1,721,331 research outputs found
Exact integrability conditions for cotangent vector fields
Full text linkIn Quantum Hydro-Dynamics the following problem is relevant: let (ρ,Λ)∈W1,2(Rd,Ld,R+)×L2(Rd,Ld,Rd) be a finite energy hydrodynamics state, i.e. Λ = 0 when ρ= 0 and E=∫Rd12|∇ρ|2+12Λ2Ld<∞.The question is under which conditions there exists a wave function ψ∈ W1 , 2(Rd, Ld, C) such that ρ=|ψ|,J=ρΛ=I(ψ ̄∇ψ).The second equation gives for ψ=ρw smooth, | w| = 1 , that iΛ=ρw ̄∇w. Interpreting ρLd as a measure in the metric space Rd, this question can be stated in generality as follows: given metric measure space (X, d, μ) and a cotangent vector field v∈ L2(T∗X) , is there a function w∈ W1 , 2(X, μ, S1) such that dw=iwv.Under some assumptions on the metric measure space (X, d, μ) (conditions which are verified on Riemann manifolds with the measure μ= ρVol or more generally on non-branching MCP (K, N)), we show that the necessary and sufficient conditions for the existence of w is that (in the case of differentiable manifold) ∫v(γ(t))·γ ̇(t)dt∈2πZfor π-a.e. γ, where π is a test plan supported on closed curves. This condition generalizes the condition that the vorticity is quantized. We also give a representation of every possible solution. In particular, we deduce that the wave function ψ=ρw is in W1 , 2(X, μ, C) whenever ρ∈W1,2(X,μ,R+)
Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems and relaxation approximation
No full textWe study two problems related to hyperbolic systems with a dissipative source. In the first part, we consider the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under a coupling condition among hyperbolic and dissipative part known as the Shizuta.Kawashima condition. Under the assumption of small initial data, these solutions approach constant equilibrium state in the Lp-norm at a rate O t.m2 (1. 1 p ) , as t �¨��, for p �¸ [min {m, 2},��]. The main tool is given by a detailed analysis of the Green function for the linearized problem. If the space dimension m = 1 or the system is rotational invariant, it is possible to give an explicit form to the main terms in the Green kernel. In the second part, we consider the hyperbolic limit of special systems of balance laws: this means to study the limit of the solution to a system of balance laws under the rescaling (t, x) �¨ (t/�Ã, x/�Ã), as �à �¨ 0. For some special dissipative systems in one space dimension, it is possible to prove the existence of the limit and to identify it as a solution to a system of conservation laws
Quadratic interaction functional for systems of conservation laws: a case study
Full text linkWe prove a quadratic interaction estimate for wavefront approximate solutions to the triangular system of conservation laws
This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme \cite{anc_mar_11_CMP}.
Our aim is to extend the analysis, done for scalar conservation laws \cite{bia_mod_13}, in the presence of transversal interactions among wavefronts of different families.
The proof is based on the introduction of a quadratic functional , decreasing at every interaction, and such that its total variation in time is bounded. %cancellations it variation is controlled by the total variation growths at most of the total variation of the solution multiplied by the amount of cancellation.
The study of this particular system is a key step in the proof of the quadratic interaction estimate for general systems: it requires a deep analysis of the wave structure of the solution and the reconstruction of the past history of each wavefront involved in an interaction
On a quadratic functional for scalar conservation laws
Full text linkWe prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme. The proof is based on the introduction of a quadratic functional (t), decreasing at every interaction, and such that its total variation in time is bounded. Differently from other interaction potentials present in the literature, the form of this functional is the natural extension of the original Glimm functional, and coincides with it in the genuinely nonlinear case
Estimates on path functionals over Wasserstein spaces
Full text linkIn this paper we consider the class a functionals (introduced by Brancolini, Buttazzo, and Santambrogio) Gr,p(γ) defined on Lipschitz curves γ valued in the p-Wasserstein space. The problem considered is the following: given a measure μ, give conditions in order to assure the existence a curve γ such that γ(0)=μ, γ(1)=δx0, and Gr,p(γ)<+∞. To this end, new estimates on Gr,p(μ) are given and a notion of dimension of a measure (called /path dimension/) is introduced: the path dimension specifies the values of the parameters (r,p) for which the answer to the previous reachability problem is positive. Finally, we compare the path dimension with other known dimensions
Quadratic interaction functional for general systems of conservation laws
Full text linkFor the Glimm scheme approximation u_\e to the solution of the system of conservation laws in one space dimension
\begin{equation*}
u_t + f(u)_x = 0, \qquad u(0,x) = u_0(x) \in \R^n,
\end{equation*}
with initial data with small total variation, we prove a quadratic (w.r.t. \TV(u_0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux are made (apart smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems.
More precisely we obtain the following results:
\begin{itemize}
\item a new analysis of the interaction estimates of simple waves;
\item a Lagrangian representation of the derivative of the solution, i.e. a map which follows the trajectory of each wave from its creation to its cancellation;
\item the introduction of the characteristic interval and partition for couples of waves, representing the common history of the two waves;
\item a new functional controlling the variation in speed of the waves w.r.t. time.
\end{itemize}
This last functional is the natural extension of the Glimm functional for genuinely nonlinear systems.
The main result is that the distribution is a measure with total mass \leq \const \TV(u_0)^2
Properties of Mixing BV Vector Fields
Full text linkWe consider the density properties of divergence-free vector fields b∈L1([0,1],BV([0,1]2)) which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow Xt is an ergodic/weakly mixing/strongly mixing measure preserving map when evaluated at t= 1 . Our main result is that there exists a Gδ -set U⊂Lt,x1([0,1]3) containing all divergence-free vector fields such that 1.the map Φ associating b with its RLF Xt can be extended as a continuous function to the Gδ -set U ;2.ergodic vector fields b are a residual Gδ -set in U ;3.weakly mixing vector fields b are a residual Gδ -set in U ;4.strongly mixing vector fields b are a first category set in U ;5.exponentially (fast) mixing vector fields are a dense subset of U . The proof of these results is based on the density of BV vector fields such that Xt=1 is a permutation of subsquares, and suitable perturbations of this flow to achieve the desired ergodic/mixing behavior. These approximation results have an interest of their own. A discussion on the extension of these results to d≥ 3 is also 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
TERRE DI LUCCA E DI VERSILIA: Pisa etrusca e romana; Gli Etruschi nella Toscana nord-occidentale; I Liguri Apuani; La romanizzazione della Toscana nord-occidentale
No full textSiti archeologici
In questa sezione troverete informazioni sulle località archeologiche.
La presentazione di un'area è, infatti, strettamente collegata alle collezioni di reperti archeologici da essa provenienti e attualmente conservati nei Musei del territorio.
L'obiettivo è di restituire virtualmente l'originaria stratificazione di numerosi siti che, per l'assenza di consistenti resti fruibili e per ragioni di conservazione o di inaccessibilità non sono di fatto visitabili.
La selezione qui presentata non ha trascurato quelle emergenze che, pur avendo una grande importanza archeologica, non sono ancora rappresentate adeguatamente in un Museo o che lo sono, ma in sedi museali esterne al territorio provinciale.
Altre informazioni sui siti archeologici sono disponibili nella sezione "approfondimenti", dedicata ad approfondimenti tematici su aspetti storico-culturali delle civiltà pre e proto storiche, etrusche e romane, in diretta relazione con l'attuale territorio della Provincia di Lucca
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