(PDF file: paper9.pdf)

5 Compactness of entropy

In this section, we show the compactness of

$$\{U(u^\Delta (t,x))_t + F(u^\Delta (t,x))_x ; \hspace{1em} (u^0_1,u^0_3,\ldots,u^0_{2L-1})\in D_L, L\geq L_0\}$$

in $\mbox{$H^{-1}_{\rm loc}((0,T)\times(0,1))$}$ for any smooth entropy pair $(U(u),F(u))$, where $L_0$ is sufficiently large integer for (19) and (22).

For simplify, we set $u^{2n}_{0,+}(x)$ as the function consists of the step values $u^{2n}_{0,j}$

$$u^{2n}_{0,+}(x) = u^{2n}_{0,j} \hspace{1em}(x\in E^{2n}_j)$$

and use notations

$$u^n_+(x)=u(n\Delta t+0,x),\hspace{2em}u^n_-(x)=u(n\Delta t-0,x),\hspace{2em} f^n_j = f(u^n_j),$$

and so on.

We consider for a particular entropy pairs $(U^\ast,F^\ast)$ defined by

$$U^\ast = \frac{u^2}{2}, \hspace{2em}F^\ast=\int_0^u uf'(u) = uf - \int_0^u f.$$

Since the approximation $u^\Delta (t,x)$ satisfies the equation (12) almost everywhere,

$$\begin{eqnarray*} %% \lefteqn{ 0 = \int_0^Tdt\int_0^1 \{U^\ast(\uD(t,x))_t 0 ... ...U^\ast)^{2n}_{0,+}\}dx \\ & & + \Sigma^\ast + I^\ast + J^\ast, \end{eqnarray*}$$

where

$$\begin{eqnarray*} J^\ast & = & \sum_{n=1}^N \sum_{j=1}^L \int_{E_{2j-1}^{2n}} ... ...\int_0^T \sum_{\mbox{shock}} (\sigma [U^\ast]-[F^\ast]) dt,\\ \end{eqnarray*}$$

$\sum_{\mbox{shock}}$ is summation for all the shock waves arise in $[0,T]\times[0,1]$, $[U^\ast]$ and $[F^\ast]$ is the difference across the wave, and $\sigma$ is the speed of it. Because of

$$\begin{eqnarray*} & & (U^\ast)'' = 1,\hspace{1em} u^{2n-1}_{+} = \frac{1}{m(E^... ...t_{E^{2n}_{2j-1}}u^{2n}_{-}dx \hspace{2em}(x\in E^{2n}_{2j-1}), \end{eqnarray*}$$

we have

$$\begin{eqnarray*} \frac{1}{2}\sum_{n=1}^{N} \int_0^1 \vert u^{2n-1}_{-} - u^{2n... ...n}_{-} - u^{2n}_{0,+}\vert^2 dx +\Sigma^\ast = -I^\ast -J^\ast. \end{eqnarray*}$$

The boundedness of $u^\Delta (t,x)$

$$\vert u^\Delta (t,x)\vert\leq M+TG_0$$

shows that

$$\begin{eqnarray*} \vert J^\ast\vert & \leq & \sum_{n=1}^N \sum_{j=1}^L \int_{E_... ...x + \int_0^1\vert U^\ast(+0,x)\vert dx \\ & \leq & (M+TG_0)^2. \end{eqnarray*}$$

Hence we obtain the inequality

$${\frac{1}{2}\sum_{n=1}^{N}\int_0^1\vert u^{2n-1}_{-}-u^{2n-1}_{+}... ...+\frac{1}{2}\sum_{n=1}^{N} \int_0^1 \vert u^{2n}_{-} - u^{2n}_{0,+}\vert^2 dx }$$

$$+ \int_0^T \sum_{\mbox{shock}} (\sigma [U^\ast]-[F^\ast]) dt \hspace{1em}\leq \hspace{1em}C(M,T,G_0),$$ (27)

where $C(M,T,G_0)$ is a constant depends on $M$, $T$ and $G_0$.

Let $\phi(t,x)$ be a function in $C_0^\infty((0,T)\times(0,1))$. The integral

$$\int\hspace{-6pt}\int _{[0,T]\times[0,1]}\{U(u^\Delta )\phi_t+F(u^\Delta )\phi_x\}dtdx$$ (28)

is divided into four parts

$$\begin{eqnarray*} \lefteqn{ \int\hspace{-6pt}\int _{[0,T]\times[0,1]}\{U(u^\Del... ...)_x\}dtdx \\ & = & \Sigma(\phi)+L_1(\phi)+L_2(\phi)+L_3(\phi), \end{eqnarray*}$$

where

$$\begin{eqnarray*} \Sigma(\phi) & = & \int_0^T \sum_{\mbox{shock}} (\sigma [U]-[... ...{2n}} (\phi-\phi_{2j-1}^{2n})(U^{2n}_{-} - U^{2n}_{0,+})dx,\\ \end{eqnarray*}$$

and $\phi^n_j=\phi(n\Delta t,j\Delta x)$. We estimate $L_j(\phi)$

$$\begin{eqnarray*} \vert L_1(\phi)\vert & \leq & \Vert\phi\Vert _{C^0}\max_{\ver... ...T,G_0,\Lambda_2) \Vert\phi\Vert _{C^\beta}(\Delta x)^{\beta-1/2} \end{eqnarray*}$$

and $\Sigma(\phi)$

$$\vert\Sigma(\phi)\vert\leq \Vert\phi\Vert _{C^0}\max_{\vert ... ...''(u)\vert\Sigma^\ast \leq C(U'',M,T,G_0)\Vert\phi\Vert _{C^0}$$

by the similar way in [3] (I). Hence the integral (28) is the sum of two linear operators $T_1(\phi)$ and $T_2(\phi)$ which satisfy

$$\vert T_1(\phi)\vert \leq C \Vert\phi\Vert _{C^0},\hspace{2e... ...\vert \leq C \Vert\phi\Vert _{C^\beta} (\Delta x)^{\beta-1/2},$$

where the constant $C$ depends on $M$, $T$, $G_0$, maximum value of $U'$ and $U''$, and $\Lambda_2$. This shows the following proposition by the argument in [3] (I).

PROPOSITION 5   For any smooth entropy pair $(U,F)$, the set

$$\{U(u^\Delta (t,x))_t + F(u^\Delta (t,x))_x ; \hspace{1em} (u^0_1,u^0_3,\ldots,u^0_{2L-1})\in D_L, L\geq L_0\}$$

is relatively compact in $\mbox{$H^{-1}_{\rm loc}((0,T)\times(0,1))$}$.

Proposition 5 and Tartar's theorem ([15]) yields the existence of subsequence which converges almost everywhere. It also valid for the initial data $(\bar{u}^0_1,\bar{u}^0_3,\ldots,\bar{u}^0_{2L-1})$ (§4).