On the convergence of stochastic transport equations to a deterministic parabolic one
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On the convergence of stochastic transport equations to a deterministic parabolic one Lucio Galeati1 Received: 3 May 2019 / Revised: 25 November 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A stochastic transport linear equation (STLE) with multiplicative space-time dependent noise is studied. It is shown that, under suitable assumptions on the noise, a multiplicative renormalization leads to convergence of the solutions of STLE to the solution of a deterministic parabolic equation. Existence and uniqueness for STLE are also discussed. Our method works in dimension d ≥ 2; the case d = 1 is also investigated but no conclusive answer is obtained. Keywords SPDE · Transport noise · Stochastic transport equation
1 Introduction Throughout this paper, we consider a stochastic transport linear equation of the form
du = b · ∇udt + ◦dW · ∇u,
(STLE)
where b = b(t, x) is a given deterministic function and W = W (t, x) is a space-time dependent noise of the form W (t, x) = σk (x)Wk (t). (1) k
Here σk are smooth, divergence free, mean zero vector fields, {Wk }k are independent standard Brownian motions and the index k might range on an infinite (countable) set; by (STLE) we mean more explicitly the identity
B 1
Lucio Galeati [email protected] Institute of Applied Mathematics, University of Bonn, Bonn, Germany
123
Stoch PDE: Anal Comp
du = b · ∇udt +
σk · ∇u ◦ dWk ,
(2)
k
where ◦ denotes Stratonovich integral. Let us explain the reasons for studying such equation. In the case of space-independent noise, it has been shown in recent years, starting with [17], that Eq. (STLE) is well posed under much weaker assumptions on b than its deterministic counterpart (i.e. with W = 0), for which essentially sharp condition are given by [1,12]. There is now an extensive literature on the topic of regularization by noise for transport equations, see the review [15] and the references in [5]. However, from the modelling point of view, space-independent noise is too simple, since formally the characteristics associated to (STLE) are given by dX t = −b(t, X t )dt − dWt . Namely, if we interpret u as an ensemble of ideal particles, the addition of such a multiplicative Stratonovich noise corresponds at the Lagrangian level to non interacting particles being transported by a drift b as well as a random, space independent noise W . There are several models, especially those arising in turbulence (see [8] and the discussion in the introduction of [9]), in which it seems more reasonable to consider all the particles to be subject to the same space-dependent, environmental noise W , which is randomly evolving over time and is not influenced by the particles; W may be interpreted as an incompressible fluid in which the particles are immersed. The formal Lagrangian description of (STLE) is dX t = −b(t, X t )dt − ◦dW (t, X t ),
(3)
where the above equation is meaningful once we consider W given by (1) and we explicit the series. Another reason to consider a more structured noise is g
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