Stochastic continuity equation with nonsmooth velocity

Abstract

In this article we study the existence and uniqueness of solutions of the stochastic continuity equation with irregular coefficients.

Introduction

Several physical phenomena arising in fluid dynamics and kinetic equations can be modeled by the continuity/ transport equation,

$$\begin{aligned} \partial _t u(t, x) + \mathrm{div} ( b(t,x) u(t,x) ) = 0, \end{aligned}$$
(1.1)

where u is the physical quantity that evolves in time. Such quantities are the vorticity of a fluid, or the density of a collection of particles advected by a velocity field which is highly irregular, in the sense that it has a derivative given by a distribution and a nonlinear dependence on the solution u. For application in the fluid dynamics see Lions’ books [18, 19] and for applications in the domain of conservation laws see Dafermos’ book [5].

Recently research activity has been devoted to study continuity equations with rough coefficients, showing a well-posedness result. We put focus in the uniqueness issue. Di Perna and Lions [7] have introduced the notion of renormalized solution to this equation: it is a solution such that

$$\begin{aligned} \partial _t \beta (u(t, x) )+ \mathrm{div} (b(t,x) \cdot \beta (u(t,x)) = 0 . \end{aligned}$$
(1.2)

for any suitable nonlinearity \(\beta \). Notice that (1.2) holds for smooth solutions, by an immediate application of the chain rule. The renormalization property asserts that nonlinear compositions of the solution are again solutions, or alternatively that the chain rule holds in this weak context. The overall result which motivates this definition is that, if the renormalization property holds, then solutions of (1.1) are unique and stable.

In the case when b has \(W^{1,1}\) spatial regularity (together with a condition of boundedness on the divergence) the commutator lemma between smoothing convolution and weak solution can be proved and, as a consequence, all \(L^{\infty }\)-weak solutions are renormalized. The theory has been generalized by Ambrosio [1] to the case of only BV regularity for b instead of \(W^{1,1}\). In the case of two-dimensional vector field, we also refer to the work of Bouchut and Desvillettes [4] that treated the case of divergence-free vector field with continuous coefficient, and to [12] in which this result is extended to vector field with \(L_{loc}^{2}\) coefficients with a condition of regularity on the direction of the vector field. We refer the readers to two excellent summaries in [2] and [6].

In recent years, much attention has been devoted to extensions of this theory under random perturbations of the drift vector field, namely considering the following stochastic linear transport/continuity equation

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t u(t, x) + \mathrm{Div} \left( \left( b(t, x) + \frac{\mathrm{d} B_{t}}{\mathrm{d}t}\right) \cdot u(t, x) \right) = 0, \\&u|_{t=0}= u_{0}. \end{aligned} \right. \end{aligned}$$
(1.3)

Here, \((t,x) \in [0,T] \times \mathbb R^d\), \(\omega \in \Omega \) is an element of the probability space \((\Omega , \mathbb P, \mathcal {F})\), \(b:\mathbb R_+ \times \mathbb R^d \rightarrow \mathbb R^d\) is a given vector field and \(B_{t} = (B_{t}^{1},\ldots ,B _{t}^{d} )\) is a standard Brownian motion in \(\mathbb {R}^{d}\). The stochastic integration is to be understood in the Stratonovich sense.

A very interesting situation is when the stochastic problem is better behaved than the deterministic one. A first result in this direction was given by Flandoli et al. [11], where they obtained well-posedness of the stochastic problem for an Hölder continuous drift term, with some integrability conditions on the divergence. Their approach is based on a careful analysis of the characteristics. Using a similar approach, Fedrizi and Flandoli in [8] obtained a well-posedness result in the class \(W_{loc}^{1,p}\)-solution under only some integrability conditions on the drift, with no assumption on the divergence, but for fairly regular initial conditions. There, it is only assumed that

$$\begin{aligned} \begin{aligned}&b\in L^{q}\big ( [0,T] ; L^{p}(\mathbb {R}^{d}) \big ), \\ \mathrm {for} \qquad&\ p,q \in [2,\infty ), \qquad \qquad \frac{\mathrm{d}}{p} + \frac{2}{q} < 1. \end{aligned} \end{aligned}$$
(1.4)

In fact, this condition (with local integrability) was first considered by Krylov and Röckner in [13], where they proved the existence and uniqueness of strong solutions for the SDE

$$\begin{aligned} X_{s,t}(x)= x + \int _{s}^{t} b(r, X_{s,r}(x)) \ \mathrm{d}r + B_{t}-B_{s}, \end{aligned}$$
(1.5)

such that

$$\begin{aligned} \mathbb {P}\left( \int _0^T |b(t,X_t)|^2 \ \mathrm{d}t< \infty \right) = 1. \end{aligned}$$

Similarly, we may consider for convenience the inverse \(Y_{s,t}:=X_{s,t}^{-1}\), which satisfies the following backward stochastic differential equations,

$$\begin{aligned} Y_{s,t}= y - \int _{s}^{t} b(r, Y_{r,t}) \ \mathrm{d}r - (B_{t}-B_{s}), \end{aligned}$$
(1.6)

for \(0\le s\le t\).

The well-posedness of Cauchy problem (1.3) under condition (1.4) for measurable initial condition was also considered in [22] and [23]. In [3], using a technique based on the regularizing effect observed on expected values of moments of the solution, well-posedness of (1.3) was obtained also for the limit cases of \(p,q=\infty \) or when the inequality in (1.4) becomes an equality. The uniqueness result in that paper is valid for solutions in weighted spaces.

We mention that other approaches have also been used to study stochastic linear transport/continuity equations. For example, Maurelli in [20] employed the Wiener chaos decomposition to deal with a weakly differentiable drift, in [21], Mohammed, Nilssen, Proske used Malliavin calculus which allows to deal with just a bounded drift, and in [9] the authors introduced a new class of solutions. We would also like to mention the generalizations to transport diffusion equations and the associated stochastic differential equations by Figalli [10] and Zhang [26].

