European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
13:30   Zuiderduinzaal: Keynote: Eduard Feireisl, Institute of Mathematics of the Czech Academy of Sciences, Czech Republic
45 mins
Solving ill posed problems: Young measures, K-convergence and Lax equivalence theorem for nonlinear systems
Eduard Feireisl
Abstract: The Euler system describing the motion of a compressible inviscid fluid represents a well known ill-posed problem in mathematical fluid dynamics. We propose a new approach how to study convergence of numerical methods for this problem based on the concept of generalized (measure--valued) solutions and the theory of K-convergence. The results can be seen as a non—linear variant of the celebrated Lax equivalence theorem: 1. If the numerical scheme is consistent and energy stable and the target Euler system admits a strong solution, then the numerical solutions converge to it pointwise and unconditionally. 2. Under certain additional hypotheses the numerical solutions converge pointwise if and only if the limit object is a weak solution to the Euler system. 3. In case the convergence is weak, then the Cesaro averages of the associated Young measure converge strongly with respect to the physical variables to the limit Young measure in a suitable Wasserstein metric.