Camillo De Lellis

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.

Estimates and regularity results for the DiPerna-Lions flow

Abstract In this paper we derive new simple estimates for ordinary differential equations with Sobolev coefficients. These estimates not only allow to recover some old and recent results in a simple direct way, but they also have some new interesting corollaries.

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 () = (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. (c) 2015 Wiley Periodicals, Inc.

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

valued functions revisited

h

Dissipative continuous Euler flows

-principle holds in several cases.

Chapter 4 – Notes on Hyperbolic Systems of Conservation Laws and Transport Equations

In this note we revisit Almgrens 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 Almgrens 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 Almgrens 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.

Well-Posedness for a Class of Hyperbolic Systems of Conservation Laws in Several Space Dimensions

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

Dissipative Euler flows and Onsager's conjecture

Contents 1. Introduction 2 1.1. The Keyfitz and Kranzer system 2 1.2. Bressans compactness conjecture 3 1.3. Ambrosios renormalization Theorem 4 1.4. Well–posedness for the Keyfitz and Kranzer system 5 1.5. Renormalization conjecture for nearly incompressible BV fields 6 1.6. Plan of the paper 7 2. Preliminaries 8 2.1. Notation 8 2.2. Measure theory 9 2.3. Approximate continuity and approximate jumps 10 2.4. BV functions 11 2.5. Caccioppoli sets and Coarea formula 12 2.6. The Volpert Chain rule 12 2.7. Albertis Rank–one Theorem 13 3. DiPerna–Lions theory for nearly incompressible flows 13 3.1. Lagrangian flows 13 3.2. Nearly incompressible fields and fields with the renormalization property 17 3.3. Existence and uniqueness of solutions to transport equations 20 3.4. Stability of solutions to transport equations 24 3.5. Existence, uniqueness, and stability of regular Lagrangian flows 26 4. Commutator estimates and Ambrosios Renormalization Theorem 30 4.

h-Principle and Rigidity for C1,α Isometric Embeddings

Abstract In this paper we consider a system of conservation laws in several space dimensions whose nonlinearity is due only to the modulus of the solution. This system, first considered by Keyfitz and Kranzer in one space dimension, has been recently studied by many authors. In particular, using standard methods from DiPerna–Lions theory, we improve the results obtained by the first and third author, showing existence, uniqueness and stability results in the class of functions whose modulus satisfies, in the entropy sense, a suitable scalar conservation law. In the last part of the paper we consider a conjecture on renormalizable solutions and show that this conjecture implies another one recently made by Bressan in connection with the system of Keyfitz and Kranzer.