We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.
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Peter Constantin, Theodore D. Drivas, Huy Q. Nguyen, Federico Pasqualotto, Compressible fluids and active potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 1, pp. 145–180DOI 10.1016/J.ANIHPC.2019.04.001