Emil Wiedemann

Existence of weak solutions for the incompressible Euler equations

Abstract Using a recent result of C. De Lellis and L. Szekelyhidi Jr. (2010) [2] we show that, in the case of periodic boundary conditions and for arbitrary space dimension d ⩾ 2 , there exist infinitely many global weak solutions to the incompressible Euler equations with initial data v 0 , where v 0 may be any solenoidal L 2 -vectorfield. In addition, the energy of these solutions is bounded in time.

Young Measures Generated by Ideal Incompressible Fluid Flows

In their seminal paper, DiPerna and Majda (Commun Math Phys 108(4):667–689, 1987) introduced the notion of a measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.

Failure of the Chain Rule for the Divergence of Bounded Vector Fields

We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. For instance, we can construct defects which are absolutely continuous with respect to Lebesgue measure, or defects which are not even measures.

Weak-strong uniqueness for measure-valued solutions of some compressible fluid models

We prove weak-strong uniqueness in the class of admissible measure-valued solutions for the isentropic Euler equations in any space dimension and for the Savage-Hutter model of granular flows in one and two space dimensions. For the latter system, we also show the complete dissipation of momentum in finite time, thus rigorously justifying an assumption that has been made in the engineering and numerical literature.

Differential Inclusions and Young Measures Involving Prescribed Jacobians

This work presents a general principle, in the spirit of convex integration, leading to a method for the characterization of Young measures generated by gradients of maps in W^{1,p} with p less than the space dimension, whose Jacobian determinant is subjected to a range of constraints. Two special cases are particularly important in the theories of elasticity and fluid dynamics: when (a) the generating gradients have positive Jacobians that are uniformly bounded away from zero and (b) the underlying deformations are incompressible, corresponding to their Jacobian determinants being constantly one. This characterization result, along with its various corollaries, underlines the flexibility of the Jacobian determinant in subcritical Sobolev spaces and gives a more systematic and general perspective on previously known pathologies of the pointwise Jacobian. Finally, we show that, for p less than the dimension, W^{1,p}-quasiconvexity and W^{1,p}-orientation-preserving quasiconvexity are both unsuitable convexity conditions for nonlinear elasticity where the energy is assumed to blow up as the Jacobian approaches zero.

On the Extension of Onsager’s Conjecture for General Conservation Laws

The aim of this work is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this work is the “universality” of the Onsager exponent,

ORIENTATION-PRESERVING YOUNG MEASURES

\alpha > 1/3

Weak-strong uniqueness in fluid dynamics

α>1/3, concerning the regularity of the solutions, say in