# Generalized plane delta shock waves for the *n*-dimensional zero-pressure gas dynamics with energy conservation law

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## Abstract

By virtue of the generalized plane wave solution, we study a type of generalized plane delta shock wave for the *n*-dimensional zero-pressure gas dynamics governed by the conservation of mass, momentum and energy. It is found that a special kind of generalized plane delta shock wave on which both state variables simultaneously contain the Dirac delta functions appears in Riemann solutions, which is significantly different from the customary ones on which only one state variable contains the Dirac delta function. The generalized Rankine-Hugoniot relation of the generalized plane delta shock wave is derived. Under a suitable entropy condition, we further solve a kind of *n*-dimensional Riemann problem with Randon measure as initial data, and four different explicit configurations of solutions are constructively established. Finally, the overtaking of two plane delta shock waves is analyzed.

## Key words

*n*-Dimensional zero-pressure gas dynamics energy conservation law generalized plane delta shock wave vacuum generalized Rankine-Hugoniot relation entropy condition

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## Notes

### Acknowledgement

Special gratefulness to the anonymous referee for his/her careful valuable suggestions, which have improved the original manuscript greatly.

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