Solution Semiflow to the Isentropic Euler System

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...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2020, Vol.235 (1), p.167-194
Main Authors: Breit, Dominic, Feireisl, Eduard, Hofmanová, Martina
Format: Article
Language:English
Subjects:
Classical Mechanics
Complex Systems
Energy conservation
Energy dissipation
Fluid- and Aerodynamics
Ill posed problems
Markov processes
Mathematical and Computational Physics
Physics
Physics and Astronomy
State variable
Theoretical
Time dependence
Well posed problems
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Summary: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.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-019-01420-6