论文标题
1D标量平衡法律的初始边界值问题严格凸出
Initial boundary value problem for 1D scalar balance laws with strictly convex flux
论文作者
论文摘要
最近，Adimurthi等人最近为纯初始值问题获得了针对1D标量平衡定律的Lax-Oleinik类型的显式公式。在[1]中。在本文中，通过引入合适的边界功能，我们为初始边界值问题建立了一个Lax-Oleinik类型公式。对于纯的初始值问题，相应的汉密尔顿 - 雅各比方程的解决方案被证明是在称为H曲线的一组曲线上的功能的最小化器。在目前的情况下，与保护法相比，与保护法相比，H-Curve在四分之一平面中连接任何两个点的部分可能会越过边界$ x = 0。此外，这使Bardos，Le Roux和Nedelec的意义上的边界条件的验证变得复杂[2]。为了验证边界条件，根据这些点处的最小化器的结构，将边界点分为三种类型。最后，通过引入特征三角形，我们构建了广义特征，并表明明确的解决方案是可允许的。
A Lax-Oleinik type explicit formula for 1D scalar balance laws has been recently obtained for the pure initial value problem by Adimurthi et al. in [1]. In this article, by introducing a suitable boundary functional, we establish a Lax-Oleinik type formula for the initial boundary value problem. For the pure initial value problem, the solution for the corresponding Hamilton-Jacobi equation turns out to be the minimizer of a functional on the set of curves known as h-curves. In the present situation, part of the h-curve joining any two points in the quarter plane may cross the boundary $x = 0.$ This phenomenon breaks the simplicity of the minimization process through the boundary functional compared to the case of conservation laws. Moreover, this complicates the verification of the boundary condition in the sense of Bardos, le Roux, and Nedelec [2]. To verify the boundary condition, the boundary points are classified into three types depending on the structure of the minimizers at those points. Finally, by introducing characteristic triangles, we construct generalized characteristics and show that the explicit solution is entropy admissible.
