Abstract. In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. We prove that the set of Hölder 1/5 − ε wild initial data is dense in L 2 , where we call an initial datum wild if it admits infinitely many admissible Hölder 1/5 − ε weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows in order to show that a general form of the h-principle applies to Hölder-continuous weak solutions of the Euler equations. Our result indicates that in a deterministic theory of 3D turbulence the Reynolds stress tensor can be arbitrary and need not satisfy any additional closure relation.

In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals dened on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by and on the metric characterization of the gradient ows generated by the functionals in the Wasserstein space.

Dedicated to Prof. Stephan Luckhaus on the occasion of his 65th birthday. AbstractWe study the functional considered in [25,26,28] and a continuous version of it, analogous to the one considered in [30]. The functionals consist of a perimeter term and a nonlocal term which are in competition. For both the continuous and discrete problem, we show that the global minimizers are exact periodic stripes. One striking feature of the functionals is that the minimizers are invariant under a smaller group of symmetries than the functional itself. In the continuous setting, to our knowledge this is the first example of a model with local/nonlocal terms in competition such that the functional is invariant under permutation of coordinates and the minimizers display a pattern formation which is one-dimensional. Such behaviour for a smaller range of exponents in the discrete setting was already shown in [28].1) where J is a positive constant, p ≥ d + 2, y ∼ x if x and y are neighbouring points in the lattice, and χ E (x) := 1 if x ∈ E, 0 otherwise. * daneri@math.fau.de † eris.runa@mis.mpg.de arXiv:1702.07334v4 [math.AP] 5 Jun 2018 c (respectively J > J c ), then the global minimizers are trivial, namely either empty or the whole domain. The critical constants J dsc c and J c are (as proven in [25] for the discrete setting and in [30] for the continuous case) J dsc c := y 1 >0, (y 2 ,...,y d )∈Z d−1 y 1 (y 2 1 + · · · + y 2 d ) p/2 and J c :=ˆR d |ζ 1 |K 1 (ζ) dζ. (1.4)When J = J dsc c − τ (resp. J = J c − τ ) with 0 < τ ≤τ for someτ > 0 small enough, it has been conjectured that the minimizers should be periodic unions of stripes of optimal period. In order to define what we mean by unions of stripes, let us fix a canonical basis {e i } d i=1 . A union of stripes in the continuous setting is a [0, L) d -periodic set which is, up to Lebesgue null sets, of the form V ⊥ i +Êe i for some i ∈ {1, . . . , d}, where V ⊥ i is the (d − 1)-dimensional subspace orthogonal to e i andÊ ⊂ R withÊ ∩ [0, L) = ∪ N k=1 (s i , t i ). A union of stripes is periodic if ∃ h > 0, ν ∈ R s.t. E ∩ [0, L) = ∪ N k=0 (2kh + ν, (2k + 1)h + ν). In the following, we will also sometimes call unions of stripes simply stripes.

We extend our previous work on a biologically inspired dynamic Monge-Kantorovich model [18] and propose it as an effective tool for the numerical solution of the L 1 -PDE based optimal transportation model. Starting from the conjecture that the dynamic model is timeasymptotically equivalent to the Monge-Kantorovich equations governing L 1 optimal transport, we experimentally analyze a simple yet effective numerical approach for the quantitative solution of these equations.The novel contributions in this paper are twofold. First, we introduce a new Lyapunov-candidate functional that better adheres to the dynamics of our proposed model. It is shown that the Lie derivative of the new Lyapunov-candidate functional is strictly negative and, more remarkably, the OT density is the unique minimizer for this new Lyapunov-candidate functional, providing further support to the conjecture of asymptotic equivalence of our dynamic model with the Monge-Kantorovich equations. Second, we describe and test different numerical approaches for the solution of our problem. The ordinary differential equation for the transport density is projected into a piecewise constant or linear finite dimensional space defined on a triangulation of the domain. The elliptic equation is discretized using a linear Galerkin finite element method defined on uniformly refined triangles. The ensuing nonlinear differential-algebraic equation is discretized by means of a first order Euler method (forward or backward) and a simple Picard iteration is used to resolve the nonlinearity. The use of two discretization levels is dictated by the need to avoid oscillations on the potential gradients that prevent convergence of the scheme.We study the experimental convergence rate of the proposed solution approaches and discuss limitations and advantages of these formulations. An extensive set of test cases, including problems that admit an explicit solution to the Monge-Kantorovich equations are appropriately designed to verify and test the expected numerical properties of the solution methods. Finally, a comparison with literature methods is performed and the ensuing transport maps are compared. The results show that optimal convergence toward the asymptotic equilibrium point is achieved for sufficiently regular forcing function, and that the proposed method is accurate, robust, and computationally efficient. 1 arXiv:1709.06765v2 [math.NA]

We show that a planar bi-Lipschitz orientation-preserving homeomorphism can be approximated in the W 1,p norm, together with its inverse, with an orientation-preserving homeomorphism which is piecewise affine or smooth.