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Elisabetta Chiodaroli

Publications and source records attributed to Elisabetta Chiodaroli.

At least 19 recordsLinked to original sources
arXiv2026

Maximal entropy production principle and the Euler system of gas dynamics

Convex integration has revealed that the Euler system of gas dynamics is ill-posed in the class of weak solutions even if the entropy inequality is imposed as an additional constraint. A natural question arises, namely, if a physically relevant solution can be selected by maximizing the entropy production rate. Firstly, we present an example of Riemann initial data in 2-D, for which the standard self-similar solution fails to satisfy the maximal entropy production principle. Hence, maximizing the entropy production rate rules out the 1-D self-similar solution which intuitively seems to be the physically relevant solution in this context. Secondly, we show for a large class of initial data that there exist entropy admissible weak solutions with an arbitrary (non-decreasing) total entropy profile.

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arXiv2024

On the long time behaviour of solutions to the Navier-Stokes-Fourier system on unbounded domains

We consider the Navier-Stokes-Fourier system on an unbounded domain in the Euclidean space $R^3$, supplemented by the far field conditions for the phase variables, specifically: $ρ\to 0,\ \vartheta \to \vartheta_\infty, \ u \to 0$ as $\ |x| \to \infty$. We study the long time behaviour of solutions and we prove that any global-in-time weak solution to the NSF system approaches the equilibrium $ρ_s = 0,\ \vartheta_s = \vartheta_\infty,\ u_s = 0$ in the sense of ergodic averages for time tending to infinity. As a consequence of the convergence result combined with the total mass conservation, we can show that the total momentum of global-in-time weak solutions is never globally conserved.

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arXiv2024

Energy conservation for 3D Euler and Navier-Stokes equations in a bounded domain. Applications to Beltrami flows

In this paper we consider the incompressible 3D Euler and Navier-Stokes equations in a smooth bounded domain. First, we study the 3D Euler equations endowed with slip boundary conditions and we prove the same criteria for energy conservation involving the gradient, already known for the Navier-Stokes equations. Subsequently, we utilise this finding, which is based on a proper approximation of the velocity (and doesn't require estimates or additional assumptions on the pressure), to explore energy conservation for Beltrami flows. Finally, we explore Beltrami solutions to the Navier-Stokes equations and demonstrate that conditions leading to energy conservation are significantly distinct from those implying regularity. This remains true even when making use of the bootstrap regularity improvement, stemming from the solution being a Beltrami vector field.

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arXiv2023

Glimm's method and density of wild data for the Euler system of gas dynamics

We adapt Glimm's approximation method to the framework of convex integration to show density of wild data for the (complete) Euler system of gas dynamics. The desired infinite family of entropy admissible solutions emanating from the same initial data is obtained via convex integration of suitable Riemann problems pasted with local smooth solutions. In addition, the wild data belong to BV class.

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arXiv2019

Ill posedness for the full Euler system driven by multiplicative white noise

We consider the Euler system describing the motion of a compressible fluid driven by a multiplicative white noise. We identify a large class of initial data for which the problem is ill posed - there exist infinitely many global in time weak solutions. The solutions are adapted to the noise and satisfy the entropy admissibility criterion.

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arXiv2019

Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data

We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant $T > 0$. In order to continue the solution after the formation of the discontinuity, we adjust and apply the theory developed by De Lellis and Székelyhidi and we construct infinitely many solutions. We introduce the notion of an admissible generalized fan subsolution to be able to handle data which are not piecewise constant and we reduce the argument to finding a single generalized subsolution.

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arXiv2019

On the energy equality for the 3D Navier-Stokes equations

In this paper we study the problem of energy conservation for the solutions of the initial boundary value problem associated to the 3D Navier-Stokes equations, with Dirichlet boundary conditions. First, we consider Leray-Hopf weak solutions and we prove some new criteria, involving the gradient of the velocity. Next, we compare them with the existing literature in scaling invariant spaces and with the Onsager conjecture. Then, we consider the problem of energy conservation for very-weak solutions, proving energy equality for distributional solutions belonging to the so-called Shinbrot class. A possible explanation of the role of this classical class of solutions, which is not scaling invariant, is also given.