The main issue of this paper is to prove uniqueness of \(L^{2}\)-weak solutions for one-dimensional stochastic continuity Eq.  (1.3) with unbounded measurable drift without assumptions on the divergence. More precisely, we assume that b satisfies

$$\begin{aligned} |b(x)|\le k (1 +|x|). \end{aligned}$$

The proof is based on the fact that one primitive V is regular and verifies the transport equation

$$\begin{aligned} \partial _t V(t, x) + \left( b(x) + \frac{\mathrm{d} B_{t}}{\mathrm{d}t}\right) \cdot \nabla V(t, x) = 0. \end{aligned}$$
(1.7)

Then using a modified version of the commutator lemma and the characteristic systems associated with SPDE (1.7) we shall show that \(V=0\) with initial condition equal to zero, which implies that \(u=0\).

Other issue in this paper is to give a well-posedness result for solutions in the Sobolev spaces \(H^{1}(\mathbb {R}^{d})\) under condition (1.4) with divergence equal to zero. In particular this result implies the persistence of the regularity for initial conditions \(u_0\in H^{1}(\mathbb {R}^{d})\). The proof is based on the commutator lemma given by Le Bris and Lions in [17] for functions with Sobolev regularity. This new result shows the uniqueness in the class of \(H^{1}\)-solutions, not covered in the previous works (see [3, 8, 22, 23]) under this condition.

Throughout this paper, we fix a stochastic basis with a d-dimensional Brownian motion \(\big ( \Omega , \mathcal {F}, \{\mathcal {F}_t: t \in [0,T] \}, \mathbb {P}, (B_{t}) \big )\).

\(L^{2}\)-solutions

In this section we assume the following hypothesis:

Hypothesis 2.1

The vector field b satisfies

$$\begin{aligned} |b(x)|\le k (1 +|x|), \end{aligned}$$
(2.1)

and the initial condition holds

$$\begin{aligned} u_0 \in L^2(\mathbb R, w\,\mathrm{d}x) \end{aligned}$$
(2.2)

where w is the weight defined by \(w(x)=e^{ 2 k_2 x^2}\) with \(k_2=2(k+99 T k^2)\).

Now, we denote by \(b^{\epsilon }\) the standard mollification of b, and let \(X_t^{\epsilon }\) be the associated flow given by SDE (1.5) replacing b by \(b^{\epsilon }\). Similarly, we consider \(Y^\epsilon _{t}\), which satisfies backward SDE (1.6). We also recall the important results in [24] (see “Appendix ”) : let \(X_{t}^{\epsilon }\) be the corresponding stochastic flows, then for all \(p\ge 1\) there are constants \(C_1=C_1(k,p,T)\) and \(C_2(k,p,T)\) such that

$$\begin{aligned} \mathbb {E}[|\partial _x X_{t}^{\epsilon }(x)|^p]\le C_1 t^{-\frac{1}{2}} e^{C_2 x^{2}}, \end{aligned}$$
(2.3)

the same results are valid for the backward flow \(Y_{t}^{\epsilon }\) since it is the solution of the same SDE driven by the drifts \(-b^{\epsilon }\). We denote \(\mu =(1+ |x|)^{2}\).

Definition of solutions

Definition 2.2

A stochastic process \(u\in L^2(\Omega \times [0,T]\times \mathbb R, \mu \mathrm{d}x) \) is called a \(L^{2}\)- weak solution of Cauchy problem (1.3) when: for any \(\varphi \in C_0^{\infty }(\mathbb R)\), the real-valued process \(\int u(t,x)\varphi (x) \mathrm{d}x\) has a continuous modification which is an \(\mathcal {F}_{t}\)-semimartingale, and for all \(t \in [0,T]\), we have \(\mathbb {P}\)-almost surely

$$\begin{aligned} \begin{aligned} \int _{\mathbb R} u(t,x) \varphi (x) \mathrm{d}x =&\int _{\mathbb R} u_{0}(x) \varphi (x) \ \mathrm{d}x + \int _{0}^{t} \!\! \int _{\mathbb R} u(s,x) b (x) \partial _x \varphi (x) \ \mathrm{d}x \mathrm{d}s \\&+ \int _{0}^{t} \!\! \int _{\mathbb R} u(s,x) \ \partial _x \varphi (x) \ \mathrm{d}x {\circ }{\mathrm{d}B_s}. \end{aligned} \end{aligned}$$
(2.4)

Remark 2.3

Using the same idea as in Lemma 13 [11], one can write problem (1.3) in Itô form as follows, a stochastic process \(u\in L^2(\Omega \times [0,T]\times \mathbb R, \mu \mathrm{d}x) \) is a \( L^{2}\)- weak solution of SPDE (1.3) iff for every test function \(\varphi \in C_{0}^{\infty }(\mathbb {R})\), the process \(\int u(t, x)\varphi (x) \mathrm{d}x\) has a continuous modification which is a \(\mathcal {F}_{t}\)-semimartingale and satisfies the following Itô’s formulation

$$\begin{aligned} \begin{aligned} \int _{\mathbb R} u(t,x) \varphi (x) \mathrm{d}x =&\int _{\mathbb R} u_{0}(x) \varphi (x) \ \mathrm{d}x + \int _{0}^{t} \!\! \int _{\mathbb R} u(s,x) \, b (x) \partial _x \varphi (x) \ \mathrm{d}x \mathrm{d}s \\&+ \int _{0}^{t} \!\! \int _{\mathbb R} u(s,x) \ \partial _x \varphi (x) \ \mathrm{d}x \, \mathrm{d}B_s \, + \frac{1}{2} \int _{0}^{t} \!\! \int _{\mathbb R} u(s,x) \ \partial _x^{2} \varphi (x) \ \mathrm{d}x \, \mathrm{d}s. \end{aligned} \end{aligned}$$

Existence

We shall here prove existence of solutions under hypothesis 2.1.

Lemma 2.4

Assume that hypothesis 2.1 holds. Then there exists \(L^2\)-weak solution of Cauchy problem (1.3).

