论文标题
可在单个点上空间不连续的通量函数的保护定律可获得的概况
Attainable profiles for conservation laws with flux function spatially discontinuous at a single point
论文作者
论文摘要
考虑具有不连续的通量的标量保护定律\ begin {equation*} \ tag {1} \ quad u_ {t}+f（x，x，x，u）_ {x} = 0，\ qquad f（x，x，x，x，x，x，u）= \ begin \ text {if} \ x> 0，\ end {cases} \ end {equation*}其中$ u = u（x，t）$是状态变量，$ f_ {l} $，$ f_ {r} $是严格的convex maps。我们从控制理论的角度研究了（1）的Cauchy问题。 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 $(A,B)$ （参见〜\ cit {MR2195983}），我们分析了（1）可以在给定时间$ t> 0 $：\ begin {equation*} \ mathcal {a}^a}^{ab}^{ab}（ab}（ab}（t）（ \ edline u \ in {\ bf l}^\ infty（\ mathbb {r}）\ right \}。 \ end {方程*}我们提供$ \ Mathcal {a}^{ab}（t）$作为$ bv_ {loc}中的一类功能的全面表征满足与界面熵标准有关的特定条件。依靠此特征，我们建立了$ {\ bf l^1} _ {loc} $ - 当初始数据$ \ edilelline u $在给定类别的均匀界限的函数中变化时，可实现的配置文件的紧凑性，在封闭的convex集中采用值。我们还讨论了这些结果的某些应用，这些应用是在多孔介质流量模型中引起的优化问题，以进行石油回收和交通流。
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.
