40 research outputs found
Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations
We study the Riemann problem for the multidimensional compressible isentropic
Euler equations. Using the framework developed by Chiodaroli, De Lellis, Kreml
and based on the techniques of De Lellis and Sz\'{e}kelyhidi, we extend our
previous results and prove that whenever the initial Riemann data give rise to
a self-similar solution consisting of one admissible shock and one rarefaction
wave and are not too far from lying on a simple shock wave, the problem admits
also infinitely many admissible weak solutions.Comment: 20 pages, 10 figure
On the energy dissipation rate of solutions to the compressible isentropic Euler system
In this paper we extend and complement some recent results by Chiodaroli, De
Lellis and Kreml on the well-posedness issue for weak solutions of the
compressible isentropic Euler system in space dimensions with pressure law
, . First we show that every Riemann
problem whose one-dimensional self-similar solution consists of two shocks
admits also infinitely many two-dimensional admissible bounded weak solutions
(not containing vacuum) generated by the method of De Lellis and Sz\'ekelyhidi.
Moreover we prove that for some of these Riemann problems and for such solutions have greater energy dissipation rate than the self-similar
solution emanating from the same Riemann data. We therefore show that the
maximal dissipation criterion proposed by Dafermos does not favour the
classical self-similar solutions