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Gianluca Crippa

Bio: Gianluca Crippa is an academic researcher from University of Basel. The author has contributed to research in topics: Vector field & Uniqueness. The author has an hindex of 22, co-authored 102 publications receiving 1991 citations. Previous affiliations of Gianluca Crippa include University of Parma & Scuola Normale Superiore di Pisa.


Papers
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Journal ArticleDOI
TL;DR: In this paper, simple estimates for ordinary differential equations with Sobolev coefficients were derived, which not only allow to recover some old and recent results in a simple direct way, but also have some new interesting corollaries.
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.

312 citations

Journal ArticleDOI
TL;DR: In this paper, the authors review many aspects of the well-posedness theory for the Cauchy problem for continuity and transport equations and for the ordinary differential equation (ODE) for velocity fields that are not smooth, but enjoy suitable "weak differentiability" assumptions.
Abstract: In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

148 citations

Book ChapterDOI
01 Jan 2008
TL;DR: In this paper, the authors provide a self-contained introduction and an up-to-date survey on many aspects of the theory of transport equations and ordinary differential equations with non-smooth velocity fields.
Abstract: The aim of this book is to provide a self-contained introduction and an up-to-date survey on many aspects of the theory of transport equations and ordinary differential equations with non-smooth velocity fields. The interest in this topic is motivated by important issues in nonlinear PDEs, in particular conservation laws and fluid mechanics. A fascinating feature of this research area, which is currently of concern in mathematics, is the interplay between PDE techniques and geometric measure theory techniques. Several masterpieces appear in the related literature, balancing completely rigorous proofs with more heuristic arguments. A consistent part of the book is based on results obtained by the author in collaboration with other mathematicians. After a short introduction to the classical smooth theory, the book is divided into two parts. The first part focuses on the PDE aspect of the problem, presenting some general tools of this analysis, many well-posedness results, an abstract characterization of the well-posedness, and some examples showing the sharpness of the assumptions made. The second part, instead, deals with the ODE aspect of the problem, respectively by an abstract connection with the PDE, and by some direct and simple (but powerful) a priori estimates.

140 citations

Journal ArticleDOI
TL;DR: In this paper, the authors prove existence and uniqueness of measure solutions for the Cauchy problem associated to the (vectorial) continuity equation with a non-local flow and give a stability result with respect to various parameters.
Abstract: In this paper, we prove existence and uniqueness of measure solutions for the Cauchy problem associated to the (vectorial) continuity equation with a non-local flow. We also give a stability result with respect to various parameters.

89 citations

Journal ArticleDOI
Abstract: We consider certain properties of maps of class C 2 from R d to R d 1 that are strictly related to Sard's theorem, and show that some of them can be extended to Lipschitz maps, while others still require some additional regularity. We also give examples showing that, in term of regularity, our results are optimal.

88 citations


Cited by
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Journal ArticleDOI
TL;DR: In this paper, it was shown that a multiplicative stochastic perturbation of Brownian type is enough to render the linear transport equation well-posed. But it was not shown that multiplicative perturbations alone are sufficient to render a deterministic PDE wellposed.
Abstract: We consider the linear transport equation with a globally Holder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of a PDE of fluid dynamics that becomes well-posed under the influence of a (multiplicative) noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of Ito-Tanaka type.

411 citations

Journal ArticleDOI
TL;DR: In this article, a novel concept of three-dimensional (3D) cellular networks, that integrate drone base stations (drone-BSs) and cellular-connected drone users (Drone-UEs), is introduced.
Abstract: In this paper, a novel concept of three-dimensional (3D) cellular networks, that integrate drone base stations (drone-BS) and cellular-connected drone users (drone-UEs), is introduced. For this new 3D cellular architecture, a novel framework for network planning for drone-BSs and latency-minimal cell association for drone-UEs is proposed. For network planning, a tractable method for drone-BSs’ deployment based on the notion of truncated octahedron shapes is proposed, which ensures full coverage for a given space with a minimum number of drone-BSs. In addition, to characterize frequency planning in such 3D wireless networks, an analytical expression for the feasible integer frequency reuse factors is derived. Subsequently, an optimal 3D cell association scheme is developed for which the drone-UEs’ latency, considering transmission, computation, and backhaul delays, is minimized. To this end, first, the spatial distribution of the drone-UEs is estimated using a kernel density estimation method, and the parameters of the estimator are obtained using a cross-validation method. Then, according to the spatial distribution of drone-UEs and the locations of drone-BSs, the latency-minimal 3D cell association for drone-UEs is derived by exploiting tools from an optimal transport theory. The simulation results show that the proposed approach reduces the latency of drone-UEs compared with the classical cell association approach that uses a signal-to-interference-plus-noise ratio (SINR) criterion. In particular, the proposed approach yields a reduction of up to 46% in the average latency compared with the SINR-based association. The results also show that the proposed latency-optimal cell association improves the spectral efficiency of a 3D wireless cellular network of drones.

388 citations

Journal ArticleDOI
TL;DR: In this paper, a novel framework for optimizing the performance of such UAV-based wireless systems in terms of the average number of bits (data service) transmitted to users as well as the UAVs' hover duration (i.e. flight time) is proposed.
Abstract: In this paper, the effective use of flight-time constrained unmanned aerial vehicles (UAVs) as flying base stations that provide wireless service to ground users is investigated. In particular, a novel framework for optimizing the performance of such UAV-based wireless systems in terms of the average number of bits (data service) transmitted to users as well as the UAVs’ hover duration (i.e. flight time) is proposed. In the considered model, UAVs hover over a given geographical area to serve ground users that are distributed within the area based on an arbitrary spatial distribution function. In this case, two practical scenarios are considered. In the first scenario, based on the maximum possible hover times of UAVs, the average data service delivered to the users under a fair resource allocation scheme is maximized by finding the optimal cell partitions associated to the UAVs. Using the powerful mathematical framework of optimal transport theory, this cell partitioning problem is proved to be equivalent to a convex optimization problem. Subsequently, a gradient-based algorithm is proposed for optimally partitioning the geographical area based on the users’ distribution, hover times, and locations of the UAVs. In the second scenario, given the load requirements of ground users, the minimum average hover time that the UAVs need for completely servicing their ground users is derived. To this end, first, an optimal bandwidth allocation scheme for serving the users is proposed. Then, given this optimal bandwidth allocation, optimal cell partitions associated with the UAVs are derived by exploiting the optimal transport theory. Simulation results show that our proposed cell partitioning approach leads to a significantly higher fairness among the users compared with the classical weighted Voronoi diagram. Furthermore, the results demonstrate that the average hover time of the UAVs can be reduced by 64% by adopting the proposed optimal bandwidth allocation scheme as well as the optimal cell partitioning approach. In addition, our results reveal an inherent tradeoff between the hover time of UAVs and bandwidth efficiency while serving the ground users.

321 citations

Journal ArticleDOI
TL;DR: In this paper, simple estimates for ordinary differential equations with Sobolev coefficients were derived, which not only allow to recover some old and recent results in a simple direct way, but also have some new interesting corollaries.
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.

312 citations

01 Jan 2016

275 citations