Multi-Dimensional Riemann Initial-Boundary Value Problems For The Scalar Nonlinear Conservation Law

In this paper, we study n-dimensional Riemann initial-boundary value problems for the scalar nonlinear conservation law, when the boundary is a smooth n-1 dimensional manifold, the initial data and the boundary data are both constants. Take two-dimensional situation as the typical research, we study the expression of boundary condition and its geometry feature. For the piecewise smooth weak entropy solutions of two-dimensional initial-boundary value problems for the scalar conservation law, we get the conditions including Rankine-Hugoniot discontinuity surface condition, internal entropy inequality, boundary entropy inequality, etc. According to the uniqueness, we construct the global weak entropy solutions of two-dimensional Riemann initial-boundary value problems for the scalar conservation law. First of all, we construct the corresponding global weak entropy solutions of two-dimensional Riemann initial value problems for the scalar conservation law, and then verify that the solutions satisfy the Rankine-Hugoniot discontinuity surface condition and internal entropy inequality, etc. Secondly, restricted in the region with boundary, the solutions are verified to satisfy the entropy inequality for boundary condition. Thus, we get the global weak entropy solutions of two-dimensional Riemann initial-boundary value problems for the scalar conservation law, which is not, self-similar. Furthermore, two examples are given that are respectively in the case of bending boundary and straight line boundary. In these examples, we not only obtain the explicit formula of the solutions, but also release the numerical simulation for each solution. Finally, we popularize the established method to n-dimensional situation.