Well/ill posedness for the Euler-Korteweg-Poisson system and related problems

0443854 - MÚ 2016 RIV US eng J - Journal Article
Donatelli, D. - Feireisl, Eduard - Marcati, P.
Well/ill posedness for the Euler-Korteweg-Poisson system and related problems.
Communications in Partial Differential Equations. Roč. 40, č. 7 (2015), s. 1314-1335. ISSN 0360-5302. E-ISSN 1532-4133
EU Projects: European Commission(XE) 320078 - MATHEF
Institutional support: RVO:67985840
Keywords : convex integration * Euler-Korteweg system * quantum hydrodynamics
Subject RIV: BA - General Mathematics
Impact factor: 1.444, year: 2015
http://www.tandfonline.com/doi/abs/10.1080/03605302.2014.972517

We consider a general Euler-Korteweg-Poisson system in R3, 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-intime 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.
Permanent Link: http://hdl.handle.net/11104/0246502