity as the scalar $v_{M}$ given by \[ v_{M}^{2}=|\hat{v}-\hat{u}|^{2}-|\hat{v}\times\hat{u}|^{2} =\frac{s(s-4)}{4v_{0}^{2}u_{0}^{2}} \] or \[ v_{M}=\frac{2g\sqrt{1+g^{2}}}{v_{0}u_{0}}\,. \] The two expressions for $v_{M}^{2}$ are equal because \begin{align*} \frac{1}{4}s(s-4) & = sg^{2} = (u_{0}v_{0}-u\cdot v + 1)(u_{0}v_{0}-u\cdot v - 1)\\ &=|u|^{2}+|v|^{2}+|u|^{2}|v|^{2}-2u_{0}v_{0}u\cdot v+(u\cdot v)^{2}\\ &= u_{0}^{2}|v|^{2}+v_{0}^{2}|u|^{2}-2u_{0}v_{0}u\cdot v-(u\times v)^{2}\\ &=u_{0}^{2}v_{0}^{2}\Bigl[\frac{|v|^{2}}{v_{0}^{2}}+ \frac{|u|^{2}}{u_{0}^{2}}- 2\frac{u}{u_{0}}\cdot\frac{v}{v_{0}}-\big|\frac{u}{u_{0}}\times \frac{v}{v_{0}}\big|^{2} \Bigr]\,. \end{align*} The relativistic equation resulting is \[ \partial_{t}F + \hat{v}\cdot\nabla_{x}F=\int_{\mathbb{R}^{3}}\int_{S^{2}} v_{M}\sigma(s,\theta)[F(u')F(v')- F(u)F(v)]d\Omega du\,, \] where $d\Omega$ is the element of surface area on $S^{2}$ and we have to write $\sigma$ as a function of $g$ and $\theta$. The Enskog equation has the same structure of the Boltzmann equation, \[ \frac{\partial f}{\partial t} +v\cdot \nabla_{x}f = E(f)\,, \] where $E(f)$ is the Enskog's collision operator defined by $E(f) = E^{+}(f)-E^{-}(f)$. The left-hand side defines the total derivative of $f$ that is equated by the Enskog's collision operator, which is expressed by the difference between the gain and loss terms respectively defined by \begin{gather*} E^{+}(f)(t,x,v)= a^{2}\int_{\mathbb{R}^{3}\times S_{+}^{2}}Y(f)\sigma(s,\theta)f(t,x+a\eta,w')d\eta dw\\ E^{-}(f)(t,x,v)= a^{2}f(t,x,v)\int_{\mathbb{R}^{3}\times S_{+}^{2}}Y(f)\sigma(s,\theta)f(t,x-a\eta,w')d\eta dw\,, \end{gather*} where $Y$ is a functional on $M$, and $S_{+}^{2} =\{\eta\in\mathbb{R}^{3}:|\eta| =1$, $\sigma(s,\theta) \geq 0\}$, and $a$ is the diameter of hard sphere. A survey of mathematical results on the existence theory for the Cauchy problem for small initial data decay to zero at infinity in the phase space is proposed in \cite{b1} as well as in papers \cite{b2,h1,i1,t1}. Several other papers have been published about this type of results. Nevertheless, the main results are contained in the papers which have been cited above. Specifically, paper \cite{i1} refers to a hard sphere gas and to initial conditions which tend exponentially to zero at infinity in the phase space. Paper \cite{h1} generalizes the result of \cite{i1}. The main result concerning the existence of solutions to the classical Boltzmann equation is a theorem by Diperna and Lions \cite{d1} that proves existence, but not uniqueness of renormalized solutions; i.e, solutions in a weak sense, which are even more general than