Stochastic transport equation with bounded and Dini continuous drift

Abstract

The results established by Flandoli, Gubinelli and Priola (2010) [17] for stochastic transport equation with bounded and Hölder continuous drift are generalized to bounded and Dini continuous drift. The uniqueness of $L^{\infty }$-solutions is established by the Itô–Tanaka trick partially solving the uniqueness problem, which is still open, for stochastic transport equation with only bounded measurable drift. Moreover the existence and uniqueness of stochastic quasi-diffeomorphisms flows for a stochastic differential equation with bounded and Dini continuous drift is obtained.

Introduction

We are concerned with the following stochastic transport equation

$$\{\begin{array}{c}{\partial }_{t}u(t,x)+b(t,x)\cdot \mathrm{\nabla }u(t,x)+{\sum }_{i=1}^d{\partial }_{x_i}u(t,x)\circ {\dot{B}}_i(t)=0,(t,x)\in (0,T)\times {\mathbb{R}}^d,\hfill \\ u(t,x){|}_{t=0}=u_0(x),x\in {\mathbb{R}}^d,\hfill \end{array}$$

where ${\{B(t)\}}_{t⩾0}={\{(B_1(t),B_2(t),{}_{\cdots },B_d(t))\}}_{t⩾0}$ is a d-dimensional standard Brownian motion defined on a stochastic basis ($\mathrm{\Omega },\mathcal{F},\mathbb{P},{({\mathcal{F}}_t)}_{t⩾0}$). The stochastic integration with notation ∘ is interpreted in Stratonovich sense. Given $T>0$, the drift coefficient $b:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ and the initial data $u_0:{\mathbb{R}}^d\to \mathbb{R}$ are measurable functions in $L^1([0,T];L_{loc}^1({\mathbb{R}}^d;{\mathbb{R}}^d))$ and $L^1([0,T];L_{loc}^1({\mathbb{R}}^d))$ respectively. We are interested in the existence and uniqueness of weak $L^{\infty }$-solutions for the stochastic equation (1.1).

There are many results on the weak $L^{\infty }$-solutions for the deterministic transport equation. The first remarkable result of the uniqueness solution in $L^{\infty }([0,T]\times {\mathbb{R}}^d)$ was obtained by DiPerna and Lions [12] under the assumption $b\in L^1([0,T];W_{loc}^{1,1}({\mathbb{R}}^d;{\mathbb{R}}^d))$ with suitable global conditions including $L^{\infty }$-bounds on spatial divergence. Then Ambrosio [1] weakened the condition $W_{loc}^{1,1}$ to $B{V}_{loc}$ and the uniqueness of weak $L^{\infty }$-solutions is obtained under the assumption that negative part of divb is in space $L^1([0,T];L^{\infty }({\mathbb{R}}^d))$. For general b just with $B{V}_{loc}$ or Hölder regularity, the uniqueness of weak $L^{\infty }$-solutions for the deterministic equation fails and counterexamples have been constructed in many works [5], [7], [9], [11], [12]. Obviously, more restrictions need to be imposed on b to overcome the obstacle of nonuniqueness of solution in deterministic case.

However the appearance of the noise makes the solution unique under very general assumptions on the drift coefficient for ordinary differential equations [20], [33], so a natural idea is to investigate the effects of noise in transport equation. The first milestone result, founded by Flandoli, Gubinelli and Priola [17], showed the uniqueness of weak $L^{\infty }$-solutions just with assuming $b\in L^{\infty }([0,T];{\mathcal{C}}_b^{\alpha }({\mathbb{R}}^d;{\mathbb{R}}^d))$ and $\mathrm{div}b\in L^q([0,T]\times {\mathbb{R}}^d)$ for some $\alpha >0$, $q>2$ for $d>1$ or $q=1$ for $d=1$. This is the first concrete example of a partial differential equation related to fluid dynamics that becomes well-posed with a suitable noise. A key step for this result is to perform differential computations on regularization of $L^{\infty }$-solutions by a commutator lemma. Unfortunately the strategy fails if b is not Sobolev differentiable. However, by observing the fact that stochastic differential equation (SDE)

