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We prove existence of globally bounded weak solutions to the compressible Stokes system with a general non-monotone pressure term. It turns out that the solution is unique for a bounded initial density. From the Lagrangian formulation of the system, we first obtain the L ∞ bounds and then construct the solution in Eulerian coordinates using the tools from the theory of transport equations. The velocity might not be Lipschitz continuous, however its gradient belongs to L ∞ ([0, ∞); BM O) and the uniqueness is shown using the key properties of the BM O space.

We investigate the existence of weak solutions to a certain system of partial differential equations, modeling the behavior of a compressible non-Newtonian fluid for small Reynolds number. We construct the weak solutions despite the lack of the L ∞ estimate on the divergence of the velocity field. The result was obtained by combining the regularity theory for singular operators with a certain logarithmic integral inequality for BMO functions, which allowed us to adjust the method from Feireisl et al. ( 2015) to more relaxed conditions on the velocity.

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