Solution Semiflow to the isentropic Euler system

0522128 - MÚ 2021 RIV DE eng J - Journal Article
Breit, D. - Feireisl, Eduard - Hofmanová, M.
Solution Semiflow to the isentropic Euler system.
Archive for Rational Mechanics and Analysis. Roč. 235, č. 1 (2020), s. 167-194. ISSN 0003-9527. E-ISSN 1432-0673
R&D Projects: GA ČR(CZ) GA18-05974S
Institutional support: RVO:67985840
Keywords : weak solution * Euler equations * convex integration
OECD category: Pure mathematics
Impact factor: 2.793, year: 2020
Method of publishing: Open access
https://doi.org/10.1007/s00205-019-01420-6

It is nowadays well understood that the multidimensional isentropic Euler system is desperately ill-posed. Even certain smooth initial data give rise to infinitely many solutions and all available selection criteria fail to ensure both global existence and uniqueness. We propose a different approach to the well-posedness of this system based on ideas from the theory of Markov semigroups: we show the existence of a Borel measurable solution semiflow. To this end, we introduce a notion of dissipative solution which is understood as time dependent trajectories of the basic state variables—the mass density, the linear momentum, and the energy—in a suitable phase space. The underlying system of PDEs is satisfied in a generalized sense. The solution semiflow enjoys the standard semigroup property and the solutions coincide with the strong solutions as long as the latter exist. Moreover, they minimize the energy (maximize the energy dissipation) among all dissipative solutions.
Permanent Link: http://hdl.handle.net/11104/0306635