arXiv 2007.044212020-07-08math.APmath.OC

通量函数在空间上单点间断的守恒律方程的渐近分布

Attainable profiles for conservation laws with flux function spatially discontinuous at a single point

Fabio Ancona, Maria Teresa Chiri

备注：: 25 pages, 7 figures
DOI：: 10.1051/cocv/2020044

摘要：

考虑一个具有间断通量的标量守恒律\begin{equation*}\tag{1} \quad u_{t}+f(x,u)_{x}=0, \qquad f(x,u)= \begin{cases} f_l(u)\ &\text{if}\ x<0,\\ f_r(u)\ & \text{if} \ x>0, \end{cases} \end{equation*}，其中$u=u(x,t)$是状态变量，$f_{l}$，$f_{r}$是严格凸映射.本文从控制论的观点出发，把初始数据看作控制量，研究了方程（1）的柯西问题。令$u(x,t)\doteq \mathcal{S}_t^{AB} \overline u(x)$表示（1）的柯西问题的解，其中初始数据$u(\cdot,0)=\overline u$在$x=0$处满足与连接$(A,B)$相关联的界面熵条件（参见~ \cite{MR2195983}），我们分析了可以通过（1）在给定时间$T>0$：\begin{equation*} \mathcal{A}^{AB}(T)=\left\{\mathcal{S}_T^{AB} \,\overline u : \ \overline u\in{\bf L}^\infty(\mathbb{R})\right\}. \end{equation*}我们提供$\mathcal{A}^{AB}(T)$作为$BV_{loc}(\mathbb{R}\setminus\{0\})$中满足适当的Ole \v nik型不等式的一类函数的充分表征，并且在$x=0$中承认满足与界面熵准则相关的特定条件的单侧限制。依赖于这个特性，我们建立了${\bf L^1}_{loc}$ -紧性的一组可达到的配置文件时，初始数据$\overline u$变化在一个给定的类的一致有界的功能，在封闭的凸集的值。我们还讨论了一些应用这些结果所产生的优化问题的多孔介质流动模型，石油开采和交通流量。

英文摘要：

Consider a scalar conservation law with discontinuous flux \begin{equation*}\tag{1} \quad u_{t}+f(x,u)_{x}=0, \qquad f(x,u)= \begin{cases} f_l(u)\ &\text{if}\ x<0,\\ f_r(u)\ & \text{if} \ x>0, \end{cases} \end{equation*} where $u=u(x,t)$ is the state variable and $f_{l}$, $f_{r}$ are strictly convex maps. We study the Cauchy problem for (1) from the point of view of control theory regarding the initial datum as a control. Letting $u(x,t)\doteq \mathcal{S}_t^{AB} \overline u(x)$ denote the solution of the Cauchy problem for (1), with initial datum $u(\cdot,0)=\overline u$, that satisfy at $x=0$ the interface entropy condition associated to a connection $(A,B)$ (see~\cite{MR2195983}), we analyze the family of profiles that can be attained by (1) at a given time $T>0$: \begin{equation*} \mathcal{A}^{AB}(T)=\left\{\mathcal{S}_T^{AB} \,\overline u : \ \overline u\in{\bf L}^\infty(\mathbb{R})\right\}. \end{equation*} We provide a full characterization of $\mathcal{A}^{AB}(T)$ as a class of functions in $BV_{loc}(\mathbb{R}\setminus\{0\})$ that satisfy suitable Ole\vınik-type inequalities, and that admit one-sided limits at $x=0$ which satisfy specific conditions related to the interface entropy criterium. Relying on this characterisation, we establish the ${\bf L^1}_{loc}$-compactness of the set of attainable profiles when the initial data $\overline u$ vary in a given class of uniformly bounded functions, taking values in closed convex sets. We also discuss some applications of these results to optimization problems arising in porous media flow models for oil recovery and in traffic flow.

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