\documentstyle{amsart} \newtheorem{Theorem}{Theorem}[section] \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Corollary}[Theorem]{Corollary} \begin{document} \noindent{\small {\sc Electronic Journal of Differential Equations}\newline Vol. {\bf 1994}(1994), No. 05, pp. 1-4. Published July 19, 1994. \newline ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{1.5cm} \thanks{\copyright 1994 Southwest Texas State University and University of North Texas \newline\indent Submitted: June 12, 1994.} \title[\hfilneg EJDE--1994/05\hfil Uniqueness of Entropy Solutions \hfil ] {A Note on the Uniqueness of Entropy Solutions to First Order Quasilinear Equations} \author[\hfil D.J. Diller\hfil EJDE--1994/05\hfilneg]{David J. Diller} \address{Lunt Hall, Northwestern University, 2033 Sheridan Rd., Evanston, IL 60208} \email{diller@@math.nwu.edu} \keywords{ Burger's Equation, Entropy Solution, Scalar Conservation Law} \subjclass{35L60,35L65} \begin{abstract} In this note, we consider entropy solutions to scalar conservation laws with unbounded initial data. In particular, we offer an extension of Kru$\check{\mbox{z}}$khov's uniqueness proof (see [1]). \end{abstract} \maketitle \newcommand{\absval}[1]{\mid\!\! {#1}\!\!\mid} \newcommand{\av}[1]{\mid {#1} \mid} \section{Introduction} We are concerned with the following Cauchy problem: $$\left\{ \begin{split} u_t +\mbox{div}F(u) =0 & \;\;\;\mbox{in} \;\;S_T =\Bbb{R}^N\times(0,T) \\ u(x,0)=u_0(x) & \;\;\; x\in\Bbb{R}^N . \end{split} \right.$$ Here $F=(F_1,\cdots,F_N)\in [C^{0,1}(\Bbb{R})]^N$, and $u_0\in L^1_{loc}(\Bbb{R}^N)$. In particular, we are interested in the entropy solutions to (1). We say that $u\in L^{\infty}_{loc}(S_T)$ is an entropy solution to (1) if $$\iint_{S_T} \mbox{sign}(u-k)\left[ (u-k)\phi_t + (F(u)-F(k))\cdot D\phi\right] \,dx\,dt\, \geq 0,$$ for all $\phi\in C_0^{\infty}(S_T)$, $\phi\geq 0$, and all $k\in\Bbb{R}$, and there exists a set $\Gamma_0 \subseteq [0,T]$ of measure zero, such that for all compact sets $K\subseteq\Bbb{R}^N$ $$\lim_{\overset{t\rightarrow 0^+}{t\notin\Gamma_0}}\|u(\cdot,t)-u_0\|_{1,K} =0.$$ In [1], Kru$\check{\mbox{z}}$khov proves existence and uniqueness of an entropy solution to (1) when $u_0$ is bounded and F is sufficiently smooth. If $u_0,v_0\in L^1(\Bbb{R}^N)\cap L^{\infty}(\Bbb{R}^N)$ with corresponding entropy solutions $u,v$ respectively then \begin{displaymath} \int_{\Bbb{R}^N} \absval{u(x,t)-v(x,t)} \,dx\,\leq \int_{\Bbb{R}^N}\absval{u_0(x)-v_0(x)}\,dx\, \end{displaymath} for a.e. $t\in[0,T]$ (see [1] equation 3.1). If $u_0\in L^1(\Bbb{R}^N)$ (but not bounded) then there is a natural candidate for an entropy solution with this initial data. This note is motivated by the following two questions: (i) Is this candidate an entropy solution? (ii) If it is an entropy solution then is it the unique entropy solution? \noindent This note is a partial answer to the second of these two questions. \section{Main Result} In proving uniqueness Kru$\check{\mbox{z}}$khov proves the following Proposition: \begin{Proposition} If $u$ and $v$ are