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Marco Romito

Publications and source records attributed to Marco Romito.

At least 37 records · Page 2Linked to original sources
arXiv2018

Limit theorems and fluctuations for point vortices of generalized Euler equations

We prove a mean field limit, a law of large numbers and a central limit theorem for a system of point vortices on the 2D torus at equilibrium with positive temperature. The point vortices are formal solutions of a class of equations generalising the Euler equations, and are also known in the literature as generalised inviscid SQG. The mean field limit is a steady solution of the equations, the CLT limit is a stationary distribution of the equations.

math.PR↗
arXiv2017

Random initial conditions for semi-linear PDEs

We analyze the effect of random initial conditions on the local well--posedness of semi--linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework.

math.PR↗
arXiv2015

Hölder regularity of the densities for the Navier--Stokes equations with noise

We prove that the densities of the finite dimensional projections of weak solutions of the Navier-Stokes equations driven by Gaussian noise are bounded and Hölder continuous, thus improving the results of Debussche and Romito [DebRom2014]. The proof is based on analytical estimates on a conditioned Fokker-Planck equation solved by the density, that has a "non-local" term that takes into account the influence of the rest of the infinite dimensional dynamics over the finite subspace under observation.

math.PR↗
arXiv2014

Time regularity of the densities for the Navier--Stokes equations with noise

We prove that the density of the law of any finite dimensional projection of solutions of the Navier--Stokes equations with noise in dimension $3$ is Hölder continuous in time with values in the natural space $L^1$. When considered with values in Besov spaces, Hölder continuity still holds. The Hölder exponents correspond, up to arbitrarily small corrections, to the expected diffusive scaling.

math.PR↗
arXiv2014

Global regularity for a logarithmically supercritical hyperdissipative dyadic equation

We prove global existence of smooth solutions for a slightly supercritical dyadic model. We consider a generalized version of the dyadic model introduced by Katz-Pavlovic [2005] and add a viscosity term with critical exponent and a supercritical correction. This model catches for the dyadic a conjecture that for Navier-Stokes equations was formulated by Tao [2009].

math.AP↗
arXiv2013

Local existence and uniqueness for a two-dimensional surface growth equation with space--time white noise

We study local existence and uniqueness for a surface growth model with space-time white noise in 2D. Unfortunately, the direct fixed-point argument for mild solutions fails here, as we do not have sufficient regularity for the stochastic forcing. Nevertheless, one can give a rigorous meaning to the stochastic PDE and show uniqueness of solutions in that setting. Using spectral Galerkin method and any other types of regularization of the noise, we obtain always the same solution.

math.PR↗
arXiv2012

Existence of densities for the 3D Navier-Stokes equations driven by Gaussian noise

We prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic Navier-Stokes equations in dimension 3. In particular, under very mild assumptions on the noise, we prove that finite dimensional projections of the solutions have densities with respect to the Lebesgue measure which have some smoothness when measured in a Besov space. This is proved thanks to a new argument inspired by an idea introduced in Fournier and Printems (2010).

math.PR↗
arXiv2011

Uniqueness and blow-up for the noisy viscous dyadic model

We consider the dyadic model with viscosity and additive Gaussian noise as a simplified version of the stochastic Navier-Stokes equations, with the purpose of studying uniqueness and emergence of singularities. We prove path-wise uniqueness and absence of blow-up in the intermediate intensity of the non-linearity, morally corresponding to the 3D case, and blow-up for stronger intensity. Moreover, blow-up happens with probability one for regular initial data.

math.PR↗
arXiv2011

Rényi Information Measures for Spectral Change Detection

Change detection within an audio stream is an important task in several domains, such as classification and segmentation of a sound or of a music piece, as well as indexing of broadcast news or surveillance applications. In this paper we propose two novel methods for spectral change detection without any assumption about the input sound: they are both based on the evaluation of information measures applied to a time- frequency representation of the signal, and in particular to the spectrogram. The class of measures we consider, the Rényi entropies, are obtained by extending the Shannon entropy definition: a biasing of the spectrogram coefficients is realized through the dependence of such measures on a parameter, which allows refined results compared to those obtained with standard divergences. These methods provide a low computational cost and are well-suited as a support for higher level analysis, segmentation and classification algorithms.

cs.SD↗
arXiv2010

On Leray's problem for almost periodic flows

We prove existence and uniqueness for fully-developed (Poiseuille-type) flows in semi-infinite cylinders, in the setting of (time) almost-periodic functions. In the case of Stepanov almost-periodic functions the proof is based on a detailed variational analysis of a linear "inverse" problem, while in the Besicovitch setting the proof follows by a precise analysis in wave-numbers. Next, we use our results to construct a unique almost periodic solution to the so called "Leray's problem" concerning 3D fluid motion in two semi-infinite cylinders connected by a bounded reservoir. In the case of Stepanov functions we need a natural restriction on the size of the flux, while for Besicovitch solutions certain limitations on the generalized Fourier coefficients are requested.

math.AP↗
arXiv2010

Smooth solutions for the dyadic model

We consider the dyadic model, which is a toy model to test issues of well-posedness and blow-up for the Navier-Stokes and Euler equations. We prove well-posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier-Stokes. Likewise we prove well-posedness for the inviscid problem (in a suitable regularity class) when the parameter corresponds to the strongest transport effect of the non-linearity.

math.AP↗
arXiv2010

Local existence and uniqueness in the largest critical space for a surface growth model

We show the existence and uniqueness of solutions (either local or global for small data) for an equation arising in different aspects of surface growth. Following the work of Koch and Tataru we consider spaces critical with respect to scaling and we prove our results in the largest possible critical space such that weak solutions are defined. The uniqueness of global weak solutions remains unfortunately open, unless the initial conditions are sufficiently small.

math.AP↗
arXiv2009

The martingale problem for Markov solutions to the Navier-Stokes equations

Under suitable assumptions of regularity and non-degeneracy on the covariance of the driving additive noise, any Markov solution to the stochastic Navier-Stokes equations has an associated generator of the diffusion and is the unique solution to the corresponding martingale problem. Some elementary examples are discussed to interpret these results.

math.PR↗