Related papers: Stochastic transport equation with singular drift
We establish the well-posedness of stochastic differential equations possessing degenerate diffusions and singular drifts. We prove that SDEs defined on the homogeneous Carnot group, whose hypoelliptic diffusion part is given by the…
We prove existence and uniqueness for the transport equation for currents (Geometric Transport Equation) when the driving vector field is time-dependent, Lipschitz in space and merely integrable in time. This extends previous work where…
We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible $C^\infty([0,T)\times \mathbb{T}^d)\cap L^1([0,T]; C^{1-}(\mathbb{T}^d))$ velocity field which…
A linear stochastic transport equation with non-regular coefficients is considered. Under the same assumption of the deterministic theory, all weak $L^\infty$-solutions are renormalized. But then, if the noise is nondegenerate, uniqueness…
We study the linear transport equation \[ \frac{\partial}{\partial t} u ( t,x ) +b ( t,x ) \cdot \nabla u ( t,x ) + \nabla u ( t,x ) \cdot \frac{\partial}{\partial t} X ( t ) =0, \hspace{2em} u ( 0,x ) =u_{0} ( x ) \] where $b$ is a…
We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be…
Stochastic differential equations with Levy motion arise the mathematical models for various phenomenon in geophysical and biochemical sciences. The Fokker Planck equation for such a stochastic differential equations is a nonlocal partial…
We consider $L^\infty_t L^p_x$ solutions of the stochastic transport equation with drift in $L^\infty_t W^{1,q}_x$. We show strong existence and pathwise uniqueness of solutions in a regime of parameters $p,q$ for which non-unique weak…
We consider stochastic differential equation $$ d X_t=b(X_t) dt +d W_t^H, $$ where the drift $b$ is either a measure or an integrable function, and $W^H$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$,…
We prove the existence and uniqueness of maximal solutions to the 3D SALT (Stochastic Advection by Lie Transport, [Holm arXiv:1410.8311]) Navier-Stokes Equation in velocity and vorticity form, on the torus and the bounded domain…
It is known that linear advection equations with Sobolev velocity fields have very poor regularity properties: Solutions propagate only derivatives of logarithmic order, which can be measured in terms of suitable Gagliardo seminorms. We…
In this paper, we extend our previous result from [16]. We prove that transport equations with rough coefficients do possess a uniqueness property. Our method relies strongly on duality and bears a strong resemblance with the well-known…
Two notions of "having a derivative of logarithmic order" have been studied. They come from the study of regularity of flows and renormalized solutions for the transport and continuity equation associated to weakly differentiable drifts.
In this article we study the existence and uniqueness of solutions of stochastic continuity equation with irregular coefficients.
In this paper, we prove the strong Feller property for stochastic delay (or functional) differential equations with singular drift. We extend an approach of Maslowski and Seidler to derive the strong Feller property of those equations. The…
The attracting inverse-square drift provides a prototypical counterexample to solvability of singular SDEs: if the coefficient of the drift is larger than a certain critical value, then no weak solution exists. We prove a positive result on…
We establish the well-posedness of linear elliptic equations with critical-order drifts in $L^d$ and positive zero-order coefficients in $L^1$ or $L^{\frac{2d}{d+2}}$, where classical methods are often too restrictive. Our approach relies…
We prove the existence and uniqueness of global, probabilistically strong, analytically strong solutions of the 2D Stochastic Navier-Stokes Equation under Navier boundary conditions. The choice of noise includes a large class of additive,…
We construct a family of velocity fields demonstrating the sharpness of the classical Zvonkin--Veretennikov--Davie strong well-posedness by noise regime. We consider stochastic differential equations driven by Brownian noise with drift $u$…
Given a bounded autonomous vector field $b \colon \mathbb R^d \to \mathbb R^d$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation \begin{equation*} \partial_t u + b \cdot \nabla u=…