samples/texts/1513018/page_16.md

where $G_{\alpha+1/2}$ is defined by (20) and $u_{\alpha+1/2}, v_{\alpha+1/2}$ are given by (21). The quantity $N_{\alpha+1/2}$ satisfies the following moment relations

$$\begin{array}{cccc}& {\int }_{{\mathbb{R}}^2}\left(\begin{array}{c}1\\ \xi \\ \gamma \end{array}\right)N_{\alpha +1\mathrm{/}2}d\xi d\gamma =\left(\begin{array}{c}{G}_{\alpha +1\mathrm{/}2}\\ u_{\alpha +1\mathrm{/}2}{G}_{\alpha +1\mathrm{/}2}\\ v_{\alpha +1\mathrm{/}2}{G}_{\alpha +1\mathrm{/}2}\end{array}\right),{\int }_{{\mathbb{R}}^2}\left(\begin{array}{c}{\xi }^2\\ \frac{\gamma }{2}\\ \frac{\gamma }{2}\end{array}\right)N_{\alpha +1\mathrm{/}2}d\xi d\gamma =\left(\left(\begin{array}{c}\frac{u_{\alpha +1\mathrm{/}2}^2}{2}+\frac{g}{4}h\\ \frac{v_{\alpha +1\mathrm{/}2}^2}{2}+\frac{g}{4}h\\ \frac{v_{\alpha +1\mathrm{/}2}^2}{2}+\frac{g}{4}h\end{array}\right)\right)G_{\alpha +1\mathrm{/}2}\mathrm{.}& & \text{(64)}\end{array}\backslash int\_\{\backslash mathbb\{R\}^2\}\; \backslash begin\{pmatrix\}\; 1\; \backslash \backslash \; \backslash xi\; \backslash \backslash \; \backslash gamma\; \backslash end\{pmatrix\}\; N\_\{\backslash alpha+1/2\}\; d\backslash xi\; d\backslash gamma\; =\; \backslash begin\{pmatrix\}\; G\_\{\backslash alpha+1/2\}\; \backslash \backslash \; u\_\{\backslash alpha+1/2\}\; G\_\{\backslash alpha+1/2\}\; \backslash \backslash \; v\_\{\backslash alpha+1/2\}\; G\_\{\backslash alpha+1/2\}\; \backslash end\{pmatrix\},\; \backslash quad\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; \backslash begin\{pmatrix\}\; \backslash xi^2\; \backslash \backslash \; \backslash frac\{\backslash gamma\}\{2\}\; \backslash \backslash \; \backslash frac\{\backslash gamma\}\{2\}\; \backslash end\{pmatrix\}\; N\_\{\backslash alpha+1/2\}\; d\backslash xi\; d\backslash gamma\; =\; \backslash left(\; \backslash begin\{pmatrix\}\; \backslash frac\{u\_\{\backslash alpha+1/2\}^2\}\{2\}\; +\; \backslash frac\{g\}\{4\}h\; \backslash \backslash \; \backslash frac\{v\_\{\backslash alpha+1/2\}^2\}\{2\}\; +\; \backslash frac\{g\}\{4\}h\; \backslash \backslash \; \backslash frac\{v\_\{\backslash alpha+1/2\}^2\}\{2\}\; +\; \backslash frac\{g\}\{4\}h\; \backslash end\{pmatrix\}\; \backslash right)\; G\_\{\backslash alpha+1/2\}.\; \backslash tag\{64\}$$