$$\{\begin{array}{c}dX(s,t)=b(t,X(s,t))dt+dB(t),0⩽s<t⩽T,\hfill \\ X(t,s){|}_{t=s}=x\in {\mathbb{R}}^d,\hfill \end{array}$$

defines a ${\mathcal{C}}^1$ stochastic diffeomorphisms flow, and along the stochastic characteristic $X(s,t)$, the integral ${\int }_s^t\mathrm{div}b(r,X(s,r,x))dr$ has a regularization, Flandoli, Gubinelli and Priola developed the commutator lemma to prove the uniqueness of solutions. On the other hand, for bounded measurable b, Mohammed, Nilssen and Proske [26] also proved the existence, uniqueness and Sobolev differentiable stochastic flows for (1.2) by employing ideas from the Malliavin calculus coupled with new probabilistic estimates on the spatial weak derivatives of solutions of (1.2). Then, as an application, they obtained the existence and uniqueness of Sobolev differentiable weak solutions for (1.1) with every ${\mathcal{C}}^1({\mathbb{R}}^d)$ initial data. Recently, by using the same approach as Mohammed, Nilssen and Proske's [26], Rezakhanlou [31] also established the Sobolev differentiable stochastic flows for (1.2) just assuming $b\in L^{\iota }([0,T];L^{\zeta }({\mathbb{R}}^d;{\mathbb{R}}^d))$ with $\iota ,\zeta \in [2,+\infty ]$ and $2/\iota +d/\zeta <1$. Notice that in one result the stochastic flow $\{X(s,t,x)\}$ is differentiable in x in the classical sense [17] and $\mathrm{\nabla }{X}^{-1}(s,t,x)$ is continuous in s almost surely, but in the other results the stochastic flow $\{X(s,t,x)\}$ are only Sobolev differentiable [26], [31], which does not imply the continuity of $\mathrm{\nabla }{X}^{-1}(s,t,x)$ in s. So the method developed by Mohammed, Nilssen and Proske, and Rezakhanlou may not be adapted to establish the uniqueness of weak $L^{\infty }$-solutions to (1.1) for bounded drift.

Recent result [3], by using a different philosophy, proved the uniqueness of weak $L^{\infty }$-solutions for (1.1) just with assuming the BV regularity for b but without the $L^{\infty }$-bounds on spatial divergence. For nonconstant diffusion, if $b\in B{V}_{lov}$ and the spatial divergence is bounded, the uniqueness is obtained by Zhang [37]. If b is in the Krylov–Röckner class, that is $b\in L^{\iota }(0,T;L^{\zeta }({\mathbb{R}}^d;{\mathbb{R}}^d))$ with $\iota ,\zeta \in (2,+\infty )$ and $2/\iota +d/\zeta <1$, by assuming that the spatial divergence for b vanishes, Neves and Olivera [28] proved the existence and uniqueness of weak $L^{\infty }$-solutions as well. If the drift and the spatial divergence are only integrable: $b\in L^{\infty }([0,T];L^{{\kappa }_1}({\mathbb{R}}^d;{\mathbb{R}}^d))$ and $\mathrm{div}b\in L^{\infty }([0,T];L^{{\kappa }_2}({\mathbb{R}}^d))$ with ${\kappa }_1>d$ and ${\kappa }_2>d/2$, Maurelli [25] also proved the uniqueness of weak $L^{\infty }$-solutions for (1.1). There are also several other related works to (1.1) with $L^p$ and $W^{1,p}$ solutions ($p\in [1,\infty )$). For example, weak $L^p$-solution with Sobolev differentiable drift [6], that is $b\in L^1(0,T;W_{loc}^{1,p^{\prime }}({\mathbb{R}}^d;{\mathbb{R}}^d))(1/p+1/p^{\prime }=1)$; ${\cap }_{p⩾1}{W}_{loc}^{1,p}$ solutions with drift in the Krylov–Röckner class [4], [15]; $W_{loc}^{1,p}$ solutions with drift in the Hölder class [35], that is $b\in L^{\iota }([0,T];{\mathcal{C}}_b^{\alpha }({\mathbb{R}}^d;{\mathbb{R}}^d))$ with $p,\iota \in [1,+\infty ]$, $\alpha \in (0,1)$ and $2/\iota <\alpha$ (also see [27] for time independent and unbounded Hölder drift); ${\mathcal{C}}^{2,{\alpha }_0}$-solutions for time independent and ${\mathcal{C}}^{2,\alpha }$ drift [13] ($0<{\alpha }_0<\alpha <1$). However, for bounded measurable b and $\mathrm{div}b\in L^q([0,T]\times {\mathbb{R}}^d)$ (for some $q\in [1,\infty )$), the uniqueness of weak $L^{\infty }$-solutions for (1.1) is still unknown.

