We demonstrate the feasibility of a scheme to obtain approximate weak solutions to the (inviscid) Burgers equation in conservation and Hamilton-Jacobi form, treated as degenerate elliptic problems. We show different variants recover non-unique weak solutions as appropriate, and also specific constructive approaches to recover the corresponding entropy solutions.

翻译：我们论证了一种将（无粘）Burgers方程视为退化椭圆问题，并获取其近似弱解方案的可行性。该方案涵盖守恒形式与Hamilton-Jacobi形式。研究表明，不同变体能够适当地恢复非唯一弱解，同时提出了恢复相应熵解的具体构造性方法。