Proof

Step 1: Regularization

Let \(\{\rho _\varepsilon \}_\varepsilon \) be a family of standard symmetric mollifiers and \(\eta \) a nonnegative smooth cutoff function supported on the ball of radius 2 and such that \(\eta =1\) on the ball of radius 1. Now, for every \(\varepsilon >0\), we introduce the rescaled functions \(\eta _\varepsilon (\cdot ) = \eta (\varepsilon \cdot )\). Thus, we define the family of regularized coefficients given by

$$\begin{aligned} b^{\epsilon }(x) = \eta _\varepsilon (x) ( b *\rho _\varepsilon (x)) \end{aligned}$$

and

$$\begin{aligned} u_0^\varepsilon (x) = \eta _\varepsilon (x) \big ( u_0 *\rho _\varepsilon (x) \big ). \end{aligned}$$

Clearly we observe that, for every \(\varepsilon >0\), any element \(b^{\varepsilon }\), \(u_0^\varepsilon \) are smooth (in space) and have compactly supported with bounded derivatives of all orders. We observe that to study the stochastic continuity equation (SCE) (1.3) is equivalent to study the stochastic transport equation given by (regularized version):

$$\begin{aligned} \left\{ \begin{aligned}&\mathrm{d} u^\varepsilon (t, x) + \nabla u^\varepsilon (t, x) \cdot \big ( b^\varepsilon (x) \mathrm{d}t + \circ d B_{t} \big ) +\mathrm{div}b^{\varepsilon }(x) \,u^\varepsilon (t,x) \mathrm{d}t = 0\, , \\&u^\varepsilon \big |_{t=0}= u_{0}^\varepsilon \end{aligned} \right. \end{aligned}$$
(2.5)

Following the classical theory of H. Kunita [14, Theorem 6.1.9] we obtain that

$$\begin{aligned} u^{\varepsilon }(t,x) = u_{0}^{\varepsilon } (\psi _t^{\varepsilon }(x)) \exp \bigg \{-\int _0^t \mathrm{div}b^{\varepsilon }(\phi _s^{\varepsilon }(\psi _t^{\varepsilon }(x))) \mathrm{d}s \bigg \} \end{aligned}$$

is the unique solution to regularized Eq. (2.5), where \(\phi _t^{\varepsilon }\) is the flow associated to the following stochastic differential equation (SDE):

$$\begin{aligned} \mathrm{d} X_t = b^\varepsilon (X_t) \, \mathrm{d}t + \mathrm{d} B_t, \quad X_0 = x . \end{aligned}$$

and \(\psi _t^{\varepsilon }\) is the inverse of \(\phi _t^{\varepsilon }\).

Step 2: Boundedness Making the change of variables \(y=\psi _t^{\varepsilon }(x)=(\phi _t^{\varepsilon }(x))^{-1}\) we have that

$$\begin{aligned} \int _{\mathbb R} \mathbb E[|u^{\varepsilon }(t,x)|^2]\, (1+ |x|)^{2} \mathrm{d}x&=\int _{\Omega }\int _{\mathbb R} |u_{0}^{\varepsilon } (y)|^2 \exp \bigg \{-2\int _0^t \mathrm{div}b^{\varepsilon }(\phi _s^{\varepsilon }(y)) \mathrm{d}s \bigg \} \\&\quad \times \frac{\mathrm{d} \phi _t^{\varepsilon }(y)}{\mathrm{d}y} \ (1+ |\phi _t^{\varepsilon }(y)|)^{2} \, \mathrm{d}y \mathbb {P}(\mathrm{d}\omega ). \end{aligned}$$

Now, if we do a minor modification in the proof of Lemma 3.6 of [24] (see “Appendix”) we obtain that there are constants \(k_1=k_1(k,T)\) and \(k_2=2(k+99 T k^2)\) such that

$$\begin{aligned} \mathbb {E}\bigg [\bigg |\frac{\mathrm{d}}{\mathrm{d}x}\phi ^{\varepsilon }_t(x)\bigg |^{-2}\bigg ] = \mathbb {E}\bigg [ \exp \bigg \{-2\int _0^t \mathrm{div}b^{\varepsilon }(\phi _s^{\varepsilon }(x)) \mathrm{d}s \bigg \} \bigg ] \le k_1 t^{-3/8} e^{k_2 x^2}. \end{aligned}$$
(2.6)

We also observe that

$$\begin{aligned} \mathbb {E}\bigg [ | \phi _t^{\varepsilon }(x) |^{4} \bigg ]\le C (|x|^{4} + T^{4}) \end{aligned}$$
(2.7)

Then we obtain

$$\begin{aligned}&\mathbb {E} \bigg [ \bigg |\frac{\mathrm{d}}{\mathrm{d}x}\phi ^{\varepsilon }_t(x)\bigg |^{-1} (1+ |\phi _t^{\varepsilon }(x)|)^{2} \bigg ]\\&\quad \le C \bigg ( \mathbb {E} \bigg |\frac{\mathrm{d}}{\mathrm{d}x}\phi ^{\varepsilon }_t(x)\bigg |^{-2} + \mathbb {E} \bigg | (1+ |\phi _t^{\varepsilon }(x)|)^{4} \bigg | \bigg ) \le C(k_1 t^{-3/8} e^{k_2 x^2} + T^4+x^4). \end{aligned}$$

Thus we deduce

$$\begin{aligned} \int _{\mathbb R}&\mathbb E[|u^{\varepsilon }(t,x)|^2](1+|x|)^{2}\, \mathrm{d}x \le \int _{\mathbb R} |u_{0}^{\varepsilon } (y)|^2 \mathbb E\bigg [\bigg |\frac{\mathrm{d} \phi _s^{\varepsilon }(y)}{\mathrm{d}y}\bigg |^{-1} (1+ |\phi _t^{\varepsilon }(y)|)^{2} \bigg ]\, \mathrm{d}y \nonumber \\&\le C \int _{\mathbb R} |u_{0}^{\varepsilon } (y)|^2 \big ( k_1 t^{-3/8} e^{k_2 x^2} + T^4+y^4 \big ) \, \mathrm{d}y \nonumber \\&\le Ck_1 t^{-3/8} \int _{\mathbb R} |u_{0}^{\varepsilon } (y)|^2 e^{k_2 y^2} \ \mathrm{d}y + C\int _{\mathbb R} |u_{0}^{\varepsilon } (y)|^2 e^{k_2 y^2} \, \mathrm{d}y . \end{aligned}$$
(2.8)

