Remark on the local well-posedness of compressible non-Newtonian fluids with initial vacuum

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Abstract: We discuss in this short note the local-in-time strong well-posedness of the compressible Navier-Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, M'{a}cha, and Nev{c}asova in doi{10.1007/s00208-021-02301-8} can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in doi{10.1016/j.matpur.2003.11.004} for compressible Navier-Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.

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