This paper intends to give a partial answer for the above problem and novelties of the work are

• The uniqueness of weak $L^{\infty }$-solutions for the Cauchy problem (1.1) with bounded and Dini-continuous drift is established due to the existence of noise, while the corresponding deterministic equation has multiple solutions.

• The existence and uniqueness of stochastic quasi-diffeomorphisms flow for singular SDE (3.1) is established without Hölder continuity or Sobolev differentiability hypotheses on b.

We follow the strategy of Flandoli, Gubinelli and Priola's [17] to establish the existence of a stochastic quasi-diffeomorphisms flow for (3.1), by the Itô–Tanaka trick, then derive a commutator estimates to get the uniqueness for weak $L^{\infty }$-solution of (1.1). The main idea of Itô–Tanaka trick is to use a parabolic partial differential equation (PDE) to transform the original SDE (3.1) with irregular drift and regular diffusion to a new SDE (3.15) with regular drift and diffusion. Then by the equivalence between (3.1) and (3.15) we show the existence of the stochastic quasi-diffeomorphisms flow for SDE (3.1). There are also some recent works on the stochastic flows and SDEs [2], [13], [14], [16], [18], [32], [34], [38].

In the following parts, we first derive the $W^{2,\infty }$ estimates for a class of second order parabolic PDEs with bounded and Dini continuous coefficients in section 2; then by using the $W^{2,\infty }$ estimates, the existence and uniqueness of stochastic quasi-diffeomorphisms flow for SDE (3.1) is shown in section 3 by the Itô–Tanaka trick; last section is concerned with the existence and uniqueness of weak $L^{\infty }$-solutions to stochastic transport equation (1.1).

Notations

The letter C denotes a positive constant, whose values may change in different places. For a parameter or a function κ, $C(\kappa )$ means the constant is only dependent on κ, and we also write it as C if there is no confusion. $\mathbb{N}$ is the set of natural numbers. For every $R>0$, $B_R:=\{x\in {\mathbb{R}}^d:|x|<R\}$. Almost surely is abbreviated to $a.s$.. Let Θ be a ${\mathbb{R}}^{d\times d}$-valued function $\mathrm{\Theta }={({\mathrm{\Theta }}_{i,j}(x))}_{d\times d}$ with norm $\Vert \mathrm{\Theta }(x)\Vert ={\max }_{1⩽i,j⩽d}|{\mathrm{\Theta }}_{i,j}(x)|$. For $\xi \in {\mathbb{R}}^d$, $|\xi |={({\sum }_{i=1}^d{\xi }_i^2)}^{1/2}$. ${\mathbb{R}}_{+}$ is the set of nonnegative real numbers and ${\overline{\mathbb{R}}}_{+}={\mathbb{R}}_{+}\cup \{+\infty \}$.