We observe that

$$\begin{aligned} \int _{\mathbb R} |u_{0}^{\varepsilon } (y)|^2 e^{k_2 y^2} \mathrm{d}y&\le \int _{\mathbb R} \bigg [ e^{k_2 y^2} \bigg (\int _{\mathbb R}\rho _{\varepsilon }(y-x)|u_0(x)|^2\mathrm{d}x\bigg )\bigg ] \mathrm{d}y \nonumber \\&= \int _{\mathbb R} \bigg [ |u_0(x)|^2 \bigg (\int _{B(x,\varepsilon )}\rho _{\varepsilon }(y-x)e^{k_2 y^2}\mathrm{d}y\bigg )\bigg ] \mathrm{d}x \nonumber \\&= \int _{\mathbb R} \bigg [ |u_0(x)|^2 \bigg (\int _{B(0,\varepsilon )}\rho _{\varepsilon }(u)e^{k_2 (x+u)^2}\mathrm{d}u\bigg )\bigg ] \mathrm{d}x \nonumber \\&\le \int _{\mathbb R} \bigg [ |u_0(x)|^2 e^{2k_2 x^2}\bigg (\int _{B(0,\varepsilon )}\rho _{\varepsilon }(u)e^{2 k_2u^2}\mathrm{d}u\bigg )\bigg ] \mathrm{d}x \nonumber \\&\le C \Vert u_0\Vert ^2_{L^2(\mathbb R, w\mathrm{d}x)} . \end{aligned}$$
(2.9)

From (2.8) and (2.9) we conclude that

$$\begin{aligned} \Vert u^{\varepsilon }\Vert ^2_{L^2(\Omega \times [0,T]\times \mathbb R, \mu \mathrm{d}x)} \le C(k,T) \Vert u_0\Vert ^2_{L^2(\mathbb R, w\mathrm{d}x)}. \end{aligned}$$

Therefore, the sequence \(\{u^{\varepsilon }\}_{\varepsilon >0}\) is bounded in \(L^2(\Omega \times [0,T]\times \mathbb R, \mu \mathrm{d}x )\). Then there exists a convergent subsequence, which we denote also by \(u^{\varepsilon }\), such that it converges weakly in \(L^2(\Omega \times [0,T]\times \mathbb R, \mu \mathrm{d}x)\) to some process \(u\in L^2(\Omega \times [0,T]\times \mathbb R, \mu \mathrm{d}x)\) .

Step 3: Passing to the Limit Now, if \(u^{\varepsilon }\) is a solution of (2.5), it is also a weak solution, that is, for any test function \(\varphi \in C_0^{\infty }(\mathbb R)\), \(u^{\varepsilon }\) satisfies (written in the Itô form):

$$\begin{aligned} \int _{\mathbb R} u^{\varepsilon }(t,x) \varphi (x) \mathrm{d}x =&\int _{\mathbb R} u^{\varepsilon }_{0}(x) \varphi (x) \ \mathrm{d}x + \int _{0}^{t} \!\! \int _{\mathbb R} u^{\varepsilon }(s,x) \, b^{\varepsilon } (x) \partial _x \varphi (x) \ \mathrm{d}x \mathrm{d}s \\&+ \int _{0}^{t} \!\! \int _{\mathbb R} u^{\varepsilon }(s,x) \ \partial _x \varphi (x) \ \mathrm{d}x \, \mathrm{d}B_s \, + \frac{1}{2} \int _{0}^{t} \!\! \int _{\mathbb R} u^{\varepsilon }(s,x) \ \partial _x^{2} \varphi (x) \ \mathrm{d}x \, \mathrm{d}s. \end{aligned}$$

Thus, for proving the existence of SCE (1.3) is enough to pass to the limit in the above equation along the convergent subsequence found. This is made through the same arguments of [11, Theorem 15]. \(\square \)

Uniqueness

Theorem 2.5

Under the conditions of hypothesis 2.1, uniqueness holds for \(L^{2}\)- weak solutions of Cauchy problem (1.3) in the following sense: if uv are \(L^{2}\)- weak solutions with the same initial data \(u_{0}\in L^2(\mathbb R, w\,\mathrm{d}x)\), then \(u= v\) almost everywhere in \( \Omega \times [0,T] \times \mathbb R\).

Proof

Step 0: Set of solutions Remark that the set of \(L^{2}\)- weak solutions is a linear subspace of \(L^{2}( \Omega \times [0, T]\times \mathbb {R} , \mu \mathrm{d}x)\), because the stochastic continuity equation is linear, and the regularity conditions are a linear constraint. Therefore, it is enough to show that a \(L^{2}\)- weak solution u with initial condition \(u_0= 0\) vanishes identically.

Step 1: Primitive of the solution We define \(V(t,x)=\int _{-\infty }^{x} u(t,y) \ \mathrm{d}y \), and we observe that \(\partial _x V(t,x)\) belong to \(L^{2}( \Omega \times [0, T]\times \mathbb {R},\mu \mathrm{d}x )\). We consider a nonnegative smooth cutoff function \(\eta \) supported on the ball of radius 2 and such that \(\eta =1\) on the ball of radius 1. For any \(R>0\), we introduce the rescaled functions \(\eta _R (\cdot ) = \eta (\frac{.}{R})\). Let \(\varphi \in C_0^{\infty }(\mathbb R)\); we observe that

$$\begin{aligned} \int _{\mathbb R} V(t,x) \varphi (x) \eta _R (x) \mathrm{d}x = - \int _{\mathbb R} u(t,x) \theta (x) \eta _R (x) \mathrm{d}x -\int _{\mathbb R} V(t,x) \theta (x) \partial _x \eta _R (x) \mathrm{d}x\,, \end{aligned}$$

where \(\theta (x) =\int _{-\infty }^{x} \varphi (y) \ \mathrm{d}y\). By the definition of the solution u, taking as test function \( \theta (x) \eta _R (x)\) we have that V(tx) verifies

$$\begin{aligned} \int _{\mathbb R}&V(t,x) \ \eta _R (x) \varphi (x) \mathrm{d}x = - \int _{0}^{t} \!\! \int _{\mathbb R} \partial _x V(s,x) \, b (x) \eta _R (x) \varphi (x) \ \mathrm{d}x \mathrm{d}s \nonumber \\&- \int _{0}^{t} \!\! \int _{\mathbb R} \partial _x V(s,x) \ \eta _R (x) \varphi (x) \ \mathrm{d}x \, {\circ }{\mathrm{d}B_s} - \int _{0}^{t} \!\! \int _{\mathbb R} \partial _x V(s,x) \, b (x) \partial _x \eta _R (x) \theta (x) \ \mathrm{d}x \mathrm{d}s\nonumber \\&- \int _{0}^{t} \!\! \int _{\mathbb R} \partial _x V(s,x) \ \partial _x \eta _R (x) \theta (x) \ \mathrm{d}x \, {\circ }{\mathrm{d}B_s}-\int _{\mathbb R} V(t,x) \theta (x) \partial _x \eta _R (x) \mathrm{d}x. \end{aligned}$$
(2.10)

