De Lellis Research Articles

On Strong Continuity of Weak Solutions to the Compressible Euler System

Let \({\mathcal {S}} = \{ \tau _n \}_{n=1}^\infty \subset (0,T)\) be an arbitrary countable (dense) set. We show that for any given initial density and momentum, the compressible Euler system admits (infinitely many) admissible weak solutions that are not strongly continuous at each \(\tau _n\), \(n=1,2,\dots \). The proof is based on a refined version of the oscillatory lemma of De Lellis and Székelyhidi with coefficients that may be discontinuous on a set of zero Lebesgue measure.

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 Szekelyhidi 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.

Effect of the COVID-19 pandemic and the lockdown measures on the local stroke network

IntroductionThe COVID-19 outbreak highly impacted the acute ischemic stroke care management. The primary end point of the study was to evaluate the impact of the COVID-19 outbreak and the following lockdown measures on our hub-and-spoke network; the secondary end point was to evaluate if the impact of the COVID-19 outbreak was different in hub-and-spoke centers.MethodsThis was a retrospective multicenter observational study conducted at the Stroke Units of Policlinico Gemelli, Ospedale San Filippo Neri, Ospedale di Belcolle, and Ospedale San Camillo de Lellis. We collected clinical reports of all consecutive patients admitted with diagnosis of acute ischemic stroke or transient ischemic attack (TIA) during the phase 1 of the lockdown period (11 March 2020–4 May 2020). As controls, we used all consecutive patients admitted for acute ischemic stroke or TIA in the same period of the previous year.ResultsA total of 156 and 142 clinical reports were collected in 2019 and 2020, respectively. During the COVID-19 outbreak, we observed a reduction of number of thrombolysis, a reduction of the length of hospitalization, and an increase of pneumonia. Regarding performance indicators, we observed an increase in onset-to-door time and in door-to-groin time. We did not observe any statistically significant interaction between year (2019 vs 2020) and facility of admission (hub vs spoke) on all variables analyzed.DiscussionOur observational study, involving hub-and-spoke stroke network of a wide regional area, indicates that the COVID-19 outbreak impacted on the acute stroke management. This impact was equally observed in hub as well as in spoke centers.

Mental illness in some Sub Saharan African communities: the perspective of Bioethics and transcultural nursing

In this paper, transcultural nursing is reviewed in light of bioethical issues arising from the interpretation of mental illness in some Western Sub-Saharan African communities. Four field studies were carried out by the authors of this paper in Sub-Saharan Africa (from 2016 to 2019), during which the traditional “treatment” of enchaining people considered “crazy” by local cultures was explored. These inhuman practices raise the attention of bioethics, which investigate ways to overcome this traditional practice not violating the cultural identity of the peoples who practice it. The model of Gregoire Ahongbonon and of his Association, “Saint Camille de Lellis”, is reported as an example of negotiation between the respect for traditions and the guarantee of human rights. The care practice in force in the Saint Camille is related to transcultural nursing as an adequate form of treatment and strategy for restoring patients’ dignity and rights.

Solutions of the (free boundary) Reifenberg Plateau problem

Abstract We solve two variants of the Reifenberg problem for all coefficient groups. We carry out the direct method of the calculus of variation and search a solution as a weak limit of a minimizing sequence. This strategy has been introduced by De Lellis, De Philippis, De Rosa, Ghiraldin and Maggi and allowed them to solve the Reifenberg problem. We use an analogous strategy proved in [C. Labourie, Weak limits of quasiminimizing sequences, preprint 2020, https://arxiv.org/abs/2002.08876] which allows to take into account the free boundary. Moreover, we show that the Reifenberg class is closed under weak convergence without restriction on the coefficient group.

A mountain pass theorem for minimal hypersurfaces with fixed boundary

In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold $$\gamma $$ , of mountain pass type, contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound $$\gamma $$ . In order to do so, we develop min–max methods similar to those of De Lellis and Ramic (Ann. Inst. Fourier 68(5): 1909 –1986, 2018) adapted to the discrete setting of Almgren and Pitts. Our approach allows one to consider the case in which the two stable hypersurfaces with boundary $$\gamma $$ intersect at interior points.