entropy solutions to (1) satisfying \begin{displaymath} \left\| \frac{F(u)-F(v)}{u-v} \right\|_{\infty,S_T}\leq M \end{displaymath} then $u=v$ almost everywhere in $S_T$. \end{Proposition} The primary result of this note is the following improvement of Proposition 2.1. \begin{Proposition} If $u$ and $v$ are entropy solutions to (1) satisfying $$\left\| \frac{F(u(\cdot,t))-F(v(\cdot,t))}{u(\cdot,t)-v(\cdot,t)} \right\|_{\infty,B_{\rho}}\leq M(t,\rho)$$ where M satisfies $$\lim_{\rho\rightarrow\infty}\left(\rho-\int_0^T M(t,\rho) \,dt\,\right) =\infty$$ then $u=v$ almost everywhere in $S_T$. \end{Proposition} The advantage of Proposition 2.2 over Proposition 2.1 is that Proposition 2.2 allows for $u_0$ to become unbounded. Set $A(u)=(F_1'(u),\cdots,F_N'(u))$. Then one can easily verify that Proposition 2.2 implies the following. \begin{Corollary} There exists at most one entropy solution to (1) satisfying \begin{displaymath} \|A(u(\cdot,t))\|_{\infty,\Bbb{R}^N}\leq M(t) \end{displaymath} where M satisfies \begin{displaymath} \int_0^T M(t)\,dt\,<\infty. \end{displaymath} \end{Corollary} As an example we apply Corollary 2.3 to the Burger's equation, i.e., $N=1$ and $F(u)=\frac{1}{2} u^2$. \begin{Lemma} If $u_0\in L^p(\Bbb{R}^N)$ with $1\leq p<\infty$ then there exists an entropy solution u of (1) satisfying $$\|u(\cdot,t)\|_{\infty,\Bbb{R}^N}\leq \frac{(2\|u_0\|_p)^{\frac{p}{p+1}}} {t^{\frac{1}{p+1}}} \in L^1([0,1]).$$ \end{Lemma} \begin{pf} Let $u$ be the almost everywhere unique minimizer of \begin{displaymath} \psi(x,t,v)=\int_x^{x-vt} u_0(s)\,ds\, +\frac{tv^2}{2}. \end{displaymath} For details see [2]. Then $u$ is an entropy solution to (1). Now $\psi(x,t,0)=0$ and by Holder's inequality \begin{align*} \psi(x,t,v)\geq & -\|u_0\|_p\absval{vt}^{\frac{p-1}{p}} +\frac{tv^2}{2} \\ =&\absval{vt}^{\frac{p-1}{p}}(-\|u_0\|_p +\frac{t^{\frac{1}{p}}\absval{v}^{\frac{p+1}{p}}}{2}) \\ >& 0 \end{align*} when \begin{displaymath} v>\frac{(2\|u_0\|_p)^{\frac{p}{p+1}}}{t^{\frac{1}{p+1}}}. \end{displaymath} This implies (6). \end{pf} Thus from Corollary 2.3 the entropy solution to Burger's equation satisfying (6) is unique. One can prove a similar estimate for solutions when $F(u)=\absval{u}^{\alpha}$ for any $\alpha>1$. \begin{pf} (of Proposition 2.2). Let $u$ and $v$ be any two entropy solutions to (1). Let $J\in C^{\infty}_{0}(-1,1)$ satisfy: \begin{equation*} \left\{ \begin{split} \int_{-1}^1 J(x) \,dx\, =1 \\ J\geq 0. \;\;\;\;\;\; \end{split} \right. \end{equation*} Any two entropy solutions $u,v$ satisfy $$\iint_{S_T} \mbox{sign}(u-v) [(u-v)\phi_t +(F(u)-F(v))\cdot D\phi] \,dx\,dt\,\geq 0$$ for all $\phi\in C_0^{\infty} (S_T)$ with $\phi\geq 0$ (see [1] equation 3.7). Set \begin{displaymath} r(t,\rho)=\rho -\int_0^t M(\tau,\rho)d\tau. \end{displaymath} Since $t\rightarrow r(t,\rho)$ is decreasing, (5) implies \begin{displaymath} \lim_{\rho\rightarrow\infty} r(t,\rho) = \infty \;\;\mbox{ for all }\;\; 0\leq t\leq T. \end{displaymath} Select $R>0$ such that if $\rho> R$ then $r(T,\rho)>0$. Fix $\rho>R.$ Set $r(t)=r(t,\rho)$. Let \$0