1,721,494 research outputs found
On the structure of weak solutions to scalar conservation laws with finite entropy production
No full textWe consider weak solutions with finite entropy production to the scalar conservation law ∂tu+divxF(u)=0in(0,T)×Rd.Building on the kinetic formulation we prove under suitable nonlinearity assumption on f that the set of non Lebesgue points of u has Hausdorff dimension at most d. A notion of Lagrangian representation for this class of solutions is introduced and this allows for a new interpretation of the entropy dissipation measure
Rectifiability of entropy defect measures in a micromagnetics model
Full text linkWe study the fine properties of a class of weak solutions u of the eikonal equation arising as asymptotic domain of a family of energy functionals introduced in [T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm. Pure Appl. Math. 54 2001, 3, 294-338]. In particular, we prove that the entropy defect measure associated to u is concentrated on a 1-rectifiable set, which detects the jump-type discontinuities of u
The rectifiability of the entropy defect measure for Burgers equation
Full text linkWe consider bounded weak solutions to the Burgers equation for which every entropy dissipation is representable by a measure and we prove that all these measures are concentrated on the graphs of countably many Lipschitz curves. The main tool is the Lagrangian representation, which is an extension of the method of characteristics to the non-smooth setting
Regularity estimates for scalar conservation laws in one space dimension
No full textWe deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function f has on the entropy solution. More precisely, if the set {w: f′′(w)≠0} is dense, the regularity of the solution can be expressed in terms of BVφ spaces, where φ depends on the nonlinearity of f. If moreover the set {w: f′′(w) = 0} is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that f′° u(t) BV loc(R) for every t > 0 and that this can be improved to SBVloc(R) regularity except an at most countable set of singular times. Finally, we present some examples that show the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces
Characterization of Minimizers of Aviles–Giga Functionals in Special Domains
Full text linkWe consider the singularly perturbed problem Fε(u, Ω) : = ∫ Ωε| ∇ 2u| 2+ ε- 1| 1 - | ∇ u| 2| 2 on bounded domains Ω ⊂ R2. Under appropriate boundary conditions, we prove that if Ω is an ellipse, then the minimizers of Fε(· , Ω) converge to the viscosity solution of the eikonal equation | ∇ u| = 1 as ε→ 0
Differentiability properties of the flow of 2d autonomous vector fields
Full text linkWe investigate under which assumptions the flow associated to autonomous planar vector fields inherits the Sobolev or BV regularity of the vector field. We consider nearly incompressible and divergence-free vector fields, taking advantage in both cases of the underlying Hamiltonian structure. Finally, we provide an example of an autonomous planar Sobolev divergence-free vector field, such that the corresponding regular Lagrangian flow has no bounded variation
“Dall’invenzione all’impresa. Marconi e la Wireless Telegraph & Signal Company”
No full textIl saggio esamina la fase iniziale di sviluppo della invenzione di Guglielmo Marconi e le vicende portarono alla costituzione della Wireless Telegraph & Signal Company (poi divenuta Marconi Wireless Telegraph Company), con particolare riguardo per la analisi dei costi sostenuti nel corso di questo processo. L’obiettivo è di individuare quali sono stati i settori in cui si sono concentrate le spese (in particolare consulenze brevettuali e legali, dimostrazioni ed esperimenti) e di fornire indicatori che suggeriscano quale possa essere stata l’entità degli investimenti che hanno consentito a Marconi e alla sua società di conquistare una posizione di leadership nella nuova tecnologia. Questa analisi consente inoltre di ricostruire con maggior precisione il ruolo di Marconi e degli altri co-protagonisti nel decollo della società, i rispettivi loro obiettivi e le loro strategie.
The essay is a study of the early stages in the development of Guglielmo Marconi’s invention and in the creation of the Wireless Telegraph & Signal Company (later to become the Marconi Wireless Telegraph Company); special attention is paid to the analysis of the costs of that process. The aim is to identify the sectors in which expenses were particularly high (patenting, legal assistance, experiments and demonstrations), and to provide indicators of the amount of the investments that allowed Marconi and his company to establish their leadership in the development of the new technology. This approach offers also new insights into the role of Marconi and of other co-protagonists in the take-off of the company, their distinctive objectives and agendas
Regularity estimates for the flow of BV autonomous divergence-free vector fields in R^2
No full textWe consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with bounded variation. We prove a Lusin-Lipschitz regularity result for X and we show that the Lipschitz constant grows at most linearly in time. As a consequence we deduce that both geometric and analytical mixing have a lower bound of order (Formula presented.) as (Formula presented.)
On the concentration of entropy for scalar conservation laws
No full textWe prove that the entropy for an L∞-solution to a scalar con-servation laws with continuous initial data is concentrated on a countably 1-rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution
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