Notice that from (20), we can give a kinetic interpretation on the exchange terms under the form

$$G_{\alpha +1\mathrm{/}2}=-\sum _{j=1}^N(\sum _{p=1}^{\alpha }{l}_p-{\mathbf{1}}_{j\le \alpha }){\int }_{{\mathbb{R}}^2}\left(\begin{array}{c}\xi \\ \gamma \end{array}\right)\cdot {\mathrm{\nabla }}_{x,y}{M}_{j}d\xi d\gamma ,(65)G\_\{\backslash alpha+1/2\}\; =\; -\; \backslash sum\_\{j=1\}^\{N\}\; \backslash left(\; \backslash sum\_\{p=1\}^\{\backslash alpha\}\; l\_p\; -\; \backslash mathbf\{1\}\_\{j\; \backslash le\; \backslash alpha\}\; \backslash right)\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; \backslash begin\{pmatrix\}\; \backslash xi\; \backslash \backslash \; \backslash gamma\; \backslash end\{pmatrix\}\; \backslash cdot\; \backslash nabla\_\{x,y\}\; M\_j\; d\backslash xi\; d\backslash gamma,\; \backslash quad\; (65)$$

for $\alpha = 1, \dots, N$.

Then we have the two following results.

Proposition 4.1 The functions u^N defined by (17) and h are strong solutions of the system (18)-(19) if and only if the sets of equilibria {M_α}^N_α=1, {N_α+1/2}^N_α=0 are solutions of the kinetic equations defined by

$$({\mathcal{B}}_{\alpha })\frac{\mathrm{\partial }{M}_{\alpha }}{\mathrm{\partial }t}+\left(\begin{array}{c}\xi \\ \gamma \end{array}\right)\cdot {\mathrm{\nabla }}_{x,y}{M}_{\alpha }-g{\mathrm{\nabla }}_{x,y}{z}_b\cdot {\mathrm{\nabla }}_{\xi ,\gamma }{M}_{\alpha }-N_{\alpha +1\mathrm{/}2}+N_{\alpha -1\mathrm{/}2}=Q_{\alpha },(66)(\backslash mathcal\{B\}\_\backslash alpha)\; \backslash quad\; \backslash frac\{\backslash partial\; M\_\backslash alpha\}\{\backslash partial\; t\}\; +\; \backslash begin\{pmatrix\}\; \backslash xi\; \backslash \backslash \; \backslash gamma\; \backslash end\{pmatrix\}\; \backslash cdot\; \backslash nabla\_\{x,y\}\; M\_\backslash alpha\; -\; g\; \backslash nabla\_\{x,y\}\; z\_b\; \backslash cdot\; \backslash nabla\_\{\backslash xi,\backslash gamma\}\; M\_\backslash alpha\; -\; N\_\{\backslash alpha+1/2\}\; +\; N\_\{\backslash alpha-1/2\}\; =\; Q\_\backslash alpha,\; \backslash quad\; (66)$$

for $\alpha = 1, \dots, N$. The quantities $Q_\alpha = Q_\alpha(t, x, y, \xi, \gamma)$ are "collision terms" equal to zero at the macroscopic level, i.e. they satisfy a.e. for values of $(t, x, y)$

$${\int }_{{\mathbb{R}}^2}{Q}_{\alpha }d\xi d\gamma ={\int }_{{\mathbb{R}}^2}\xi Q_{\alpha }d\xi d\gamma ={\int }_{{\mathbb{R}}^2}\gamma Q_{\alpha }d\xi d\gamma =0.(67)\backslash int\_\{\backslash mathbb\{R\}^2\}\; Q\_\backslash alpha\; d\backslash xi\; d\backslash gamma\; =\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; \backslash xi\; Q\_\backslash alpha\; d\backslash xi\; d\backslash gamma\; =\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; \backslash gamma\; Q\_\backslash alpha\; d\backslash xi\; d\backslash gamma\; =\; 0.\; \backslash qquad\; (67)$$

