Stochastic regularization for transport equations
type: Item
instance of: scholarly article
title: Stochastic regularization for transport equations
zbMATH Open document ID: 1477.60103
author: Guangying Lv
author: Jinlong Wei
author: Jinqiao Duan
author: Hongjun Gao
published in: Stochastic and Partial Differential Equations. Analysis and Computations
publication date: 2021-08-12T00:00:00Z
review text: In this article, the authors deal with the stochastic transport equation \[ \partial_t u(t,x) + b(t,x) \cdot \nabla u(t,x) + \sum_{i=1}^d \partial_{x_i} u(t,x) \circ \dot{B}_i(t) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^d, \] \[ u(t,x)|_{t=0} = u_0(x), \,\, x \in \mathbb{R}^d, \] where \(B\) is a \(d\)-dimensional standard Brownian motion. Their first main result (Theorem 1.1) is an existence and uniqueness result for this equation with drift coefficients \(b \in L^q(0,T;\mathcal{C}_b^{\alpha}(\mathbb{R}^d))\), where \(\alpha > 2/q\). Then the authors deal with the question whether the noise has a regularizing effect for the stochastic transport equation. This is indeed the case: As Theorem 1.2 shows, for a suitably chosen drift \(b\) one has nonexistence of solutions for the corresponding deterministic transport equation. Theorem 1.3 presents a nonexistence result for the stochastic transport equation in case \(\alpha + 1 < 2/q\) and \(d \geq 2\). The authors also provide new results on existence and uniqueness of stochastic flows of diffeomorphisms on stochastic differential equations, which are used for the proofs of the above mentioned main results.
reviewed by: Stefan Tappe
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35L02
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 7381497
zbMATH Keywords: stochastic transport equations
zbMATH Keywords: stochastic strong solution
zbMATH Keywords: uniqueness
zbMATH Keywords: existence
zbMATH Keywords: nonexistence
MaRDI profile type: Publication
full work available at URL: s40072-020-00171-w
OpenAlex ID: W3023023447
cites work: Stochastic flows of diffeomorphisms for one-dimensional SDE with discontinuous drift
cites work: Renormalized Solutions for Stochastic Transport Equations and the Regularization by Bilinear Multiplicative Noise
cites work: Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness
cites work: An ill-posed Cauchy problem for a hyperbolic system in two space dimensions
cites work: \(L^p\)-solutions of the stochastic transport equation
cites work: Q4831550
cites work: Nearly Lipschitzean divergence free transport propagates neither continuity nor BV regularity
cites work: Oscillatory solutions to transport equations
cites work: Stochastic partial differential equations and applications II. Proceedings of a conference held in Trento, Italy, February 1-6, 1988
cites work: Noise prevents collapse of Vlasov-Poisson point charges
cites work: Nonuniqueness of bounded solutions for some BV outside a hyperplane vector field
cites work: Q2930843
cites work: Stochastic regularization effects of semi-martingales on random functions
cites work: Q4399897
cites work: Flow of Homeomorphisms and Stochastic Transport Equations
cites work: Pathwise uniqueness and continuous dependence for SDEs with non-regular drift
cites work: Noise prevents singularities in linear transport equations
cites work: Hölder Flow and Differentiability for SDEs with Nonregular Drift
cites work: REGULARIZING PROPERTIES OF BROWNIAN PATHS AND A RESULT OF DAVIE
cites work: Well-posedness of the transport equation by stochastic perturbation
cites work: Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift
cites work: Remarks on the stochastic transport equation with H\"{o}lder drift
cites work: Noise prevents infinite stretching of the passive field in a stochastic vector advection equation
cites work: Critical non-Sobolev regularity for continuity equations with rough velocity fields
cites work: Q4436177
cites work: Elliptic and Parabolic Second-Order PDEs with Growing Coefficients
cites work: Strong solutions of stochastic equations with singular time dependent drift
cites work: Stochastic nonlinear Schrödinger equation. I: A priori estimates
cites work: Holder Estimates for Solutions of Parabolic SPDEs
cites work: Q3217380
cites work: Q3995203
cites work: Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition
cites work: Hölder estimates of mild solutions for nonlocal SPDEs
cites work: Stochastic flows of SDEs with irregular coefficients and stochastic transport equations
DOI: 10.1007/S40072-020-00171-W
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description: scientific article; zbMATH DE number 7381497