SDLThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.https://preprints.sissa.it:443/xmlui2017-09-12T01:21:36Z2017-09-12T01:21:36ZGlobal, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentialsAntonelli, PaoloMichelangeli, AlessandroScandone, Raffaelehttp://preprints.sissa.it/handle/1963/352942017-09-12T00:00:39Z2017-01-01T00:00:00ZGlobal, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials
Antonelli, Paolo; Michelangeli, Alessandro; Scandone, Raffaele
We prove the existence of weak solutions in the space of energy for
a class of non-linear Schördinger equations in the presence of a external rough
magnetic potential. Under our assumptions it is not possible to study the
problem by means of usual arguments like resolvent techniques or Fourier integral
operators, for example. We use a parabolic regularization and we solve
the approximating Cauchy problem. This is achieved by obtaining suitable
smoothing estimates for the dissipative evolution. The total mass and energy
bounds allow to extend the solution globally in time. We then infer suffcient
compactness properties in order to produce a global-in-time finite energy weak
solution to our original problem.
2017-01-01T00:00:00ZOn fractional powers of singular perturbations of the LaplacianGeorgiev, VladimirMichelangeli, AlessandroScandone, Raffaelehttp://preprints.sissa.it/handle/1963/352932017-09-07T00:00:38Z2017-01-01T00:00:00ZOn fractional powers of singular perturbations of the Laplacian
Georgiev, Vladimir; Michelangeli, Alessandro; Scandone, Raffaele
We qualify a relevant range of fractional powers of the so-called
Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator, and, when applicable, of the decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms.
Partially supported by the 2014-2017 MIUR-FIR grant \Cond-Math: Condensed Matter and
Mathematical Physics" code RBFR13WAET.
2017-01-01T00:00:00ZAnalysis of a dynamic peeling test with speed-dependent toughnessLazzaroni, GiulianoNardini, Lorenzohttp://preprints.sissa.it/handle/1963/352922017-09-05T00:00:35Z2017-01-01T00:00:00ZAnalysis of a dynamic peeling test with speed-dependent toughness
Lazzaroni, Giuliano; Nardini, Lorenzo
We analyse a one-dimensional model of dynamic debonding for a thin film, where
the local toughness of the glue between the film and the substrate also depends on the debonding speed. The wave equation on the debonded region is strongly coupled with Griffth's criterion for the evolution of the debonding front. We provide an existence and uniqueness result and find explicitly the solution in some concrete examples. We study the limit of solutions as inertia tends to zero, observing phases of unstable propagation, as well as time discontinuities, even though the toughness diverges at a limiting debonding speed.
2017-01-01T00:00:00ZRegularity estimates for scalar conservation laws in one space dimensionMarconi, Eliohttp://preprints.sissa.it/handle/1963/352912017-08-17T00:00:37Z2017-08-01T00:00:00ZRegularity estimates for scalar conservation laws in one space dimension
Marconi, Elio
In this paper we deal with the regularizing effect that, in a scalar conservation laws in one space dimension, the nonlinearity of the flux function ƒ has on the entropy solution. More precisely, if the set ⟨w : ƒ " (w) ≠ 0⟩ is dense, the regularity of the solution can be expressed in terms of BV Ф spaces, where Ф depends on the nonlinearity of ƒ. If moreover the set ⟨w : ƒ " (w) = 0⟩ is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that ƒ' 0 u(t) ∈ BVloc (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 shows the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.
2017-08-01T00:00:00Z