Global solutions to one-dimensional shallow water magnetohydrodynamic equations
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.