Well-posedness of the transport equation by stochastic perturbation

Abstract

We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, 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 partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.

Addition of new parts

References in corpus (1)

• Global flows for stochastic differential equations without global Lipschitz conditions

Cited by in corpus (125)

• Noise Prevents Singularities in Linear Transport Equations
• Solution properties of a 3D stochastic Euler fluid equation
• Pathwise uniqueness and continuous dependence for SDEs with nonregular drift
• Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift
• Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation
• High mode transport noise improves vorticity blow-up control in 3D Navier-Stokes equations
• Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness
• On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift
• Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations
• 2D Euler equations with Stratonovich transport noise as a large scale stochastic model reduction
• From additive to transport noise in 2D fluid dynamics
• Multidimensional SDE with distributional drift and Lévy noise
• Well-posedness by noise for scalar conservation laws
• Stochastic Navier-Stokes Equations and Related Models
• A sufficient condition for the Kolmogorov 4/5 law for stationary martingale solutions to the 3D Navier-Stokes equations
• Nonlinear Young integrals and differential systems in Hölder media
• On a Stochastic Leray-α model of Euler equations
• Quantifying a convergence theorem of Gyöngy and Krylov
• The Hunter-Saxton equation with noise
• Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise
• Scattering for stochastic nonlinear Schrödinger equations
• Noise prevents infinite stretching of the passive field in a stochastic vector advection equation
• Pathwise uniqueness for singular SDEs driven by stable processes
• Nonlinear PDEs with modulated dispersion II: Korteweg-de Vries equation
• Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations
• Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli
• Degenerate SDE with Hölder-Dini Drift and Non-Lipschitz Noise Coefficient
• On a stochastic Camassa-Holm type equation with higher order nonlinearities
• Averaging along irregular curves and regularisation of ODEs
• LDP and CLT for SPDEs with Transport Noise
• SDEs with critical time dependent drifts: strong solutions
• On the Boltzmann Equation with Stochastic Kinetic Transport: Global Existence of Renormalized Martingale Solutions
• Non-uniqueness in law of stochastic 3D Navier--Stokes equations
• Stochastic flows for Lévy processes with Hölder drifts
• Nonlinear PDEs with modulated dispersion I: Nonlinear Schrödinger equations
• Variability of paths and differential equations with $BV$-coefficients
• Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier-Stokes equations perturbed by transport noise
• $L^q(L^p)$-theory of stochastic differential equations
• Nonlinear Anisotropic Degenerate Parabolic-Hyperbolic Equations with Stochastic Forcing
• Regularization by transport noise for 3D MHD equations
• Remarks on the stochastic transport equation with Hölder drift
• The density of the solution to the stochastic transport equation with fractional noise
• Transport and continuity equations with (very) rough noise
• Quantitative stability estimates for Fokker-Planck equations
• Well-posedness for a stochastic Camassa-Holm type equation with higher order nonlinearities
• Rough linear transport equation with an irregular drift
• Nonlocal elliptic equation in Hölder space and the martingale problem
• Stochastic Differential Equations with Critical Drifts
• Fokker-Planck equation for dissipative 2D Euler equations with cylindrical noise
• No blow-up by nonlinear Itô noise for the Euler equations
• Kalman filtering and smoothing for linear wave equations with model error
• Stochastic differential equations with Sobolev diffusion and singular drift
• Exponential Convergence in $L^p$-Wasserstein Distance for Diffusion Processes without Uniformly Dissipative Drift
• Non-explosion by Stratonovich noise for ODEs
• Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients
• Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing
• Regularized vortex approximation for 2D Euler equations with transport noise
• Solution theory of fractional SDEs in complete subcritical regimes
• On weak solutions of stochastic differential equations with sharp drift coefficients
• Stochastic transport equations with unbounded divergence
• Regularization by noise and stochastic Burgers equations
• The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics at fixed Reynolds number
