Ricerche di Matematica https://doi.org/10.1007/s11587-020-00515-7 Exact solutions and conservation laws in dissipative fluid dynamics Natale Manganaro 1 Received: 15 January 2020 / Accepted: 6 April 2020 © Università degli Studi di Napoli "Federico II" 2020 Abstract In this paper the 1D Euler system with a source term is considered in the case of isentropic flows. Classes of exact solutions of the equations under interest have been determined within the framework of the differential constraint method and a Riemann problem was solved. Finally Conservation Laws of the system here considered have been determined following the Direct Method. Keywords Exact solutions · Conservation laws · Ideal fluids Mathematics Subject Classification 35L40 · 35L45 · 35N10 1 Introduction Many mathematical methods have been proposed along the years for determining exact solutions of partial differential equations (PDEs). Among others, the method of differential constraints results to be particularly useful for solving problems of interest in nonlinear wave propagation. Such an approach was first proposed and applied to gas-dynamics in [1–3]. The main idea is to add to the governing system under interest some further differential equations which play the role of constraints because they select the class of special exact solutions admitted by the overderdetermined set of equations consisting of the original equations along with the additional differential constraints. The method is develop on two steps: first the compatibility of such an overdetermined system must be studied, next exact solutions of the full set of equations can be determined. On this subject many contributions have been given [4–9] as well This paper is dedicated to Professor G. Toscani and to Professor M. Sugiyama on the occasion of their 70 years. B Natale Manganaro nmanganaro@unime.it 1 MIFT, University of Messina, Messina, Italy 123