arXiv subjects

Camillo De Lellis

Publications and source records attributed to Camillo De Lellis.

90 records · Page 5Linked to original sources

arXiv2013

Global ill-posedness of the isentropic system of gas dynamics

We consider the isentropic compressible Euler system in 2 space dimensions with pressure law $p(ρ) = ρ^2$ and we show the existence of classical Riemann data, i.e. pure jump discontinuities across a line, for which there are infinitely many admissible bounded weak solutions (bounded away from the void). We also show that some of these Riemann data are generated by a 1-dimensional compression wave: our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions.

math.AP↗

arXiv2012

Dissipative Euler Flows and Onsager's Conjecture

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

math.AP↗

arXiv2012

Dissipative continuous Euler flows in two and three dimensions

Recently, two of these authors construct dissipative continuous (weak) solutions to the incompressible Euler equations on the three-dimensional torus $\mathbb T^3$. The building blocks in their proof are Beltrami flows, which are inherently three-dimensional. The purpose of this note is to show that the techniques can nevertheless be adapted to the two-dimensional case.

math.AP↗

arXiv2012

Dissipative continuous Euler flows

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

math.AP↗

arXiv2011

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 Gromov's $h$-principle holds in several cases.

math.AP↗

arXiv2011

Almost-Schur lemma

Schur's lemma states that every Einstein manifold of dimension $n\geq 3$ has constant scalar curvature. Here $(M,g)$ is defined to be Einstein if its traceless Ricci tensor $$\Rico:=\Ric-\frac{R}{n}g$$ is identically zero. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci tensor is assumed to be \emph{small} rather than identically zero.

math.DG↗

arXiv2011

Center Manifold: a case study

Following Almgren's construction of the "center manifold" in his Big regularity paper, we show the C^{3,α} regularity of area-minimizing currents in the neighborhood of points of density one without using the nonparametric theory. This study is intended as a first step towards the understanding of Almgren's construction in its full generality.

math.AP↗

arXiv2011

Q-valued functions revisited

In this note we revisit Almgren's theory of Q-valued functions, that are functions taking values in the space of unordered Q-tuples of points in R^n. In particular: 1) we give shorter versions of Almgren's proofs of the existence of Dir-minimizing Q-valued functions, of their Hoelder regularity and of the dimension estimate of their singular set; 2) we propose an alternative intrinsic approach to these results, not relying on Almgren's biLipschitz embedding; 3) we improve upon the estimate of the singular set of planar Dir-minimizing functions by showing that it consists of isolated points.

math.AP↗

arXiv2011

Lower semicontinuous functionals for Almgren's multiple valued functions

We consider general integral functionals on the Sobolev spaces of multiple valued functions, introduced by Almgren. We characterize the semicontinuous ones and recover earlier results of Mattila as a particular case. Moreover, we answer positively to one of the questions raised by Mattila in the same paper.

math.AP↗

arXiv2009

Weak-strong uniqueness for measure-valued solutions

We prove the weak-strong uniqueness for measure-valued solutions of the incompressible Euler equations. These were introduced by R.DiPerna and A.Majda, and in particular global existence to any L2 initial data was proven. Whether measure-valued solutions agree with classical solutions if the latter exist has apparently remained open. We also show that DiPerna's measure-valued solutions to systems of conservation laws have the weak-strong uniqueness property.

math.AP↗

arXiv2009

The existence of embedded minimal hypersurfaces

We give a shorter proof of the existence of nontrivial closed minimal hypersurfaces in closed smooth $(n+1)$--dimensional Riemannian manifolds, a theorem proved first by Pitts for $2\leq n\leq 5$ and extended later by Schoen and Simon to any $n$.

math.AP↗

arXiv2009

Genus bounds for minimal surfaces arising from min-max constructions

In this paper we prove genus bounds for closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubistein but to our knowledge its proof has never been published. Our proof follows ideas of Simon and uses an extension of a famous result of Meeks, Simon and Yau on the convergence of minimizing sequences of isotopic surfaces. This result is proved in the second part of the paper.

math.AP↗

arXiv2009

h-Principle and Rigidity for $C^{1,α}$ Isometric Embeddings

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper 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 $α>2/3$. On the other hand he announced in 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 this paper we provide analytic proofs of all these statements, for general dimension and general metric.

math.DG↗

arXiv2008

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 we show bounded initial data for which admissible solutions to the p-system of isentropic gas dynamics in Eulerian coordinates are not unique in more than one space dimension.

math.AP↗

arXiv2007

The Euler equations as a differential inclusion

In this paper we propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy--decreasing solutions. Our results are stronger because they work in any dimension and yield bounded velocity and pressure.

math.AP↗

arXiv2007

Three circles theorems for Schrodinger operators on cylindrical ends and geometric applications

We show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends the space of solutions which grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently for a surface, for a fixed potential and a dense set of metrics), the constant function zero is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity. These results follow from a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution $u$ to a Schrödinger equation on a product $N\times [0,T]$, where $N$ is a closed manifold with a certain spectral gap. Examples of such $N$'s are all (round) spheres $\SS^n$ for $n\geq 1$ and all Zoll surfaces. Finally, we discuss some examples arising in geometry of such manifolds and Schrödinger operators.

math.DG↗

arXiv2003

The min--max construction of minimal surfaces

In this paper we survey with complete proofs some well--known, but hard to find, results about constructing closed embedded minimal surfaces in a closed 3-dimensional manifold via min--max arguments. This includes results of J. Pitts, F. Smith, and L. Simon and F. Smith.

math.AP↗

arXiv2002

Singular limit laminations, Morse index, and positive scalar curvature

For any 3-manifold M and any nonnegative integer g, we give here examples of metrics on M each of which has a sequence of embedded minimal surfaces of genus g and without Morse index bounds. On any spherical space form S^3/Gamma we construct such a metric with positive scalar curvature. More generally we construct such a metric with Scal>0 (and such surfaces) on any 3-manifold which carries a metric with Scal>0. In all but one of these examples the Hausdorff limit will be a singular minimal lamination. The singularities being in each case exactly two points lying on a closed leaf (the leaf is a strictly stable sphere).

math.DG↗