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Multiplication of distributions and travelling wave solutions for the Keyfitz-Kranzer system. (English) Zbl 1412.46049

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.

MSC:

46F10 Operations with distributions and generalized functions
35D99 Generalized solutions to partial differential equations
35L67 Shocks and singularities for hyperbolic equations
35F50 Systems of nonlinear first-order PDEs
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