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 = 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 ε ) , 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.