Publications SISSA

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  • Pubblicazione
    Long-time stability of the quantum hydrodynamic system on irrational tori
    ( 2022)
    Roberto Feola
    ;
    Felice Iandoli
    ;
    Federico Murgante
    We consider the quantum hydrodynamic system on a d-dimensional irrational torus with d = 2, 3. We discuss the behaviour, over a “non-trivial” time interval, of the Hs-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving form εsmall initial conditions, remain bounded in Hs for a time scale of order O(ε−1−1/(d−1)+), which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schrödinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising form three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes. This is due to the irrationality of the torus which prevents to have “good separation” properties of the eigenvalues of the linearized operator at zero. The main steps of the proof are: (i) to prove precise lower bounds on small divisors; (ii) to construct a modified energy by means of a suitable high/low frequencies analysis, which gives an a priori estimate on the solutions.
  • Pubblicazione
    The relaxed area of S1-valued singular maps in the strict BV-convergence
    ( 2022)
    Bellettini, G.
    ;
    Carano, S.
    ;
    Scala, R.
    Given a bounded open set Ω ⊂ R2, we study the relaxation of the nonparametric area functional in the strict topology in BV(Ω; R2). and compute it for vortex-type maps, and more generally for maps in W1,1(Ω;S1) having a finite number of topological singularities. We also extend the analysis to some specific piecewise constant maps in BV(Ω;S1), including the symmetric triple junction map.
  • Pubblicazione
    Entropy preserving low dissipative shock capturing with wave-characteristic based sensor for high-order methods
    ( 2020)
    Tonicello N.
    ;
    Lodato G.
    ;
    Vervisch L.
    Shock capturing procedures are required to stabilise numerical simulations of gas dynamics problems featuring non-isentropic discontinuities. In the present work, particular attention is focused on the expected non-monotonicity of the entropy profile across shock waves. A peculiar physical property which was not considered so far in the evaluation of shock capturing techniques. In the context of high-order spectral difference methods and using most recent discontinuity sensors based on the decay rate of the modes of the amplitude of characteristic waves, results show how the choice of a physical-based procedure (additional viscosity) returns a better description of shocks compared to approaches relying on the direct addition of a Laplacian term in the solved equations. Various canonical compressible flows are simulated, in one-, two-, and three-dimensional setups, to illustrate the performance and flexibility of the proposed approach. It is shown that the addition of a well-calibrated bulk viscosity is capable of smoothing out discontinuities without an excessive damping of vortical structures, preserving also specific compressible flow physics, as the non-monotonic entropy profiles through the shocks.
  • Pubblicazione
    Nonclassical minimizing surfaces with smooth boundary
    ( 2022)
    De Lellis C.
    ;
    De Philippis G.
    ;
    Hirsch J.
    We construct a Riemannian metric g on R4(arbitrarily close to the euclidean one) and a smooth simple closed curve Γ ⊂ R4such that the unique area minimizing surface spanned by Γ has infinite topology. Furthermore the metric is almost Kähler and the area minimizing surface is calibrated.
  • Pubblicazione
    Supertranslations and Superrotations at the Black Hole Horizon
    ( 2016)
    Donnay, Laura
    ;
    Giribet, Gaston
    ;
    González, Hernán A
    ;
    Pino, Miguel
    We show that the asymptotic symmetries close to nonextremal black hole horizons are generated by an extension of supertranslations. This group is generated by a semidirect sum of Virasoro and Abelian currents. The charges associated with the asymptotic Killing symmetries satisfy the same algebra. When considering the special case of a stationary black hole, the zero mode charges correspond to the angular momentum and the entropy at the horizon.