Weak-Strong Uniqueness for Measure-Valued Solutions

By Yann Brenier, Camillo De Lellis, L. Sz'ekelyhidi. 2009

171 citations recorded by the source corpus.

We prove the weak-strong uniqueness for measure-valued solutions of the incompressible Euler equations. These were introduced by DiPerna and Majda in their landmark paper (Commun Math Phys 108(4):667–689, 1987), where in particular global existence to any L2 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.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0ae75ef722e4e9f714c6d986c02a95b197e640dc

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In Applied Mathematics

In Computational Mechanics

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Citations in the corpus

Convex integration and phenomenologies in turbulence (2019, background): “See [12, 202] for the weak-strong uniqueness property in this class.”
Dissipative measure-valued solutions to the compressible Navier–Stokes system (2015, methodology): “Let us also point out that an analogue of our dissipative measure-valued solutions could be considered also for the incompressible and compressible Euler system and might thus lead to a slight simpliﬁcation and generalization of the results in [2] and [17].”
The $h$-principle and the equations of fluid dynamics (2011, background): “ ≤ Z Rn |v0|2(x)dx for a.e. t. (18) Then (ν,λ,ν∞) coincides with vas long as the latter exists, i.e. νx,t = δv(x,t) for a.a. (x,t) ∈ Rn ×(0,T) and λ≡ 0 on Rn ×(0,T). This theorem recently appeared in [15], building upon ideas of [14, 16], where the authors dealt with the energy of measure-valued sol”
Non-uniqueness and h-Principle for Hölder-Continuous Weak Solutions of the Euler Equations (2016, background): “- this is known as weak-strong uniqueness, and holds even for measurevalued solutions [2].”
Onsager's Conjecture for Admissible Weak Solutions (2017, background): “In fact, the latter is a relevant point for at least two reasons: dissipative solutions satisfy the weak-strong uniqueness property [1, 30], and Onsager, in his work, conjectures the existence of dissipative solutions.”