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arXiv2018

Concentration Compactness for Critical Radial Wave Maps

We consider radially symmetric, energy critical wave maps from (1 + 2)-dimensional Minkowski space into the unit sphere $\mathbb{S}^m$, $m \geq 1$, and prove global regularity and scattering for classical smooth data of finite energy. In addition, we establish a priori bounds on a suitable scattering norm of the radial wave maps and exhibit concentration compactness properties of sequences of radial wave maps with uniformly bounded energies. This extends and complements the beautiful classical work of Christodoulou-Tahvildar-Zadeh [3, 4] and Struwe [31, 33] as well as of Nahas [22] on radial wave maps in the case of the unit sphere as the target. The proof is based upon the concentration compactness/rigidity method of Kenig-Merle [6, 7] and a "twisted" Bahouri-Gerard type profile decomposition [1], following the implementation of this strategy by the second author and Schlag [17] for energy critical wave maps into the hyperbolic plane as well as by the last two authors [16] for the energy critical Maxwell-Klein-Gordon equation.

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arXiv2017

On contact discontinuities in multi-dimensional isentropic Euler equations

In this short note we partially extend the recent nonuniqueness results on admissible weak solutions to the Riemann problem for the 2D compressible isentropic Euler equations. We prove nonuniqueness of admissible weak solutions that start from the Riemann initial data allowing a contact discontinuity to emerge.

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arXiv2017

Non-uniqueness of admissible weak solutions to the Riemann problem for the isentropic Euler equations

We study the Riemann problem for the multidimensional compressible isentropic Euler equations. Using the framework developed by Chiodaroli, De Lellis, Kreml and based on the techniques of De Lellis and Székelyhidi, we extend our previous results and prove that whenever the initial Riemann data give rise to a self-similar solution consisting of one admissible shock and one rarefaction wave and are not too far from lying on a simple shock wave, the problem admits also infinitely many admissible weak solutions.

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arXiv2016

Existence and non-uniqueness of global weak solutions to inviscid primitive and Boussinesq equations

We consider the initial value problem for the inviscid Primitive and Boussinesq equations in three spatial dimensions. We recast both systems as an abstract Euler-type system and apply the methods of convex integration of De Lellis and Székelyhidi to show the existence of infinitely many global weak solutions of the studied equations for general initial data. We also introduce an appropriate notion of dissipative solutions and show the existence of suitable initial data which generate infinitely many dissipative solutions.

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arXiv2016

A class of large global solutions for the Wave--Map equation

In this paper we consider the equation for equivariant wave maps from $R^{3+1}$ to $S^3$ and we prove global in forward time existence of certain $C^\infty$-smooth solutions which have infinite critical Sobolev norm $\dot{H}^{\frac{3}{2}}(R^3)\times \dot{H}^{\frac{1}{2}}(R^3)$. Our construction provides solutions which can moreover satisfy the additional size condition $\|u(0, \cdot)\|_{L^\infty(|x|\geq 1)}>M$ for arbitrarily chosen $M>0$. These solutions are also stable under suitable perturbations. Our method is based on a perturbative approach around suitably constructed approximate self--similar solutions.

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arXiv2015

$\mathcal{A}$-free rigidity and applications to the compressible Euler system

Can every measure-valued solution to the compressible Euler equations be approximated by a sequence of weak solutions? We prove that the answer is negative: Generalizing a well-known rigidity result of Ball and James to a more general situation, we construct an explicit measure-valued solution for the compressible Euler equations which can not be generated by a sequence of distributional solutions. We also give an abstract necessary condition for measure-valued solutions to be generated by weak solutions, relying on work of Fonseca and Müller. This difference between weak and measure-valued solutions in the compressible case is in contrast with the incompressible situation, where every measure-valued solution can be approximated by weak solutions, as shown by Székelyhidi and Wiedemann.

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arXiv2013

On the energy dissipation rate of solutions to the compressible isentropic Euler system

In this paper we extend and complement some recent results by Chiodaroli, De Lellis and Kreml on the well-posedness issue for weak solutions of the compressible isentropic Euler system in $2$ space dimensions with pressure law $p(ρ)=ρ^γ$, $γ\geq 1$. First we show that every Riemann problem whose one-dimensional self-similar solution consists of two shocks admits also infinitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Székelyhidi. Moreover we prove that for some of these Riemann problems and for $1\leq γ< 3$ such solutions have greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos does not favour the classical self-similar solutions.

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arXiv2013

On the weak solutions to the equations of a compressible heat conducting gas

We consider the weak solutions to the Euler-Fourier system describing the motion of a compressible heat conducting gas. Employing the method of convex integration, we show that the problem admits infinitely many global-in-time weak solutions for any choice of smooth initial data. We also show that for any initial distribution of the density and temperature, there exists an initial velocity such that the associated initial-value problem possesses infinitely many solutions that conserve the total energy.

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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.

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