distributional solutions. An analogous result holds in the relativistic case, as was shown by Dudynsky and Ekiel-Jezewska \cite{d2}. Regarding classical solutions, Illner and Shinbrot \cite{i1} have shown global existence of solutions to the nonrelativistic Boltzmann equation for small initial data(close to the vacuum), Galeano, Vasquez and Orozco \cite{g1} shown a result for the relativistic Boltzmann equation. When the data are close to equilibrium, global existence of classical solutions has been proved by Glassey and Strauss \cite{g3} in the relativistic case and by Ukay \cite{u1} in the nonrelativistic case. In the case of the relativistic Enskog equation we don't known results and this would be a first one. The paper is divide in two sections, we build the functional setting in the first, and we prove a lemma and the theorem of existence and uniqueness in the second one. \section{Functional Setting} For a given $\beta >0$, let \begin{align*} M=\big\{&f\in C([0,\infty)\times\mathbb{R}^{3}\times\mathbb{R}^{3}):\text{there exists $c>0$ such that}\\ & |f(t,x,v)|\leq ce^{-\beta(\sqrt{1+|v|^{2}}+|x+tv|^{2})}\big\}\,. \end{align*} This space is a Banach space (See \cite{d1}) \[ \|f\|=\sup_{t,x,v}e^{\beta(\sqrt{1+|v|^{2}}+|x+tv|^{2})}|f(t,x,v)|\,. \] We introduce the notation \[ f^{\#}(t,x,v)=f(t,x+tv,v),. \] Then the Enskog equation can be written as \[ \frac{d}{dt}f^{\#}(t,x,v)=E^{\#}(f)\,. \] Therefore $f^{\#}(t,x,v)=f_{0}(x,v)+\int_{0}^{t}E^{\#}(f) d\tau$\,. Now \[ E^{+}(f)(t,x,v) = a^{2}\int_{\mathbb{R}^{3}\times S_{+}^{2}}Y(f)\sigma(s,\theta)f(t,x,v')f(t,x+a\eta,w')d\eta dw \] and \begin{align*} &E^{+}(f^{\#})(t,x,v)\\ &= a^{2}\int_{\mathbb{R}^{3}\times S_{+}^{2}}Y(f^{\#})\sigma(s,\theta)f^{\#}(t,x,v')f^{\#}(t,x+a\eta,w')d\eta dw\\ &= a^{2}\int_{\mathbb{R}^{3}\times S_{+}^{2}}Y(f^{\#})\sigma(s,\theta)f(t,x+tv,v')f(t,x+a\eta+tv,w')d\eta dw\\ &= a^{2}\int_{\mathbb{R}^{3}\times S_{+}^{2}} \! Y(f^{\#})\sigma(s,\theta)f^{\#}(t,x+t(v-v'),v') f^{\#}(t,x+a\eta+t(v-w'),w')d\eta dw\,. \end{align*} Analogously \begin{align*} &E^{-}(f^{\#})(t,x,v)\\ &= a^{2}f(t,x+tv,v)\int_{\mathbb{R}^{3}\times S_{+}^{2}}Y(f^{\#})\sigma(s,\theta)f(t,x-a\eta+tv,w)d\eta dw\\ & =a^{2}f^{\#}(t,x,v)\int_{\mathbb{R}^{3}\times S_{+}^{2}}Y(f^{\#})\sigma(s,\theta) f^{\#}(t,x-a\eta+t(v-w),w)d\eta dw\,. \end{align*} \section{Relativistic Enskog Equation} \begin{lemma} \label{lem1} Suppose that $\sigma(s,\theta)\in L_{\rm loc}^{1}(\Omega)$ and that there is a constant $c > 0$ such that $|Y(f^{\#})|\leq c\|f^{\#}\|$ for every $f^{\#}\in M$. Then for some constant $L>0$, \begin{gather*} \int_{0}^{t}|E^{+}(f^{\#})|d\tau \leq \frac{4acL\pi^{2}}{\beta^{4}|v|}e^{\beta\sqrt{1+|v|^{2}}}\|f^{\#}\|^{3}\\ \int_{0}^{t}|E^{-}(f^{\#})|d\tau \leq \frac{4acL\pi^{2}}{\beta^{4}|v|}e^{\beta\sqrt{1+|v|^{2}}}\|f^{\#}\|^{3}\,. \end{gather*} \end{lemma} \begin{proof} Note that \begin{align*} |E^{+}(f^{\#})| &\leq a^{2}\int_{\mathbb{R}^{3}\times S_{+}^{2}}c\|f^{\#})\|\, |\sigma(s,\theta)|\, |f^{\#}(t,x+t(v-v'),v')| e^{\beta(\sqrt{1+|v'|^{2}}+|x+tv|^{2})}\\ &\quad\times |f^{\#}(t,x+a\eta+t(v-w'),w')| e^{\beta(\sqrt{1+|w'|^{2}}+|x+a\eta +tv|^{2})}\\ &\quad\times e^{-\beta(\sqrt{1+|v'|^{2}}+|x+tv|^{2})}e^{-\beta(\sqrt{1+|w'|^{2}}+|x+a\eta +tv|^{2})}d\eta dw\,. \end{align*} Since $\sigma(s,\theta)\in L_{\rm loc}^{1}(\Omega)$, there is a constant $L>0$ such that \begin{align*} &|E^{+}(f^{\#})|\\ &\leq a^{2}\int_{\mathbb{R}^{3}\times S_{+}^{2}}cL\|f^{\#})\|^{3}e^{-\beta(\sqrt{1+|v'|^{2}}+|x+tv|^{2})} e^{-\beta(\sqrt{1+|w'|^{2}}+|x+a\eta +tv|^{2})}d\eta dw\,. \end{align*} Applying the conservation of energy law, we obtain \begin{align*} &\int_{\mathbb{R}^{3}\times S_{+}^{2}}cL\|f^{\#})\|^{3}e^{-\beta(\sqrt{1+|v'|^{2}}+|x+tv|^{2})} e^{-\beta(\sqrt{1+|w'|^{2}}+|x+a\eta +tv|^{2})}d\eta dw \\ &= cL\|f^{\#})\|^{3}\int_{\mathbb{R}^{3}\times S_{+}^{2}}e^{-\beta(\sqrt{1+|v|^{2}}+|x+tv|^{2})} e^{-\beta(\sqrt{1+|w|^{2}}+|x+a\eta +tv|^{2})}d\eta dw\,. \end{align*} Moreover, \begin{align*} & cL\|f^{\#})\|^{3}\int_{\mathbb{R}^{3}\times S_{+}^{2}}e^{-\beta(\sqrt{1+|v|^{2}}+|x+tv|^{2})} e^{-\beta(\sqrt{1+|w|^{2}}+|x+a\eta +tv|^{2})}d\eta dw \\ &= cL\|f^{\#})\|^{3}e^{-\beta\sqrt{1+|v|^{2}}}e^{-\beta|x+tv|^{2}} \int_{\mathbb{R}^{3}\times S_{+}^{2}}e^{-\beta(\sqrt{1+|w|^{2}}+|x+a\eta +tv|^{2})}d\eta dw\,. \end{align*} By Fubbini's theorem, \begin{align*} \int_{\mathbb{R}^{3}\times S_{+}^{2}}e^{-\beta(\sqrt{1+|w|^{2}}+|x+a\eta +tv|^{2})}d\eta dw &=\int_{\mathbb{R}^{3}}e^{-\beta\sqrt{1+|w|^{2}}}\bigl( \int_{S_{+}^{2}}e^{-\beta|x+a\eta +tv|^{2}}d\eta\bigr)dw\\ &\leq \frac{4\pi}{\beta^{3}}\sqrt{\frac{\pi}{\beta}}\cdot\frac{1}{a}\,. \end{align*} Therefore, \[ |E^{+}(f^{\#})|\leq \frac{4a^{2}cL\pi^{\frac{3}{2}}}{\beta^{\frac{7}{2}} a}\|f^{\#})\|^{3} e^{-\beta\sqrt{1+|v|^{2}}}e^{-\beta|x+tv|^{2}}\,. \] Hence \begin{align*} \int_{0}^{t}|E^{+}(f^{\#})|d\tau &\leq \frac{4acL\pi^{\frac{3}{2}}}{\beta^{\frac{7}{2}}}\|f^{\#})\|^{3} e^{-\beta\sqrt{1+|v|^{2}}}\int_{0}^{\infty}e^{-\beta|x+tv|^{2}}d\tau\\ &\leq \frac{4acL\pi^{2}}{\beta^{4}|v|}\|f^{\#})\|^{3} e^{-\beta\sqrt{1+|v|^{2}}}\,. \end{align*} Then \begin{align*} |E^{-}(f^{\#})| &\leq a^{2} |f(t,x+tv,v)|e^{\beta(\sqrt{1+|v|^{2}}+ |x+tv|^{2}) }e^{-\beta(\sqrt{1+|v|^{2}}+ |x+tv|^{2})}\\ &\quad\times \int_{\mathbb{R}^{3}\times