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We study the initial value problem for a kind of Euler equation with a source term. Our main result is the existence of a globally-in-time weak solution whose total variation is bounded on the domain of definition, allowing the existence of shock waves. Our proof relies on a well-balanced random choice method called Glimm method which preserves the fluid equilibria and we construct a sequence of approximate weak solutions which converges to the exact weak solution of the initial value problem, based on the construction of exact solutions of the generalized Riemann problem associated with initially piecewise steady state solutions.
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Shuyang Xiang, Yangyang Cao, Global existence for a one-dimensional non-relativistic Euler model with relaxation. Port. Math. 77 (2020), no. 1, pp. 45–71DOI 10.4171/PM/2044