Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems |
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Authors: | Donatella Donatelli Eduard Feireisl Pierangelo Marcati |
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Affiliation: | 1. Department of Information Engineering, Computer Science and Mathematics , University of L'Aquila , L'Aquila , Italy;2. Institute of Mathematics of the Academy of Sciences of the Czech Republic , Praha , Czech Republic;3. Department of Information Engineering, Computer Science and Mathematics , University of L'Aquila , L'Aquila , Italy;4. GSSI - Gran Sasso Science Institute , L'Aquila , Italy |
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Abstract: | We consider a general Euler-Korteweg-Poisson system in R 3, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any sufficiently smooth initial data including the case of a vanishing initial density - the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solutions satisfying the energy inequality. Finally, we establish the weak-strong uniqueness property in a class of solutions without vacuum. In this paper we show that, even in presence of a dispersive tensor, we have the same phenomena found by De Lellis and Székelyhidi. |
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Keywords: | Convex integration Euler-Korteweg system Quantum hydrodynamics Weak solution |
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