Abstract
We discuss the problem of well-posedness of the Euler-Poisson system arising, for example, in the theory of semi-conductors, models of plasma and gaseous stars in astrophysics. We introduce the concept of dissipative weak solution satisfying, in addition to the standard system of integral identities replacing the original system of partial differential equations, the balance of total energy, together with the associated relative entropy inequality. We show that strong solutions are unique in the class of dissipative solutions (weak-strong uniqueness). Finally, we use the method of convex integration to show that the Euler-Poisson system may admit even infinitely many weak dissipative solutions emanating from the same initial data.
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Acknowledgements
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement 320078.
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Feireisl, E. (2016). On Global Well/Ill-Posedness of the Euler-Poisson System. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds) Recent Developments of Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0939-9_12
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DOI: https://doi.org/10.1007/978-3-0348-0939-9_12
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