這篇論文主要在研究單一非線性平衡律黎曼問題廣義解的存在性。而這個方程式有別於一般的平衡律，方程式有加上來源項(source term)，而這來源項是奇異函數（singular function），來源項的型式為delta函數和不連續函數的乘積，所以在分佈（distribution）下是沒有定義的。 我們先把這來源項的delta 函數光滑化，使整個來源項在分佈（distribution）下有定義，進而造出擾動黎曼問題(perturbed Riemann problem)的廣義解，我們稱這廣義解為 perturbed Riemann solutions 。 而且，perturbed Riemann solutions 取極值時( 趨近於零時)，就能逼近黎曼問題廣義解的自相似性（self-similarity），同時，這個結果也能讓我們用Lax的方法去探討非線性平衡律。 We study the existence of generalized solutions to the Riemann problem for some scalar nonlinear balance law. The source term of equation is singular in the sense of a product of delta function and discontinuous function (so that it is undefined in distribution). We construct the generalized solutions based on a limiting process of measurable solutions (so-called perturbed Riemann solutions) for associated perturbed Riemann problem. The characteristic method is applied to study the behavior of perturbed Riemann solutions. Furthermore, the self-similarity of generalized solutions to our Riemann problem can be obtained from the limiting behavior of perturbed Riemann solutions, and this enables us to apply Lax's method to nonlinear balance laws.
