1,720,999 research outputs found
Noiseless regularisation by noise
Full text linkWe analyse the effect of a generic continuous additive perturbation to the well-posedness of ordinary differential equations. Genericity here is understood in the sense of prevalence. This allows us to discuss these problems in a setting where we do not have to commit ourselves to any restrictive assumption on the statistical properties of the perturbation. The main result is that a generic continuous perturbation renders the Cauchy problem well-posed for arbitrarily irregular vector fields. Therefore we establish regularisation by noise “without probability”
On the global evolution of vortex filaments, blobs, and small loops in 3D ideal flows
No full textWe consider a wide class of approximate models of evolution of singular
distributions of vorticity in three dimensional incompressible fluids and we show that
they have global smooth solutions. The proof exploits the existence of suitable Hamilto-
nian functions. The approximate models we analyze (essentially discrete and continuous
vortex filaments and vortex loops) are related to some problem of classical physics con-
cerning turbulence and also to the numerical approximation of flows with very high
Reynolds number. Finally, we apply our strategy to discrete models for filaments used
in numerical methods
Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations
No full textThe motion of a finite number of point vortices on a two-dimensional periodic domain is considered. In the deterministic case it is known to be well posed only for almost every initial configuration. Coalescence of vortices may occur for certain initial conditions. We prove that when a generic stochastic perturbation compatible with the Eulerian description is introduced, the point vortex motion becomes well posed for every initial configuration, in particular coalescence disappears
A Note on Supersymmetry and Stochastic Differential Equations
No full textWe obtain a dimensional reduction result for the law of a class of stochastic differential equations using a supersymmetric representation first introduced by Parisi and Sourlas
Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift
No full textWe consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Hölder continuous drift term. We prove the existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Itô–Tanaka type. The proof requires non-standard elliptic estimates in Hölder spaces. As an application of the stochastic flow, we obtain a Bismut–Elworthy–Li type formula for the first derivatives of the associated diffusion semigroup
Rigorous remarks about scaling laws in turbulent fluids
No full textA definition of scaling law for suitable families of measures is given and investigated. First, a number of necessary conditions are proved. They imply the absence of scaling laws for 2D stochastic Navier-Stokes equations and for the stochastic Stokes (linear) problem in any dimension, while they imply a lower bound on the mean vortex stretching in 3D. Second, for the 3D stochastic Navier-Stokes equations, necessary and sufficient conditions for scaling laws to hold are given, translating the problem into bounds for energy and enstrophy of high and low modes respectively. Unlike in the 2D case, the validity or invalidity of such conditions in 3D remains open
On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model
No full textStochastic currents, namely random vector valued distributions, related to Ito integrals are investigated. Their regularity as distributions is identified
MIXING FOR GENERIC ROUGH SHEAR FLOWS
No full textWe study mixing and diffusion properties of passive scalars driven by generic rough shear flows. Genericity is here understood in the sense of prevalence, and (ir)regularity is measured in the Besov-Nikolskii scale B\alpha 1,\infty, \alpha \in (0,1). We provide upper and lower bounds, showing that, in general, inviscid mixing in H1/2 holds sharply with rate r(t) \sim t1/(2\alpha ), while enhanced dissipation holds with rate r(\nu) \sim \nu\alpha /(\alpha +2). Our results in the inviscid mixing case rely on the concept of \rho-irregularity, first introduced by Catellier and Gubinelli [Stochastic Process. Appl., 126 (2016), pp. 2323-2366], and provide some new insights compared to the behavior predicted by Colombo, Zelati, and Widmayer [Ars Inveniendi Anal. (2021)].AMC
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