Distributional products and global solutions for nonconservative inviscid Burgers equation (Q1399355)

The paper deals with the existence of global distributional solutions for the nonconservative Burgers equation (NCB) \(\partial _{t}u+u\partial _{x}u=0, \) \(t\geq 0\) and \(x\in \mathbb{R}\). First the author introduces his definition of the product of distributions used in the paper, called \(\alpha -\)product, and sketches some properties of this product; then he defines the concept of global \(\alpha -\)solution for the (NCB) equation. The following sections of the paper are devoted to the conditions on \(C^{1}\) functions \(u(x,t)\) having a jump discontinuity along a \(C^{1}\)\ curve \( \gamma \)\ of the \((x,t)-\)plane,\ to be global \(\alpha -\)solutions for the (NCB) equation. The relationship with the global weak solutions of the conservative Burgers equation \(\partial _{t}u+\partial _{x}(\frac{1}{2} u^{2})=0\) is also considered.