Taking the limit as \(R\rightarrow \infty \) we obtain

$$\begin{aligned} \begin{aligned}&\int _{\mathbb R} V(t,x) \varphi (x) \mathrm{d}x \\&\quad =- \int _{0}^{t} \!\! \int _{\mathbb R} \partial _x V(s,x) \, b (x) \varphi (x) \ \mathrm{d}x \mathrm{d}s - \int _{0}^{t} \!\! \int _{\mathbb R} \partial _x V(s,x) \ \varphi (x) \ \mathrm{d}x \, {\circ }{\mathrm{d}B_s} . \end{aligned} \end{aligned}$$
(2.11)

Step 2: Smoothing Let \(\{\rho _{\varepsilon }(x)\}_\varepsilon \) be a family of standard symmetric mollifiers. For any \(\varepsilon >0\) and \(x\in \mathbb R^d\) we use \(\rho _\varepsilon (x-\cdot )\) as test function; then we get

$$\begin{aligned} \begin{aligned} \int _{\mathbb R} V(t,y) \rho _\varepsilon (x-y) \, \mathrm{d}y =&\, - \int _{0}^{t} \int _{\mathbb R} \big ( b(y) \partial _y V(s,y) \big ) \rho _\varepsilon (x-y) \ \mathrm{d}y \mathrm{d}s \\&- \int _{0}^{t} \!\! \int _{\mathbb R} \partial _y V(s,y) \, \rho _\varepsilon (x-y) \, \mathrm{d}y \circ \mathrm{d}B_s \end{aligned} \end{aligned}$$

We set \(V_\varepsilon (t,x)= (V*\rho _\varepsilon )(x)\), \(b_\varepsilon (x)= (b *\rho _\varepsilon )(x)\) and \((bV)_\varepsilon (t,x)= (b.V*\rho _\varepsilon )(x)\). Then we deduce

$$\begin{aligned} \begin{aligned}&V_{\varepsilon }(t,x) + \int _{0}^{t} b_{\epsilon }(x) \partial _x V_{\varepsilon }(s,x) \, \mathrm{d}s + \int _{0}^{t} \partial _{x} V_{\varepsilon }(s,x) \, \circ \mathrm{d}B_s \\ {}&\quad =\,\int _{0}^{t} \big (\mathcal {R}_{\epsilon }(V,b) \big ) (x,s) \, \mathrm{d}s , \end{aligned} \end{aligned}$$

where we denote \( \mathcal {R}_{\epsilon }(V,b) = b_\varepsilon \ \partial _x V_\varepsilon - (b\partial _x V)_\varepsilon \).

Step 3: Method of Characteristics Applying the Itô–Wentzell–Kunita formula to \( V_{\varepsilon }(t,X_{t}^{\epsilon })\), see Theorem 8.3 of [15], we have

$$\begin{aligned} V_{\varepsilon }(t,X_{t}^{\epsilon }) = \int _{0}^{t} \big (\mathcal {R}_{\epsilon }(V,b) \big ) (X_s^{\epsilon },s) \mathrm{d}s . \end{aligned}$$

Then, considering that \(X_{t}^{\epsilon }=X_{0,t}^{\epsilon }\) and \(Y_{t}^{\epsilon }=Y_{0,t}^{\epsilon }=(X_{0,t}^{\epsilon })^{-1}\) we have that

$$\begin{aligned} V_{\varepsilon }(t,x) = \int _{0}^{t} \big (\mathcal {R}_{\epsilon }(V,b) \big ) (X_{0,s}^{\epsilon }(Y_{0,t}^{\epsilon }),s) \mathrm{d}s=\int _{0}^{t} \big (\mathcal {R}_{\epsilon }(V,b) \big ) (Y_{s,t}^{\epsilon },s) \mathrm{d}s . \end{aligned}$$

Multiplying by the test functions \(\varphi \) and integrating in \(\mathbb R\) we get

$$\begin{aligned} \int V_{\varepsilon }(t,x) \ \varphi (x) \mathrm{d}x = \int _{0}^{t} \int \big (\mathcal {R}_{\epsilon }(V,b) \big ) (Y_{s,t}^{\epsilon },s) \ \ \varphi (x) \ \, \mathrm{d}x \ \mathrm{d}s . \end{aligned}$$
(2.12)

We observe that

$$\begin{aligned} \int _{0}^{t} \int \big (\mathcal {R}_{\epsilon }(V,b) \big ) (Y_{s,t}^{\epsilon },s) \ \varphi (x) \ \, \mathrm{d}x \ \mathrm{d}s = \int _{0}^{t} \int \big (\mathcal {R}_{\epsilon }(V,b) \big ) (x,s) \ JX_{s,t}^{\epsilon } \varphi (X_{s,t}^{\epsilon }) \ \, \mathrm{d}x \ \mathrm{d}s . \end{aligned}$$
(2.13)

Step 4: Convergence of the commutator Now, we observe that \(\mathcal {R}_{\epsilon }(V,b)\) converge to zero in \(L^{2}([0,T]\times \mathbb R)\). In fact,

$$\begin{aligned} (b \ \partial _x V)_{\varepsilon } \rightarrow b \ \partial _x V \ in \ L^{2}([0,T]\times \mathbb R), \end{aligned}$$

and by the dominated convergence theorem we obtain

$$\begin{aligned} b_{\epsilon } \partial _x V_{\varepsilon } \rightarrow b \ \partial _x V \ in \ L^{2}([0,T]\times \mathbb R). \end{aligned}$$

