2025/03/14 by Lasarzik, Robert · 1 citation
#35D99 #35Q35 #35Q74 #35R35 #35R45 #76T06 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.11438
We consider different measure-valued solvability concepts from the literature and show that they could be simplified by using the energy-variational structure of the underlying system of partial differential equations. In the considered examples, we prove that a certain class of improved measure-valued solutions can be equivalently expressed as an energy-variational solution. The first concept represents the solution as a high-dimensional Young measure, whether for the second concept, only a scalar auxiliary variable is introduced and the formulation is relaxed to an energy-variational inequality. We investigate four examples: the two-phase Navier--Stokes equations, a quasilinear wave equation, a system stemming from polyconvex elasticity, and the Ericksen--Leslie equations equipped with the Oseen--Frank energy. The wide range of examples suggests that this is a recurrent feature in evolution equations in general.