On the selection of measure-valued solutions for the isentropic Euler system

Measure-valued solutions to fluid equations arise naturally, for instance as vanishing viscosity limits, yet exhibit non-uniqueness to a vast extent. In this paper, we show that some measure-valued solutions to the two-dimensional isentropic compressible Euler equations, although they are energy admissible, can be discarded as unphysical, as they do not arise as vanishing viscosity limits. In fact, these measure-valued solutions also do not arise from a sequence of weak solutions of the Euler equations, in contrast to the incompressible case. Such a phenomenon has already been observed by Chiodaroli, Feireisl, Kreml, and Wiedemann using an A-free rigidity argument, but only for non-deterministic initial datum. We develop their rigidity result to the case of non-constant states and combine this with a compression wave solution evolving into infinitely many weak solutions, constructed by Chiodaroli, De Lellis, and Kreml. Hereby, we show that there exist infinitely many generalized measure-valued solutions to the two-dimensional isentropic Euler system with quadratic pressure law, which behave deterministically up to a certain time and which cannot be generated by weak solutions with bounded energy or by vanishing viscosity sequences.

A probabilistic Takens theorem

Let be a Borel set, μ a Borel probability measure on X and T : X → X a locally Lipschitz and injective map. Fix strictly greater than the (Hausdorff) dimension of X and assume that the set of p-periodic points of T has dimension smaller than p for p = 1, …, k − 1. We prove that for a typical polynomial perturbation of a given locally Lipschitz function , the k-delay coordinate map is injective on a set of full μ-measure. This is a probabilistic version of the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We also provide a non-dynamical probabilistic embedding theorem of similar type, which strengthens a previous result by Alberti, Bölcskei, De Lellis, Koliander and Riegler. In both cases, the key improvements compared to the non-probabilistic counterparts are the reduction of the number of required coordinates from 2 dim X to dim X and using Hausdorff dimension instead of the box-counting one. We present examples showing how the use of the Hausdorff dimension improves the previously obtained results.

Behavioral Teleporting of Individual Ethograms onto Inanimate Robots: Experiments on Social Interactions in Live Zebrafish.

SummarySocial behavior is widespread in the animal kingdom, and it remarkably influences human personal and professional lives. However, a thorough understanding of the mechanisms underlying social behavior is elusive. Integrating the seemingly different fields of robotics and preclinical research could bring new insight on social behavior. Toward this aim, we established “behavioral teleporting” as an experimental solution to independently manipulate multiple factors underpinning social interactions. Behavioral teleporting consists of real-time transfer of the complete ethogram of a live zebrafish onto a remotely-located robotic replica. Through parallel and simultaneous behavioral teleporting, we studied the interaction between two live fish swimming in remotely-located tanks: each live fish interacted with an inanimate robot that mirrored the behavior of the other fish, and the morphology of each robot was independently tailored. Our results indicate that behavioral teleporting can preserve natural interaction between two live animals, while allowing fine control over morphological features that modulate social behavior.

Convex integration and phenomenologies in turbulence

In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier–Stokes equations. These results build on the groundbreaking works of De Lellis and Székelyhidi Jr., who extended Nash’s fundamental ideas on C^1 flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration , have fundamental analogies with the phenomenological theories of hydrodynamic turbulence [51,54,55,200]. Mathematical problems arising in turbulence(such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions. First, we give an elementary construction of nonconservative C^{0+}_{x,t} weak solutions of the Euler equations, first proven by De Lellis–Székelyhidi Jr. [52,53]. Second, we present Isett’s [108] recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work [21] of De Lellis–Székelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class C^{1/3-}_{x,t} are constructed, attaining any energy profile. Third, we give a concise proof of the authors’ recent result [23], which proves the existence of infinitely many weak solutions of the Navier–Stokes in the regularity class C^0_t L^{2+}_x \cap C^0_t W^{1,1+}_x . We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.

