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arXiv2017

Piecewise constant subsolutions for the Muskat problem

We show the existence of infinitely many admissible weak solutions for the incompressible porous media equations for all Muskat-type initial data with $C^{3,α}$-regularity of the interface in the unstable regime and for all non-horizontal data with $C^{3,α}$-regularity in the stable regime. Our approach involves constructing admissible subsolutions with piecewise constant densities. This allows us to give a rather short proof where it suffices to calculate the velocity and acceleration at time zero - thus emphasizing the instantaneous nature of non-uniqueness due to discontinuities in the initial data.

math.AP↗

arXiv2004

Lieb-Thirring inequalities for higher order differential operators

We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order gamma >= 3/4 for fourth order Schrödinger operators in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of such operators. For the critical case gamma = 1 - 1/2l in dimension d=1 with differential order 2l >= 4 we prove the strict inequality L^0(l,gamma,d) < L(l,gamma,d), which holds in contrast to current conjectures.

math-ph↗