A shallow-water system with vanishing buoyancy

ABSTRACT

In this work, a shallow-water system for interfacial waves in the case of a neutrally buoyant two-layer fluid system is considered. Such a situation arises in the case of large underwater lakes of compressible liquids such as CO${}_2$ in the deep ocean which may happen naturally or may be man-made. Depending on temperature and depth, such deposits may be either stable, unstable or neutrally stable, and in the current contribution, the neutrally stable case is considered. The motion of the long waves at the interface can be described by a shallow-water system which becomes triangular in the neutrally stable case. In this case, the system ceases to be strictly hyperbolic, and the standard theory of hyperbolic conservation laws may not be used to solve the initial value or even the Riemann problem. It is shown that the Riemann problem can still be solved uniquely using singular shocks containing Dirac delta distributions traveling with the shock. We characterize the solutions in integrated form, so that no measure-theoretic extension of the solution concept is needed. Uniqueness follows immediately from the construction of the solution. We characterize solutions in terms of the complex vanishing viscosity method, and show that the two solution concepts coincide.

Keywords:

• Rankine–Hugoniot deficit
• singular solutions
• weak asymptotics
• traveling waves
• Riemann problem

2010 Mathematics Subject Classifications:

• 35L65
• 35L67
• 35L80

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the Research Council of Norway under grant no. 213747/F20 and grant no. 239033/F20. HK would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program Mathematics of sea ice phenomena when work on this paper was undertaken. This work was also supported by the Simons Foundation.