Asymptotic Estimates for the Willmore Flow With Small Energy

Abstract Kuwert and Schätzle showed in 2001 that the Willmore flow converges to a standard round sphere, if the initial energy is small. In this situation, we prove stability estimates for the barycenter and the quadratic moment of the surface. Moreover, in codimension one, we obtain stability bounds for the enclosed volume and averaged mean curvature. As direct applications, we recover a quasi-rigidity estimate due to De Lellis and Müller (2006) and an estimate for the isoperimetric deficit by Röger and Schätzle (2012), whose original proofs used different methods.

Global weak solutions to the three-dimensional inviscid Boussinesq system in the presence of magnetic field

In this paper, we consider the inviscid Boussinesq system in the presence of magnetic field in three space dimensions. We prove that the system admits infinitely many global weak solutions with large initial data. Our main tool is based on the convex integration method developed by De Lellis and Szekelyhidi in the context of incompressible Euler system.

Nonuniqueness of Weak Solutions to the SQG Equation

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 of De Lellis and Székelyhidi in 2012. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. In view of the results of Marchand in 2008, we establish that for the dissipative SQG equation, weak solutions may be constructed in the same function space both via classical weak compactness arguments and via convex integration. © 2019 Wiley Periodicals, Inc.

SUN-507 Parathyroid Carcinoma and Incidental Finding of Hashimoto's Thyroiditis in a Patient with End-Stage Renal Disease: A Case Report

Background: Parathyroid carcinoma is a very rare endocrine neoplasia. The incidence reported is less than 1% in primary hyperparathyroidisim (1). The presence of this rare carcinoma is even less likely in the setting of chronic kidney disease. Unfortunately, parathyroid carcinoma is difficult to differentiate clinically with parathyroid adenoma and hyperplasia pre-operatively. Clinical Case: A case of a 48-year-old male, diagnosed with end-stage renal disease secondary to IgA Nephropathy and maintained on chronic hemodialysis for 8 years, presented with progressive decreasing height and occasional bone pain. Despite compliance to hemodialysis, calcimimetics, phosphate binders, calcium and vitamin D supplementation to suppress the high PTH levels, patient remained persistently hyperparathyroid with intact PTH of 4793pg/ml (NV: 15-65 pg/ml). Other laboratory findings showed vitamin D sufficiency (Vitamin D 87 nmol/L (NV: >75 nmol/L), hypocalcemia (ionized calcium 1.08 mmol/L (NV: 1.10-1.35) and hyperphosphatemia (inorganic phosphate 2.15 mmol/L (NV: 0.01-1.45 mmol/L). With the failure of medications to suppress the high PTH levels, surgical removal of the parathyroid glands was then the next possible option. A dual phase sestamibi scan of the parathyroid revealed a single focal activity at the mid aspect of the right thyroid bed on the wash-out phase. Patient underwent total parathyroidectomy and total thyroidectomy. Intraoperative findings revealed enlarged bilateral parathyroid glands with largest measuring 2x2 cm and incidental findings of multiple nodules in bilateral thyroid glands. Final histopathology revealed parathyroid carcinoma in 3 out of 4 glands and Hashimoto’s thyroiditis with multi-nodular adenomatous goiter. Conclusion: It is a challenge to clinically differentiate parathyroid carcinoma and other causes of hyperparathyroidism among patients undergoing chronic maintenance hemodialysis due to the effect of renal insufficiency on calcium metabolism. This case highlights an unusual case of parathyroid carcinoma in a patient with ESRD and incidental finding of hashimoto’s thyroiditis, who presented with extremely elevated intact parathyroid hormone accompanied with hypocalcemia. Reference: (1) Ronald A. De Lellis, Ricardo V. Lloyd, Philipp U. Heitz, CharisEng: World Health Organization Classification of Tumors. Pathology and Genetics, Tumors of Endocrine Organs.77, 2004

Very weak solutions to the two-dimensional Monge-Ampére equation

In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Amp\'ere equation with H\"older-continuous first derivatives of exponent $\beta<1/5$. Our approach is based on combining the approach of Lewicka-Pakzad \cite{Lewicka:2015gz} with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by the second author with De Lellis and Inauen in \cite{DeLellis:2015wm} for the isometric immersion problem.

