Solution semiflow to the isentropic Euler system

Abstract

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 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.

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Cited by in corpus (8)

• Dissipative solutions and semiflow selection for the complete Euler system
• On the long--time behavior of dissipative solutions to models of non-Newtonian compressible fluids
• Statistical solutions to the barotropic Navier-Stokes system
• Ergodic theory for energetically open compressible fluid flows
• Semiflow selection for the compressible Navier-Stokes system
• Probabilistic Descriptions of Fluid Flow: A Survey
• Semiflow selection to models of general compressible viscous fluids
• Maximal turbulence as a selection criterion for measure-valued solutions