S_{+}^{2}}|Y(f^{\#})\|\sigma(s,\theta)|f^{\#}(t,x-a\eta + tv -tw, w)\\ &\quad\times e^{\beta(\sqrt{1+|w|^{2}}+ |x-a\eta+tv|^{2}) }e^{-\beta(\sqrt{1+|w|^{2}}+ |x-a\eta+tv|^{2})}d\eta dw\\ &\leq a^{2}\|f^{\#}\|^{3}cL e^{-\beta\sqrt{1+|v|^{2}}}e^{-\beta|x+tv|^{2}} \int_{\mathbb{R}^{3}\times S_{+}^{2}}e^{-\beta(\sqrt{1+|w|^{2}}+ |x-a\eta+tv|^{2})}d\eta dw\\ &\leq a^{2}\|f^{\#}\|^{3}cL e^{-\beta\sqrt{1+|v|^{2}}}e^{-\beta|x+tv|^{2}} \int_{\mathbb{R}^{3}}e^{-\beta\sqrt{1+|w|^{2}}}dw\int_{S_{+}^{2}}\! e^{-\beta|x-a\eta+tv|^{2}}d\eta\\ &\leq \frac{4a^{2}cL\pi}{\beta^{3}}e^{-\beta\sqrt{1+|v|^{2}}}e^{-\beta|x+tv|^{2}} \sqrt{\frac{\pi}{\beta}}\frac{1}{a}\|f^{\#}\|^{3}\\ &\leq \frac{4acL\pi^{3/2}}{\beta^{7/2}}e^{-\beta\sqrt{1+|v|^{2}}}e^{-\beta|x+tv|^{2}} \|f^{\#}\|^{3} \end{align*} and \begin{align*} \int_{0}^{t}|E^{-}(f^{\#})|d\tau &\leq\frac{4acL\pi^{3/2}}{\beta^{7/2}} e^{-\beta\sqrt{1+|v|^{2}}}\int_{0}^{t}e^{-\beta|x+\tau v|^{2}}d\tau \|f^{\#}\|^{3}\\ &\leq\frac{4acL\pi^{2}}{\beta^{4}|v|} e^{-\beta\sqrt{1+|v|^{2}}}\|f^{\#}\|^{3}\,. \end{align*} So that \[ \int_{0}^{t}|E^{-}(f^{\#})|d\tau \leq \frac{4acL\pi^{2}}{\beta^{4}|v|}\|f^{\#})\|^{3} e^{-\beta\sqrt{1+|v|^{2}}} \] which completes the proof\end{proof} \begin{theorem} \label{thm1} Suppose that $\sigma(s,\theta)\in L_{\rm loc}^{1}(\Omega)$ and there exists $c > 0$ such that $|Y(f^{\#}|\leq c\|f^{\#})\|$ for every $f^{\#}\in M_{R}=\{f\in M:\|f\|\leq R\}$ with $R^{2} < \frac{\beta^{4}|v|}{16\pi^{2}cLa}$ and $\|f_{0}\|<\frac{R}{2e^{-\beta|x|^{2}}}$. Then the Enskog relativistic equation has solution in $M_{R}$. \end{theorem} \begin{proof} We define the operator $\boldsymbol{\mathcal{F}}$ on $M$ by \[\boldsymbol{\mathcal{F}} f^{\#}=f_{0}(x,v) +\int_{0}^{t}|E^{\#}(f)|d\tau\,. \] Then \begin{align*} |\boldsymbol{\mathcal{F}} f^{\#}|&\leq |f_{0}(x,v)| +\big|\int_{0}^{t}E^{\#}(f)d\tau\big|\\ &\leq |f_{0}(x,v)|e^{\beta(\sqrt{1+|v|^{2}}+|x|^{2})}e^{-\beta(\sqrt{1+|v|^{2}}+|x|^{2})} +\big|\int_{0}^{t}E^{+}(f^{\#})- E^{-}(f^{\#})d\tau\big|\\ &\leq \|f_{0}\|e^{-\beta\sqrt{1+|v|^{2}}}e^{-\beta|x|^{2}}+\int_{0}^{t}|E^{+}(f^{\#})- E^{-}(f^{\#})|d\tau\\ &\leq \|f_{0}\|e^{-\beta\sqrt{1+|v|^{2}}}e^{-\beta|x|^{2}}+ \frac{8acL\pi^{2}}{\beta^{4}|v|}\|f^{\#})\|^{3} e^{-\beta\sqrt{1+|v|^{2}}}\\ &\leq \big[\frac{R}{2}+\frac{8acL\pi^{2}}{\beta^{4}|v|}R^{3}\big] e^{-\beta\sqrt{1+|v|^{2}}}\\ &=R e^{-\beta\sqrt{1+|v|^{2}}} \big[\frac{1}{2}+\frac{8acL\pi^{2}}{\beta^{4}|v|}R^{2}\big]\\ &\leq Re^{-\beta\sqrt{1+|v|^{2}}}\big[\frac{1}{2}+\frac{8acL\pi^{2}}{\beta^{4}|v|} \frac{\beta^{4}|v|}{16\pi^{2}cLa}\big]\\ &\leq Re^{-\beta\sqrt{1+|v|^{2}}}\big[\frac{1}{2} + \frac{1}{2}\big]\\ &= Re^{-\beta\sqrt{1+|v|^{2}}}