László Székelyhidi

On Admissibility Criteria for Weak Solutions of the Euler Equations

We consider solutions to the Cauchy problem for the incompressible Euler equations satisfying several additional requirements, like the global and local energy inequalities. Using some techniques introduced in an earlier paper, we show that, for some bounded compactly supported initial data, none of these admissibility criteria singles out a unique weak solution. As a byproduct, in more than one space dimension, we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique.

Convolution Type Functional Equations on Topological Abelian Groups

This book is devoted to the possible applications of spectral analysis and spectral synthesis for convolution type functional equations on topological abelian groups. The solution space of convolution type equations has been synthesized in the sense that the general solutions are built up from exponential monomial solutions. In particular, equivalence of systems of functional equations can be tested. This leads to a unified treatment of classical equations and to interesting new results.

The ℎ-principle and the equations of fluid dynamics

In this note we survey some recent results for the Euler equations in compressible and incompressible fluid dynamics. The main point of all these theorems is the surprising fact that a suitable variant of Gromovs

Dissipative continuous Euler flows

h

Dissipative Euler flows and Onsager's conjecture

-principle holds in several cases.

h-Principle and Rigidity for C1,α Isometric Embeddings

We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy.

Young Measures Generated by Ideal Incompressible Fluid Flows

For any \theta<1/10 we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Holder-continuous with exponent \theta. A famous conjecture of Onsager states the existence of such dissipative solutions with any Holder exponent \theta<1/3. Our theorem is the first result in this direction.

Spectral synthesis on discrete Abelian groups

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper (Nash in Ann. Math. 60:383–396, 1954; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58:545–556, 1955; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58:683–689, 1955) says that any short embedding in codimension one can be uniformly approximated by C 1 isometric embeddings. This statement clearly cannot be true for C 2 embeddings in general, due to the classical rigidity in the Weyl problem. In fact Borisov extended the latter to embeddings of class C 1,α with α>2/3 in (Borisov in Vestn. Leningr. Univ. 14(13):20–26, 1959; Borisov in Vestn. Leningr. Univ. 15(19):127–129, 1960). On the other hand he announced in (Borisov in Doklady 163:869–871, 1965) that the Nash–Kuiper statement can be extended to local C 1,α embeddings with α<(1+n+n 2)−1, where n is the dimension of the manifold, provided the metric is analytic. Subsequently a proof of the 2-dimensional case appeared in (Borisov in Sib. Mat. Zh. 45(1):25–61, 2004). In this paper we provide analytic proofs of all these statements, for general dimension and general metric.

Functional Equations on Hypergroups

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.

Weak Solutions to the Stationary Incompressible Euler Equations

We prove that spectral synthesis holds on a discrete Abelian group G if and only if the torsion free rank of G is finite.