Global solutions to one-dimensional shallow water magnetohydrodynamic equations

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=0h=0) since the nonlinear flux function P(h)P(h) is singular and the definition of solution is not clear near h=0h=0. First, assuming that hh 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,fε)(c,fε), and obtain a global weak entropy solution. Second, under some technical conditions on the initial data such that the Riemann invariants (w,z)(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)P(h) is for the polytropic gas or for the Chaplygin gas.