We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general right-hand sides in different functional setting, including weighted Sobolev spaces Hs δ. An essential tool to achieve the continuity of the flow map is a new type of energy estimate, which we call a low regularity energy estimate. We then apply these results to the Euler-Poisson system which describes various systems of physical interest.
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Dedicated to Lawrence Zalcman
U. B. gratefully acknowledges support from Grant MTM2016-75465 and Grant PID2019-103860GB-I00 by MINECO, Spain and UCM-GR17-920894.
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Brauer, U., Karp, L. Continuity of the flow map for symmetric hyperbolic systems and its application to the Euler-Poisson system. JAMA 141, 113–163 (2020). https://doi.org/10.1007/s11854-020-0125-4