Step 5: Conclusion From step 3 we obtain

$$\begin{aligned} \int V_{\varepsilon }(t,x) \ \varphi (x) \mathrm{d}x = \int _{0}^{t} \int \big (\mathcal {R}_{\epsilon }(V,b) \big ) (x,s) \ JX_{s,t}^{\epsilon } \varphi (X_{s,t}^{\epsilon }) \ \, \mathrm{d}x \ \mathrm{d}s , \end{aligned}$$
(2.14)

Using Hölder’s inequality we have

$$\begin{aligned}&\mathbb E\bigg |\int _{0}^{t} \int \bigg (\mathcal {R}_{\epsilon }(V,b) \bigg ) (x,s) \ JX_{s,t}^{\epsilon } \varphi (X_{s,t}^{\epsilon }) \ \, \mathrm{d}x \ \mathrm{d}s \bigg |\\&\le \bigg (\mathbb E\int _{0}^{t} \int |\big (\mathcal {R}_{\epsilon }(V,b) \big ) (x,s)|^{2} \ \, \mathrm{d}x \ \mathrm{d}s \bigg )^{\frac{1}{2}} \bigg (\mathbb E\int _{0}^{t} \int | JX_{s,t}^{\epsilon } \varphi (X_{s,t}^{\epsilon })|^{2} \ \, \mathrm{d}x \ \mathrm{d}s \bigg )^{\frac{1}{2}} \end{aligned}$$

From step 4 result

$$\begin{aligned} \bigg (\mathbb E\int _{0}^{t} \int |\big (\mathcal {R}_{\epsilon }(V,b) \big ) (x,s)|^{2} \ \, \mathrm{d}x \ \mathrm{d}s \bigg )^{\frac{1}{2}}\rightarrow 0. \end{aligned}$$

From formula (2.3) we obtain

$$\begin{aligned} \bigg (\mathbb E\int _{0}^{t} \int | JX_{s,t}^{\epsilon } \varphi (X_{s,t}^{\epsilon })|^{2} \ \, \mathrm{d}x \ \mathrm{d}s \bigg )^{\frac{1}{2}} \le C \bigg (\mathbb E\int _{0}^{t} |\varphi (x)|^{2} \ \, \mathrm{d}x \ \mathrm{d}s \bigg )^{\frac{1}{2}}, \end{aligned}$$

Passing to the limit in equation (2.14) we conclude that \(V=0\). Then we deduce that \(u=0\). \(\square \)

\(H^{1}(\mathbb {R}^{d})\) solutions

We will be considered the divergence-free condition, that is

$$\begin{aligned} \mathrm{div}\, b= 0 \end{aligned}$$
(3.1)

(understood in the sense of distributions).

Definition 3.1

A stochastic process \(u\in L^{2}( \Omega \times [0, T], H^{1}(\mathbb {R}^{d}) ) \cap L^{\infty }( \Omega \times [0, T] \times \mathbb {R}^{d} )\) is called a \( H^{1}\)- weak solution of Cauchy problem (1.3) when: for any \(\varphi \in C_0^{\infty }(\mathbb R^d)\), the real-valued process \(\int u(t,x)\varphi (x) \mathrm{d}x\) has a continuous modification which is an \(\mathcal {F}_{t}\)-semimartingale, and for all \(t \in [0,T]\), we have \(\mathbb {P}\)-almost surely

$$\begin{aligned} \begin{aligned} \int _{\mathbb R^d} u(t,x) \varphi (x) \mathrm{d}x =&\int _{\mathbb R^d} u_{0}(x) \varphi (x) \ \mathrm{d}x -\int _{0}^{t} \!\! \int _{\mathbb R^d} \partial _i u(s,x) \cdot \, b^{i} (s,x) \varphi (x) \ \mathrm{d}x \mathrm{d}s \\&- \int _{0}^{t} \!\! \int _{\mathbb R^d} \partial _{i} u(s,x) \ \varphi (x) \ \mathrm{d}x \, {\circ }{\mathrm{d}B^i_s}. \end{aligned} \end{aligned}$$
(3.2)

Remark 3.2

Analogously, as it was done in remark 2.3 we can write problem (1.3) in the Itô form as follows, a stochastic process \(u \in L^{2}( \Omega \times [0, T], H^{1}(\mathbb {R}^{d}) ) \cap L^{\infty }( \Omega \times [0, T] \times \mathbb {R}^{d} )\) is a \( H^{1}\)- weak solution of SPDE (1.3) iff for every test function \(\varphi \in C_{0}^{\infty }(\mathbb {R}^{d})\), the process \(\int u(t, x)\varphi (x) \mathrm{d}x\) has a continuous modification which is a \(\mathcal {F}_{t}\)-semimartingale and satisfies the following Itô’s formulation

$$\begin{aligned} \int u(t,x) \varphi (x) \mathrm{d}x =&\int u_{0}(x) \varphi (x) \ \mathrm{d}x \\&-\int _{0}^{t} \int b^{i}(s,x)\cdot \varphi (x) \partial _i u(s,x) \ \mathrm{d}x\, \mathrm{d}s \\&- \int _{0}^{t} \int \varphi (x) \partial _{i} u(s,x) \ \mathrm{d}x \ \mathrm{d}B_{s}^{i} \\&+\frac{1}{2} \int _{0}^{t} \int \Delta \,\varphi (x) u(s,x) \ \mathrm{d}x\, \mathrm{d}s. \end{aligned}$$

Existence

Lemma 3.3

We assume that \(u_0\in H^{1}(\mathbb R^{d}) \cap L^{\infty }(\mathbb R^{d})\) and conditions (1.4) and (3.1). Then there exists \(H^{1}\)- weak solution u of Cauchy problem (1.3).

Proof

Let \(\{\rho _\varepsilon \}_\varepsilon \) be a family of standard symmetric mollifiers. Consider a nonnegative smooth cutoff function \(\eta \) supported on the ball of radius 2 and such that \(\eta =1\) on the ball of radius 1. For every \(\varepsilon >0\) introduce the rescaled functions \(\eta _\varepsilon (\cdot ) = \eta (\varepsilon \cdot )\). Using these two families of functions we define the family of regularized coefficient as \(b^{\epsilon }(t,x) = \eta _\varepsilon (x) \big ( [ b(t,\cdot ) *\rho _\varepsilon (\cdot ) ] (x) \big ) \). Similarly, we define the family of regular approximations of the initial condition \(u_0^\varepsilon (x) = \eta _\varepsilon (x) \big ( [ u_0(\cdot ) *\rho _\varepsilon (\cdot ) ] (x) \big ) \).

