Well-posedness of the transport equation by stochastic perturbation

By Franco Flandoli, Massimiliano Gubinelli, Enrico Priola. 2008

473 citations recorded by the source corpus.

We consider the linear transport equation with a globally Hölder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of a PDE of fluid dynamics that becomes well-posed under the influence of a (multiplicative) noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of Itô-Tanaka type.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/b95eb189cbcebf9c7df33500f15c9be95d66ed7d

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Citations in the corpus

Selection of equilibria in a linear quadratic mean-field game (2018, background): “Similar questions have been also addressed in the framework of infinite dimensional stochastic differential equations, see for instance Flandoli, Gubinelli and Priola [23] and the monograph of Flandoli [22].”
Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient (2017, background): “Flandoli and co- authors, see for instance [23].”
Averaging along irregular curves and regularisation of ODEs (2012, background): “Note that even in the case of the Brownian motion this was an open problem [6] since the usual approach cannot be applied in this case.”
On the convergence of stochastic transport equations to a deterministic parabolic one (2019, background): “than the initial data, at least at the level of trajectories. However, when considering the corresponding Ito formulation, the Ito-Stratonovich corrector gives rise to a Laplacian. It was intuited in [14] that equation (STLE) has some parabolic features at the mean level; this has become clear in [4”
Degenerate parabolic stochastic partial differential equations (2013, unknown): “Remark, that unlike [23], working with the”
Long‐Time Behavior, Invariant Measures, and Regularizing Effects for Stochastic Scalar Conservation Laws (2014, methodology): “Moreover, it has been proven, see, for example, Fedrizzi and Flandoli [24], Flandoli, Gubinelli and Priola [28] and Mohammed, Nilssen and Proske [44], that solutions to (1.15) satisfy higher regularity than in the deterministic case.”
On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift (2018, background): “heorem 1.1. Clearly it suﬃces to prove the theorem for suﬃciently small (but n-independent) T, this choice of Twill be speciﬁed later. Let T0 = T0(kbkD,d) be given by Lemma 2.3. We invoke the idea of [FGP10] (see also [PT17]): for any T ∈ (0,T0] and for each i= 1,...d, by Lemma 2.3 and a simple time”
Hölder Flow and Differentiability for SDEs with Nonregular Drift (2013, methodology): “These results are issued from a partially new and more quantitative approach to the problem, based on an idea of Flandoli, Gubinelli and Priola [3].”
Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift (2011, methodology): “work generalizes [DF10] devoted to the case where Bis bounded and in addition H¨older continuous, but with no restriction on the initial conditions. To prove our result we use some ideas from [KR05], [FGP10], [DF10] and [FF10]. The extension of Veretennikov’s result [Ve80] and also of [KR05] to inﬁn”
Wellposedness for stochastic continuity equations with Ladyzhenskaya–Prodi–Serrin condition (2013, background): “More precisely, the condition assumed in [14] is Hölder continuity and boundedness of b, and an integrability condition on the divergence.”
Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014, methodology): “6) from [41] in the case b ∈ L∞(0, T ;C b (R)); (6.”
$L^q(L^p)$-theory of stochastic differential equations (2019, background): “1) through the stochastic inverse flow induced by the strong solution, see [3, 11] and references therein.”
Renormalized Solutions for Stochastic Transport Equations and the Regularization by Bilinear Multiplicative Noise (2010, methodology): “That noise could improve the theory of transport equations was first discovered by [14].”
Noise Prevents Singularities in Linear Transport Equations (2012, background): “The Itô formulation (as explained in detail also in [15]) is du+ b · ∇u dt+ σ∇u dW = σ(2) 2 ∆u dt , u|t=0 = u0 .”
Flow of diffeomorphisms for SDEs with unbounded H (2009, methodology): “In [10] in order to study a linear stochastic transport equation with a bounded vector field b̃(t, x) which is Hölder continuous in x, uniformly in time, we have showed that if in (1) σ = (σi) is constant and non-degenerate and b = b̃, then there exists a stochastic flow of C1-”