5,143 research outputs found
A uniqueness result for the continuity equation in two dimensions
No full textWe characterize the autonomous, divergence-free vector fields on the plane such that the Cauchy problem for the continuity equation admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential associated to .
As a corollary we obtain uniqueness under the assumption that the curl of is a measure. This result can be extended to certain non- autonomous vector fields with bounded divergence
Renormalized solutions to the continuity equation with an integrable damping term
No full textWe consider the continuity equation with a nonsmooth vector field and a damping term. In their fundamental paper, DiPerna and Lions (Invent Math 98:511–547, 1989) proved that, when the damping term is bounded in space and time, the equation is well posed in the class of distributional solutions and the solution is transported by suitable characteristics of the vector field. In this paper, we prove existence and uniqueness of renormalized solutions in the case of an integrable damping term, employing a new logarithmic estimate inspired by analogous ideas of Ambrosio et al. (Rendiconti del Seminario Fisico Matematico di Padova 114:29–50, 2005), Crippa and De Lellis (J Reine Angew Math 616:15–46, 2008) in the Lagrangian case
On the Lp-differentiability of certain classes of functions
No full textWe prove the -differentiability at almost every point for convolution products on of the form , where is bounded measure and is a homogeneous kernel of degree .
From this result we derive the -differentiability for vector fields on whose curl and divergence are measures, and also for vector fields with bounded deformation
Structure of level sets and Sard-type properties of Lipschitz maps
No full textWe consider certain properties of maps of class C2 from Rd to Rd−1 that are strictly related to Sard’s theorem, and we show that some of them can be extended to Lipschitz maps, while others require some additional regularity. We also give examples showing that, in terms of regularity, our results are optimal
L'arte paleocristiana. Visione e spazio dalle origini a Bisanzio,
No full text-il testo di architettura è integralmente di M.A. Crippa
-l'ed. francese è Desclée de Bouwer, Parigi. l'ed. spagnola è Lunwerg, Madrid-Barcellon
La fantasia è un dovere. L'arte murale come strumento politico di riscoperta dei centri urbani
No full textLuoghi e modernità. Pratiche e saperi dell'architettura
No full textAlcuni docenti italiani e stranieri vengono provocati da M. A. Crippa a rispondere ad alcune domande su pratiche e saperi dell'architettura contemporanea: in quale avventura di senso trascina la convinzione di alcuni studiosi che l'architettura 'moderna' abbia sostanzialmente prodotto non-luoghi? quali sono gli abiti, mentali e operativi, di carattere storiografico in grado di fornire strumenti di indagine adeguati al progetto moderno? quale è lo specifico del 'fare' architettura? quali le componenti o 'stazioni di posta' più rilevanti dell'attuale ricerca storiografica? Sono individuabili continuità tra passato e presente? Gli studiosi, coordinati da M. A. Crippa, orientano le riflessioni in risposta a tali interrogativi rispondendo sia a tematiche specificamente storigrafiche che a questioni di progetto contemporaneo
Continuity equations and ODE flows with non-smooth velocity
No full textIn 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
Flows of singular vector fields and applications to fluid and kinetic equations
No full textSeveral physical phenomena arising in fluid dynamics and kinetic equations can be modeled by nonlinear transport PDE. Such quantities are the vorticity of a fluid, or the density of a collection of particles advected by a velocity field which is highly irregular. The theory of characteristics provides a link between this PDE and the ODE dX/dt=b(t,X(t,x)), where b is the velocity field. When b has Sobolev or BV regularity and bounded divergence, the theory of DiPerna-Lions and Ambrosio gives a good notion of solution to the ordinary differential equation using the concept of regular Lagrangian flow. Extending the results of Crippa-DeLellis, and more recently Bouchut-Crippa, we study Lagrangian flows associated to velocity fields with anisotropic regularity: those with gradient given by the singular integral of an L^1 function in some directions, and the singular integral of a measure in others. We exploit an anisotropic version of the previous arguments and estimate the difference quotients in this context, thereby gaining quantitative estimates in terms of the given regularity bounds. One then recovers well-posedness for the ordinary differential equation. This answers positively the question of existence of Lagrangian solutions to the Vlasov Poisson and Euler equations with L^1 data
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