Marco Romito

Partial regularity for the stochastic Navier-Stokes equations

The effects of random forces on the emergence of singularities in the Navier-Stokes equations are investigated. In spite of the presence of white noise, the paths of a martingale suitable weak solution have a set of singular points of one-dimensional Hausdorff measure zero. Furthermore statistically stationary solutions with finite mean dissipation rate are analysed. For these stationary solutions it is proved that at any time t the set of singular points is empty. The same result holds true for every martingale solution starting from μ 0 -a.e. initial condition u 0 , where μ 0 is the law at time zero of a stationary solution. Finally, the previous result is non-trivial when the noise is sufficiently non-degenerate, since for any stationary solution, the measure μ 0 is supported on the whole space H of initial conditions.

A probabilistic representation for the vorticity of a three-dimensional viscous fluid and for general systems of parabolic equations

A probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term, is given. On this basis, an implicit probabilistic representation for the vorticity in a three-dimensional viscous fluid (described by the Navier–Stokes equations) is carefully analysed, and a theorem of local existence and uniqueness is proved. The aim of the probabilistic representation is to provide an extension of the Lagrangian formalism from the non-viscous (Euler equations) to the viscous case. As an application, a continuation principle, similar to the Beale–Kato–Majda blow-up criterion, is proved.

Analysis of Equilibrium States of Markov Solutions to the 3D Navier-Stokes Equations Driven by Additive Noise

We prove that every Markov solution to the three dimensional Navier-Stokes equations with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast.Moreover, we give a well-posedness criterion for the equations in terms of invariant measures. We also analyse the energy balance and identify the term which ensures equality in the balance.

MARKOVIANITY AND ERGODICITY FOR A SURFACE GROWTH PDE

The paper analyzes a model in surface growth where the uniqueness of weak solutions seems to be out of reach. We prove existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under nondegeneracy conditions on the noise, we establish that any such solution is strong Feller and has a unique invariant measure.

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.

Automatic Adaptation of the Time-Frequency Resolution for Sound Analysis and Re-Synthesis

We present an algorithm for sound analysis and re-synthesis with local automatic adaptation of time-frequency resolution. The reconstruction formula we propose is highly efficient, and gives a good approximation of the original signal from analyses with different time-varying resolutions within complementary frequency bands: this is a typical case where perfect reconstruction cannot in general be achieved with fast algorithms, which provides an error to be minimized. We provide a theoretical upper bound for the reconstruction error of our method, and an example of automatic adaptive analysis and re-synthesis of a music sound.

Global regularity for a slightly supercritical hyperdissipative Navier–Stokes system

We prove global existence of smooth solutions for a slightly supercritical hyperdissipative Navier--Stokes under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao [Tao2009].

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.

Existence of martingale and stationary suitable weak solutions for a stochastic Navier–Stokes system

The existence of suitable weak solutions of 3D Navier–Stokes equations, driven by a random body force, is proved. These solutions satisfy a local balance of energy. Existence of statistically stationary solutions is also proved.

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.