Existence of a weak solution to a generalized Riemann-type hydrodynamical equation

Abstract

In this paper, we study a generalized Riemann-type hydrodynamical equation and establish the existence of a weak solution to the equation in a lower order Sobolev space $(H^s(\mathbb{R}){)}^N$ with $1<s\le \frac{3}{2}$ by using a modified pseudo-parabolic regularization method. Although the absence of some important conservation laws leads to the failure of global existence of the approximate solutions, it has been addressed by making good use of the structure of the system.

Keywords:

• Generalized Riemann equation
• weak solution
• pseudo-parabolic regularization

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35L60
• 35Q35

Acknowledgements

The author was grateful to the reviewers and editors for the valuable comments and suggestions, which greatly improved the contents of this paper. And the author would like to thank Professor Long Wei, her supervisor, for introducing this problem to her and providing guidance and support throughout the research period.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was supported by Zhejiang Provincial Natural Science Foundation of China [grant number LY21A010008].