• Strong regularization by Brownian noise propagating through a weak H{ö}rmander structure
• Stochastic transport equation in bounded domains
• Regularization by noise for stochastic Hamilton-Jacobi equations
• Stochastic Hamiltonian flows with singular coefficients
• Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient
• On the rate of convergence of strong Euler approximation for SDEs driven by Levy processes
• Schauder's estimate for nonlocal kinetic equations and its applications
• Strong solutions of a stochastic differential equation with irregular random drift
• On a stochastic Hardy-Littlewood-Sobolev inequality with application to Strichartz estimates for the white noise dispersion
• A stochastic sewing lemma and applications
• Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift
• The three dimensional stochastic Zakharov system
• Density Analysis for coupled forward-backward SDEs with non-Lipschitz drifts and Applications
• Uniqueness for a Stochastic Inviscid Dyadic Model
• Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations
• Gaussian invariant measures and stationary solutions of 2D Primitive Equations
• On the Construction and Malliavin Differentiability of Levy Noise Driven SDEs with Singular Coefficients
• Existence for stochastic 2D Euler equations with positive $H^{-1}$ vorticity
• Harnack Inequalities for SDEs with Multiplicative Noise and Non-regular Drift
• Blow-up for a Stochastic Model of Chemotaxis Driven by Conservative Noise on ${\mathbb{R}}^2$
• A local-in-time theory for singular SDEs with applications to fluid models with transport noise
• Existence and uniqueness of $W_{loc}^{1,r}$-solutions for stochastic transport equations
• Transport equation driven by a stochastic measure
• A solution selection problem with small symmetric stable perturbations
• Renormalized Solutions to Stochastic Continuity Equations with Rough Coefficients
• Irregular Stochastic differential equations driven by a family of Markov processes
• Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients
• Well-posedness of nonlinear transport equation by stochastic perturbation
• Strong uniqueness for stochastic evolution equations with unbounded measurable drift term
• SDEs with random and irregular coefficients
• Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty }$-vector fields
• Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise
• Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplicative noise
• Generalized Navier-Stokes flows and applications to incompressible viscous fluids
• Strong solutions of stochastic differential equations with square integrable drift
• Smoothing of Boundary Behaviour in Stochastic Planar Evolutions
• Stochastic Flows of SDEs with Irregular Coefficients and Stochastic Transport Equations
• Pathwise uniqueness for stochastic reaction-diffusion equations in Banach spaces with an Hölder drift component
• Stochastic continuity equation with non-smooth velocity
• Strong solution for stochastic transport equations with irregular drift: existence and non-existence
• Well-posedness for SDEs driven by different type of noises
• Noise and Stability in Reaction-diffusion Equations
• Scaling Limits for Stochastic SQG Equations and 2D Inviscid Critical Boussinesq Equations with Transport Noises
• The viscous variational wave equation with transport noise
• Convergent finite difference schemes for stochastic transport equations
• Finite time mixing and enhanced dissipation for 2D Navier--Stokes equations by Ornstein--Uhlenbeck flow
• Large time behavior of strong solutions for stochastic Burgers equation
• Well-posedness by noise for linear advection of $k$-forms
• Collective stochastic dynamics of the Cucker-Smale ensemble under uncertain communications
• Hypoelliptic diffusions with singular drifts
• Density for solutions to stochastic differential equations with unbounded drift
• Probabilistic representation for mild solution of the Navier-Stokes equations
• The effect of noise intensity on stochastic parabolic equations
• Regularization by noise in (2x 2) hyperbolic systems of conservation law
• The Itô SDEs and Fokker--Planck equations with Osgood and Sobolev coefficients
• On Weak Solutions of SDEs with Singular Time-Dependent Drift and Driven by Stable Processes
• Rescaling nonlinear noise for 1D stochastic parabolic equations
• Uniqueness of strong solutions for SDEs with Hölder diffusions
• Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation
• Existence of stochastic entropy solutions for stochastic scalar balance laws with Lipschitz vector fields
• A comparison theorem for stochastic differential equations under a Novikov-type condition
• Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition
• Stochastic differential equations with coefficients in Sobolev spaces