We give sufficient conditions on the regularity of solutions to the
inhomogeneous incompressible Euler and the compressible isentropic Euler
systems in order for the energy to be conserved. Our strategy relies on
commutator estimates similar to those employed by P. Constantin et al. for the
homogeneous incompressible Euler equations
We introduce a new concept of dissipative measure-valued solution to the compressible Navier-Stokes system satisfying, in addition, a relevant form of the total energy balance. Then we show that a dissipative measure-valued and a standard smooth classical solution originating from the same initial data coincide (weak-strong uniqueness principle) as long as the latter exists. Such a result facilitates considerably the proof of convergence of solutions to various approximations including certain numerical schemes that are known to generate a measure-valued solution. As a byproduct we show that any measure-valued solution with bounded density component that starts from smooth initial data is necessarily a classical one.
ABSTRACT. We prove weak-strong uniqueness in the class of admissible measure-valued solutions for the isentropic Euler equations in any space dimension and for the Savage-Hutter model of granular flows in one and two space dimensions. For the latter system, we also show the complete dissipation of momentum in finite time, thus rigorously justifying an assumption that has been made in the engineering and numerical literature.
In their seminal paper [11] R. DiPerna and A. Majda introduced the notion of measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.
Using a recent result of C. De Lellis and L. Székelyhidi Jr. ( [2]) we show that, in the case of periodic boundary conditions and for arbitrary dimension d ≥ 2, there exist infinitely many global weak solutions to the incompressible Euler equations with initial data v0, where v0 may be any solenoidal L 2 -vectorfield. In addition, the energy of these solutions is bounded in time.
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