# Preprint 2012-015

# Global solutions to one-dimensional shallow water magnetohydrodynamic equations

## Yun-guang Lu

**Abstract:**
In this paper, we study the Cauchy problem
for the one-dimensional shallow water magnetohydrodynamic equations.
The main difficulty is the case of zero depth ($h = 0$)
since the nonlinear flux function $P (h)$ is singular
and the definition of solution is not clear near $h = 0$.
First, assuming that $h$ has a positive and lower bound,
we establish the pointwise convergence of the viscosity solutions
by using the div–curl lemma from the compensated compactness theory
to special pairs of functions $(c,f^\epsilon)$,
and obtain a global weak entropy solution.
Second, under some technical conditions on the initial data
such that the Riemann invariants $(w,z)$ are monotonic and increasing,
we introduce a “variant” of the vanishing artificial viscosity
to select a weak solution.
Finally, we extend the results to two special cases,
where $P(h)$ is for the polytropic gas or for the Chaplygin gas.