1,720,992 research outputs found
Non-explosion by Stratonovich noise for ODEs
Full text linkWe show that the addition of a suitable Stratonovich noise prevents the explosion for ODEs with drifts of super-linear growth, in dimension d≥2. We also show the existence of an invariant measure and the geometric ergodicity for the corresponding SDE
Wiener chaos and uniqueness for stochastic transport equation = Chaos de Wiener et unicité pour l’équation de transport stochastique
Full text linkWe prove a uniqueness result for the stochastic transport linear equation (STLE), without any W(1.1) or BV hypothesis on the coefficient, which is needed for the corresponding deterministic equation. We use Wiener chaos decomposition to pass from the STLE to a deterministic second-order transport equation with uniqueness property
No blow-up by nonlinear Itô noise for the Euler equations
Full text linkBy employing a suitable multiplicative Itô noise with radial structure and with more than linear growth, we show the existence of a unique, global-in-time, strong solution for the stochastic Euler equations in two and three dimensions. More generally, we consider a class of stochastic partial differential equations (SPDEs) with a superlinear growth drift and suitable nonlinear, multiplicative Itô noise, with the stochastic Euler equations as a special case within this class. We prove that the addition of such a noise effectively prevents blow-ups in the solution of these SPDEs
Uniform approximation of 2D Navier-Stokes equations with vorticity creation by stochastic interacting particle systems
Full text linkWe consider a stochastic interacting particle system in a bounded domain with reflecting boundary, including creation of new particles on the boundary prescribed by a given source term. We show that such particle system approximates 2D Navier-Stokes equations in vorticity form and impermeable boundary, the creation of particles modeling vorticity creation at the boundary. Kernel smoothing, more specifically smoothing by means of the Neumann heat semigroup on the space domain, allows to establish uniform convergence of regularized empirical measures to (weak solutions of) Navier-Stokes equations
Well-posedness by noise for scalar conservation laws
Full text linkWe consider stochastic scalar conservation laws with spatially inhomogeneous flux. The regularity of the flux function with respect to its spatial variable is assumed to be low, so that entropy solutions are not necessarily unique in the corresponding deterministic scalar conservation law. We prove that perturbing the system by noise leads to well-posedness
Regularization by rough Kraichnan noise for the generalised SQG equations
No full textWe consider the generalised Surface Quasi-Geostrophic (gSQG) equations in
with parameter , an active scalar model
interpolating between SQG () and the 2D Euler equations () in
vorticity form. Existence of weak -valued solutions in the
deterministic setting is known, but their uniqueness is open. We show that the
addition of a rough Stratonovich transport noise of Kraichnan type regularizes
the PDE, providing strong existence and pathwise uniqueness of solutions for
initial data , for suitable values
related to the regularity degree of the noise and the singularity
degree of the velocity field; in particular, we can cover any for suitable and and we can reach a suitable ("critical")
threshold. The result also holds in the presence of external forcing and solutions are shown to depend continuously on the data
of the problem; furthermore, they are well approximated by vanishing viscosity
and regular approximations
Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity
Full text linkThe strong existence and the pathwise uniqueness of solutions with (Formula presented.) -vorticity of the 2D stochastic Euler equations are proved. The noise is multiplicative and it involves the first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. The stability under regularizations is also proved
Global well posedness and ergodic results in regular Sobolev spaces for the nonlinear Schrödinger equation with multiplicative noise and arbitrary power of the nonlinearity
No full textWe consider the nonlinear Schrödinger equation on the d-dimensional torus Td, with the nonlinearity of polynomial type |u|2σu. For any σ ∈ N and s > d2 we prove that adding to this equation a suitable stochastic forcing term there exists a unique global solution for any initial data in Hs(Td). The effect of the noise is to prevent blow-up in finite time, differently from the deterministic setting. Moreover, we prove the existence of an invariant measure and its uniqueness under more restrictive assumptions on the noise term
Large deviations for singularly interacting diffusions
Full text linkIn this paper we prove a large deviation principle (LDP) for the empirical measure of a general system of mean-field interacting diffusions with singular drift (as the number of particles tends to infinity) and show convergence to the associated McKean–Vlasov equation. Along the way, we prove an extended version of the Varadhan Integral Lemma for a discontinuous change of measure and subsequently a LDP for Gibbs and Gibbs-like measures with singular potentials.</p
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