Remark that any element \(b^{\varepsilon }\), \(u_0^\varepsilon \), \(\varepsilon >0\) of the two families we have defined is smooth (in space) and compactly supported, therefore with bounded derivatives of all orders. Then, for any fixed \(\varepsilon >0\), the classical theory of Kunita, see [14] or [16], provides the existence of an unique solution \(u^{\varepsilon }\) to regularized equation

$$\begin{aligned} \left\{ \begin{aligned}&\mathrm{d} u^\varepsilon (t, x,\omega ) + \nabla u^\varepsilon (t, x,\omega ) \cdot \big ( b^\varepsilon (t, x) \mathrm{d}t + \circ \mathrm{d} B_{t}(\omega ) \big ) = 0, \\&u^\varepsilon \big |_{t=0}= u_{0}^\varepsilon \end{aligned} \right. \end{aligned}$$
(3.3)

together with the representation formula

$$\begin{aligned} u^\varepsilon (t,x) = u_0^\varepsilon \big ( (\phi _t^\varepsilon )^{-1} (x) \big ) \end{aligned}$$
(3.4)

in terms of the (regularized) initial condition and the inverse flow \((\phi _t^\varepsilon )^{-1}\) associated with the equation of characteristics of (3.3), which reads

$$\begin{aligned} \mathrm{d} X_t = b^\varepsilon (t, X_t) \, \mathrm{d}t + \mathrm{d} B_t, \quad X_0 = x . \end{aligned}$$

Now, by Lemma 5 of [8] we have that for every \(p \ge 1\), there exists \(C_{d,p,T}> 0\) such that

$$\begin{aligned} \sup _{t \in [0,T]} \sup _{x \in \mathbb R^d} \mathbb {E}[| D(\phi _{t}^\varepsilon ) |^p] \le C_{d,p,T}, \quad \text {uniformly in}\,\epsilon > 0. \end{aligned}$$
(3.5)

Then, we can use the random change of variables \((\phi _t^\varepsilon )^{-1}(x) \mapsto x\) to obtain that

$$\begin{aligned} \int _{\mathbb R^d} \mathbb E\big | u^{\epsilon }(t,x) \big |^{2} \mathrm{d}x&= \int _{\mathbb R^d} \mathbb E\big | u_{0}^{\epsilon }\big ( (\phi _{t}^\varepsilon )^{-1} (x,\omega )\big ) \big |^{2} \mathrm{d}x = \int _{\mathbb R^d} \big |u_{0}^{\epsilon }(x) \big |^2 \, \mathrm{d}x . \end{aligned}$$
(3.6)

Moreover, we have

$$\begin{aligned} \int _{\mathbb R^d} \mathbb E\big | \nabla u^{\epsilon }(t,x) \big |^{2} \mathrm{d}x =&\int _{\mathbb R^d} \mathbb E\big | \nabla [ u_{0}^{\epsilon }\big ( (\phi _{t}^\varepsilon )^{-1} (x,\omega )\big ) ] \big |^{2} \mathrm{d}x \nonumber \\ =&\int _{\mathbb R^d} \big |\nabla u_{0}^{\epsilon }((\phi _{t}^\varepsilon )^{-1}) D(\phi _{t}^\varepsilon )^{-1} (x,\omega ) \big |^2 \mathrm{d}x \nonumber \\ =&\int _{\mathbb R^d} \big |\nabla u_{0}^{\epsilon }(x) \big |^2 \mathbb E|D(\phi _{t}^\varepsilon )^{-1} (\phi _{t}^\varepsilon ,\omega )|^{2} \, \mathrm{d}x. \end{aligned}$$
(3.7)

We observe that

$$\begin{aligned} D(\phi _{t}^\varepsilon )^{-1} (\phi _{t}^\varepsilon ,\omega )= D^{-1}(\phi _{t}^\varepsilon ), \end{aligned}$$

and

$$\begin{aligned} D^{-1}(\phi _{t}^\varepsilon )= Cof (D \phi _{t}^\varepsilon )^{T} \end{aligned}$$

where Cof denoted the cofactor matrix of \(D\phi _{t}^\varepsilon \). By inequality (3.5) we deduce that \( Cof (D\phi _{t}^\varepsilon )^{T} \in L^\infty \big ( \mathbb R^d\times [0,T], L^{2} (\Omega ) \big ) \). Thus we obtain that

$$\begin{aligned} \int _{\mathbb R^d} \mathbb E\big | \nabla u^{\epsilon }(t,x) \big |^{2} \mathrm{d}x&= \int _{\mathbb R^d} \left| \nabla u_{0}^{\epsilon }(x) \right| ^2 \mathbb E| |D(\phi _{t}^\varepsilon )^{-1} (\phi _{t}^\varepsilon ,\omega )|^{2}| \, \mathrm{d}x \nonumber \\&\le C \int _{\mathbb R^d} \big |\nabla u_{0}^{\epsilon }(x) \big |^2 \, \mathrm{d}x. \end{aligned}$$
(3.8)

If \(u^\varepsilon \) is a solution of (3.3), it is also a weak solution, which means that for any test function \(\varphi \in C_0^\infty (\mathbb R^d)\), \(u^\varepsilon \) satisfies the following equation (written in Itô form)

$$\begin{aligned} \begin{aligned} \int _{\mathbb R^d} u^\varepsilon (t,x)&\varphi (x) \, \mathrm{d}x= \int _{\mathbb R^d} u^\varepsilon _{0}(x) \varphi (x) \, \mathrm{d}x -\int _{0}^{t} \!\! \int _{\mathbb R^d} \partial _i u^\varepsilon (s,x) \, b^{i,\varepsilon }(s,x) \varphi (x) \, \mathrm{d}x \mathrm{d}s \\&- \int _{0}^{t} \!\! \int _{\mathbb R^d} \partial _{i} u^\varepsilon (s,x) \, \varphi (x) \, \mathrm{d}x \, \mathrm{d}B^i_s + \frac{1}{2} \int _{0}^{t} \!\!\int _{\mathbb R^d} u^\varepsilon (s,x) \Delta \varphi (x) \, \mathrm{d}x \mathrm{d}s. \end{aligned} \end{aligned}$$
(3.9)

To prove the existence of weak solutions to (1.3) we can extract subsequence (for simplicity the whole sequence), which converges weakly to some u in \(L^{2}( \Omega \times [0, T], H^{1}(\mathbb {R}^{d}) )\) and weak star in \( L^{\infty }( \Omega \times [0, T] \times \mathbb {R}^{d} )\). Then via classical arguments, we can pass to the limit in the above equation along this subsequence. This is done following the classical argument of [25, Sect. II, Chap. 3], [11, Theorem 15] and [3, Theorem 23]. \(\square \)

Uniqueness

Before starting and proving the main theorem of this subsection, we shall introduce some further notation and the key lemma on commutators.

