Nonlinear algebraic analysis of delta shock wave solutions to Burgers’ equation. (English) Zbl 1059.35122

Marti, J.-A.

Pac. J. Math. 210, No. 1, 165-187 (2003).

This paper uses an abstract algebraic analysis to study delta shock wave solutions to Burgers’ equation. It uses three fundamental structures in order to define in a very general way a sheaf of differential algebras that contains within it most of the special cases that are encountered in the theory of generalized functions.
By making a suitable choice of structures, Burgers’ equation with a Dirac \(\delta\) measure as initial data is considered, and a generalized \(\delta\)-shock wave is constructed that forms an approximate solution that is self-similar to the initial data.

Reviewer: Alan Jeffrey (Newcastle upon Tyne)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35L60	First-order nonlinear hyperbolic equations
58J47	Propagation of singularities; initial value problems on manifolds
76L05	Shock waves and blast waves in fluid mechanics
46F99	Distributions, generalized functions, distribution spaces

Keywords:

Burgers equation; generalized \(\delta\)-shock; self-similarity; sheaf of differential algebras; generalized functions