Proposition 4.2 The solutions of (66) are entropy solutions if

$$\frac{\mathrm{\partial }H(M_{\alpha })}{\mathrm{\partial }t}+\left(\begin{array}{c}\xi \\ \gamma \end{array}\right)\cdot {\mathrm{\nabla }}_{x,y}H(M_{\alpha })-g{\mathrm{\nabla }}_{x,y}{z}_b\cdot {\mathrm{\nabla }}_{\xi ,\gamma }H(M_{\alpha })\le (H(N_{\alpha +1\mathrm{/}2})-H(N_{\alpha -1\mathrm{/}2})),(68)\backslash frac\{\backslash partial\; H(M\_\backslash alpha)\}\{\backslash partial\; t\}\; +\; \backslash begin\{pmatrix\}\; \backslash xi\; \backslash \backslash \; \backslash gamma\; \backslash end\{pmatrix\}\; \backslash cdot\; \backslash nabla\_\{x,y\}\; H(M\_\backslash alpha)\; -\; g\; \backslash nabla\_\{x,y\}\; z\_b\; \backslash cdot\; \backslash nabla\_\{\backslash xi,\backslash gamma\}\; H(M\_\backslash alpha)\; \backslash leq\; (H(N\_\{\backslash alpha+1/2\})\; -\; H(N\_\{\backslash alpha-1/2\})),\; \backslash quad\; (68)$$

with the notation $H(M) = H(M, \xi, \gamma, z_b)$ and $H$ defined by (61). The integration in $\xi, \gamma$ of relation (68) gives

$$\frac{\mathrm{\partial }{E}_{\alpha }}{\mathrm{\partial }t}+{\mathrm{\nabla }}_{x,y}\mathrm{.}{u}_{\alpha }(E_{\alpha }+\frac{g}{2}{h}_{\alpha }h)\le l_{\alpha }(\frac{\mathrm{\mid }{u}_{\alpha +1\mathrm{/}2}{\mathrm{\mid }}^2}{2}+g{z}_b)G_{\alpha +1\mathrm{/}2}-l_{\alpha }(\frac{\mathrm{\mid }{u}_{\alpha -1\mathrm{/}2}{\mathrm{\mid }}^2}{2}+g{z}_b)G_{\alpha -1\mathrm{/}2}\mathrm{.}\backslash frac\{\backslash partial\; E\_\{\backslash alpha\}\}\{\backslash partial\; t\}\; +\; \backslash nabla\_\{x,y\}\; .\; u\_\{\backslash alpha\}\; (E\_\{\backslash alpha\}\; +\; \backslash frac\{g\}\{2\}\; h\_\{\backslash alpha\}\; h)\; \backslash leq\; l\_\{\backslash alpha\}\; \backslash left(\; \backslash frac\{|u\_\{\backslash alpha+1/2\}|^2\}\{2\}\; +\; gz\_b\; \backslash right)\; G\_\{\backslash alpha+1/2\}\; -\; l\_\{\backslash alpha\}\; \backslash left(\; \backslash frac\{|u\_\{\backslash alpha-1/2\}|^2\}\{2\}\; +\; gz\_b\; \backslash right)\; G\_\{\backslash alpha-1/2\}.$$

Proof of proposition 4.1 The proof relies on averages w.r.t the variables $\xi, \gamma$ of Eq. (66) against the vector $(1, \xi, \gamma)^T$. Using relations (60), (63), (64) and the properties of the collision terms (67), the quantities

$${\int }_{{\mathbb{R}}^2}({\mathcal{B}}_{\alpha })d\xi d\gamma ,{\int }_{{\mathbb{R}}^2}\xi ({\mathcal{B}}_{\alpha })d\xi d\gamma ,\text{and}{\int }_{{\mathbb{R}}^2}\gamma ({\mathcal{B}}_{\alpha })d\xi d\gamma ,\backslash int\_\{\backslash mathbb\{R\}^2\}\; (\backslash mathcal\{B\}\_\backslash alpha)\; d\backslash xi\; d\backslash gamma,\; \backslash quad\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; \backslash xi\; (\backslash mathcal\{B\}\_\backslash alpha)\; d\backslash xi\; d\backslash gamma,\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash int\_\{\backslash mathbb\{R\}^2\}\; \backslash gamma\; (\backslash mathcal\{B\}\_\backslash alpha)\; d\backslash xi\; d\backslash gamma,$$

respectively give Eqs. (28) and (19). The sum for $\alpha = 1$ to $N$ of Eqs. (28) with (20) gives (18) that completes the proof. $\blacksquare$