Multiplication of distributions and travelling wave solutions for the Keyfitz-Kranzer system
Since the impossibility result on the multiplication of distributions, it is well known that the space of distributions is not suitable for studying nonlinear operations, in particular, nonlinear partial differential equations. Nevertheless, certain authors have introduced an intrinsic multiplication of distributions which works for some classes of distributions. The author of this paper first recalls the definition of an intrinsic multiplication called \(\alpha\)-product of distributions depending on a regularizing test function \(\alpha\) satisfying \(\int \alpha =1,\) and gives some properties of this product. Rephrasing the following system of nonlinear partial differential equations \[ \begin{aligned} u_{t}+(u^{2}-v)_{x}&=0, \\ v_{t}+(u^{3}/3-u)_{x}&=0,\end{aligned} \] on the basis of his definition, he introduces the concept of \(\alpha\)-solutions for the system. A~comparison with classical solutions is given. In the last section, travelling wave solutions with distributional profiles for the cited system are studied as well as necessary and sufficient conditions for the propagation of distributional wave profiles.
scientific article; zbMATH DE number 6965392
scientific article