Nursing care organizations in Western Europe from the 16th to 18th centuries

From the 16th to 18th centuries in Western Europe care and nursing institutions for the sick were created by the faithful of the Catholic Church. The greatest successes in that field were achieved by three persons: Juan de Dios, Camillo de Lellis and Vincent de Paul. They established charity, care and nursing congregations, orders and convents which conducted wide charity activities in Europe before the French Revolution.

Degraded mixing solutions for the Muskat problem

We prove the existence of infinitely many mixing solutions for the Muskat problem in the fully unstable regime displaying a linearly degraded macroscopic behaviour inside the mixing zone. In fact, we estimate the volume proportion of each fluid in every rectangle of the mixing zone. The proof is a refined version of the convex integration scheme submitted in De Lellis and Szekelyhidi Jr. (Arch Ration Mech Anal 195:225–260, 2010), Szekelyhidi (Ann Sci Ec Norm Super 45(3):491–509, 2012) applied to the subsolution in Castro et al. (Mixing solutions for the Muskat problem, arXiv:1605.04822 , 2016). More generally, we obtain a quantitative h-principle for a class of evolution equations which shows that, in terms of weak*-continuous quantities, a generic solution in a suitable metric space essentially behaves like the subsolution. This applies of course to linear quantities, and in the case of IPM to the power balance $$\mathbf {P}$$ (14) which is quadratic. As further applications of such quantitative h-principle we discuss the case of vortex sheet for the incompressible Euler equations.

On uniqueness of solutions to conservation laws verifying a single entropy condition

For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see [E. Yu. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mat. Z. 55(5) (1994) 116–129 (in Russian), Math. Notes 55(5) (1994) 517–525]. This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Minimal entropy conditions for Burgers equation, Quart. Appl. Math. 62(4) (2004) 687–700]. These two proofs both rely on the special connection between Hamilton–Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In this paper, we prove the single entropy result for scalar conservation laws without using Hamilton–Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case.

Infinitely many Leray–Hopf solutions for the fractional Navier–Stokes equations

We prove the ill-posedness for the Leray–Hopf weak solutions of the incompressible and ipodissipative Navier–Stokes equations, when the power of the diffusive term is . We construct infinitely many solutions, starting from the same initial datum, which belong to and strictly dissipate their energy in small time intervals. The proof exploits the “convex integration scheme” introduced by C. De Lellis and L. Székelyhidi for the incompressible Euler equations, joining these ideas with new stability estimates for a class of non-local advection-diffusion equations and a local (in time) well-posedness result for the fractional Navier–Stokes system. Moreover, we show the existence of dissipative Hölder continuous solutions of Euler equations that can be obtained as a vanishing viscosity limit of Leray–Hopf weak solutions of suitable fractional Navier–Stokes equations.

Non-uniqueness of energy-conservative solutions to the isentropic compressible two-dimensional Euler equations

We consider the 2-d isentropic compressible Euler equations. It was shown in [E. Chiodaroli, C. De Lellis and O. Kreml, Global ill-posedness of the isentropic system of gas dynamics, Comm. Pure Appl. Math. 68(7) (2015) 1157–1190] that there exist Riemann initial data as well as Lipschitz initial data for which there exist infinitely many weak solutions that fulfill an energy inequality. In this paper, we will prove that there is Riemann initial data for which there exist infinitely many weak solutions that conserve energy, i.e. they fulfill an energy equality. As in the aforementioned paper, we will also show that there even exist Lipschitz initial data with the same property.