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

A class of large global solutions for the Wave--Map equation

In this paper we consider the equation for equivariant wave maps from $R^{3+1}$ to $S^3$ and we prove global in forward time existence of certain $C^\infty$-smooth solutions which have infinite critical Sobolev norm $\dot{H}^{\frac{3}{2}}(R^3)\times \dot{H}^{\frac{1}{2}}(R^3)$. Our construction provides solutions which can moreover satisfy the additional size condition $\|u(0, \cdot)\|_{L^\infty(|x|\geq 1)}>M$ for arbitrarily chosen $M>0$. These solutions are also stable under suitable perturbations. Our method is based on a perturbative approach around suitably constructed approximate self--similar solutions

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 $2$ space dimensions with pressure law $p(\rho)=\rho^\gamma$, $\gamma \geq 1$. 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 $1\leq \gamma < 3$ 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

An overview of some recent results on the Euler system of isentropic gas dynamics

In this overview we discuss some recent results of non--uniqueness for the isentropic Euler equations of gas dynamics with particular attention to the role of some admissibility criteria proposed in the literature.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1312.471

A counterexample to well-posedness of entropy solutions to the compressible Euler system

We deal with entropy solutions to the Cauchy problem for the isentropic compressible Euler equations in the space-periodic case. In more than one space dimension, the methods developed by De Lellis-Sz\'ekelyhidi enable us to show failure of uniqueness on a finite time-interval for entropy solutions starting from any continuously differentiable initial density and suitably constructed bounded initial linear momenta.Comment: 29 page

Existence and Non-uniqueness of Global Weak Solutions to Inviscid Primitive and Boussinesq Equations

We consider the initial value problem for the inviscid Primitive and Boussinesq equations in three spatial dimensions. We recast both systems as an abstract Euler-type system and apply the methods of convex integration of De Lellis and Szekelyhidi to show the existence of infinitely many global weak solutions of the studied equations for general initial data. We also introduce an appropriate notion of dissipative solutions and show the existence of suitable initial data which generate infinitely many dissipative solutions

Contact discontinuities in multi-dimensional isentropic euler equations

In this note we partially extend the recent nonuniqueness results on admissible weak solutions to the Riemann problem for the 2D compressible isentropic Euler equations. We prove non-uniqueness of admissible weak solutions that start from the Riemann initial data allowing a contact discontinuity to emerge