Weak-Strong Uniqueness for Measure-Valued Solutions

Abstract

Abstract. We prove the weak-strong uniqueness for measure-valued solutions of the incompressible Euler equations. These were introduced by R.DiPerna and A.Majda in their landmark paper [10], where in particular global existence to any L 2 initial data was proven. Whether measure-valued solutions agree with classical solutions if the latter exist has apparently remained open.We also show that DiPerna's measure-valued solutions to systems of conservation laws have the weak-strong uniqueness property.

Weak-strong uniqueness for measure-valued solutions of some compressible fluid models

“…Owing to commutator estimates analogous to the ones provided by Constantin, E and Titi for the incompressible Euler system in [CET94], the result of Theorem 1.1 can be obtained assuming only Lipschitz continuity of U , and Sobolev regularity of H. In fact, as in [BDLS11] it is sufficient to assume only that the symmetric part of ∇U be bounded. We omit details.…”

“…It is therefore surprising that, in the case of the incompressible Euler equations, the so-called weak-strong uniqueness property was proved, on the whole space, for admissible measure-valued solutions by Brenier, De Lellis, and Székelyhidi [BDLS11]. This means that if there exists a sufficiently regular (classical) solution, then every admissible measure-valued solution with the same initial data will coincide with the classical solution.…”

Weak solutions of the Euler equations: non-uniqueness and dissipation

“…This theorem was proved in [15], building upon ideas of [14,16], where the authors dealt with the energy of measure-valued solutions to the Vlasov-Poisson system. More precisely, the proof of [15] yields the following information: if ν x,t satisfies (1.5), then…”

Remarks on the Contributions of Constantine M. Dafermos to the Subject of Conservation Laws

“…It has been applied in various directions, e.g. for asymptotic stability problems in conservation laws [6,7], for relaxation or kinetic limits [1,38], or for comparing entropic measure-valued solutions and strong solutions for conservation laws [3].…”