Let \(\{\rho _{\varepsilon } \}\) be a family of standard positive symmetric mollifiers. Given two functions \(f:\mathbb R^d \mapsto \mathbb R^d\) and \(g:\mathbb R^d \mapsto \mathbb R\), the commutator \(\mathcal {R}_\varepsilon (f,g)\) is defined as

$$\begin{aligned} \mathcal {R}_{\varepsilon }(f,g):= (f \cdot \nabla ) (\rho _{\epsilon }*g )- \rho _{\varepsilon }*(f\cdot \nabla g ). \end{aligned}$$
(3.10)

The following lemma is due to Le Bris and Lions [17, Lemma 5.1.].

Lemma 3.4

(C. Le Bris - P. L.Lions ) Let \(f \in L^2_\text {loc}(\mathbb R^d ) \), \(g \in H^{1}(\mathbb R^d) \). Then, passing to the limit as \(\varepsilon \rightarrow 0\)

$$\begin{aligned} \mathcal {R}_{\varepsilon }(f,g) \rightarrow 0 \qquad in \qquad L^1_\text {loc}(\mathbb R^d). \end{aligned}$$

We can finally state our uniqueness result.

Theorem 3.5

Under conditions ( 1.4) and (3.1), uniqueness holds for \(H^{1}\)- weak solutions of Cauchy problem (1.3) in the following sense: if uv are \(H^{1}\)- weak solutions with the same initial data \(u_{0}\in H^{1}(\mathbb {R}^{d}) \cap L^{\infty }( \mathbb {R}^{d} )\), then \(u= v\) almost everywhere in \( \Omega \times [0,T] \times \mathbb R^d \).

Proof

The proof is essentially based on energy-type estimates on u. However, to rigorously obtain it two preliminary technical steps of regularization and localization are needed, where above Lemma 3.4 will be used to deal with the commutators appearing in the regularization process.

Step 0: Set of solutions Remark that the set of \(H^{1}\)- weak solutions is a linear subspace of \(L^{2}( \Omega \times [0, T], H^{1}(\mathbb {R}^{d}) )\cap L^{\infty }( \Omega \times [0, T] \times \mathbb {R}^{d} )\), because the stochastic transport equation is linear, and the regularity conditions are a linear constraint. Therefore, it is enough to show that a \(H^{1}\)- weak solution u with initial condition \(u_0= 0\) vanishes identically.

Step 1: Smoothing Let \(\{\rho _{\varepsilon }(x)\}_\varepsilon \) be a family of standard symmetric mollifiers. For any \(\varepsilon >0\) and \(x\in \mathbb R^d\) we can use \(\rho _\varepsilon (x-\cdot )\) as test function, we get

$$\begin{aligned} \begin{aligned} \int _{\mathbb R^d} u(t,y) \rho _\varepsilon (x-y) \, \mathrm{d}y =&\, - \int _{0}^{t} \int _{\mathbb R^d} \big ( b^{i}(s,y) \cdot \partial _i u(s,y) \big ) \rho _\varepsilon (x-y) \ \mathrm{d}y \mathrm{d}s \\&- \frac{1}{2}\int _{0}^{t} \int _{\mathbb R^d} \! \partial _i u(s,y) \, \partial _i \, \rho _\varepsilon (x-y) \, \mathrm{d}y \mathrm{d}s \\&+ \int _{0}^{t} \!\! \int _{\mathbb R^d} u(s,y) \, \partial _i \rho _\varepsilon (x-y) \, \mathrm{d}y \, \mathrm{d}B^i_s. \end{aligned} \end{aligned}$$

Set \(u_\varepsilon (t,x)= u(t,x) *_x \rho _\varepsilon (x)\). Using definition (3.10) of the commutator \(\big (\mathcal {R}_{\epsilon }(f,g)\big ) (s)\) with \(f=b(s, \cdot )\) and \(g=u(s, \cdot )\), we have for each \(t \in [0,T]\)

$$\begin{aligned} \begin{aligned} u_{\varepsilon }(t,x) + \int _{0}^{t} b^{i}(s,x) \cdot \partial _i u_{\varepsilon }(s,x) \, \mathrm{d}s&- \frac{1}{2}\int _{0}^{t} \Delta u_{\varepsilon }(s,x) \, \mathrm{d}s \\&+ \int _{0}^{t} \partial _{i} u_{\varepsilon }(s,x) \, \mathrm{d}B^i_s \\&= \int _{0}^{t} \big (\mathcal {R}_{\epsilon }(b,u) \big ) (s) \, \mathrm{d}s. \end{aligned} \end{aligned}$$

By Itô formula we have

$$\begin{aligned} u_{\epsilon }^2(t,x) =&-\int _0^t b^{i}(s,x)\cdot \partial _i u^2_{\epsilon }(s,x) \mathrm{d}s -\int _{0}^{t} \int _{\mathbb R^d}\partial _{i} u_{\epsilon }^2(s,x) \ \mathrm{d}x \, {\mathrm{d}B^i_s} \nonumber \\&\, +\frac{1}{2}\int _0^t \triangle u^2_{\epsilon }(s,x) \mathrm{d}x + \int _0^t 2u_{\epsilon }(s,x)\mathcal {R}_{\epsilon } (b,u)(s,x) \mathrm{d}s \end{aligned}$$
(3.11)

Step 2: Localization Consider a nonnegative smooth cutoff function \(\eta \) supported on the ball of radius 2 and such that \(\eta =1\) on the ball of radius 1. For any \(R>0\) introduce the rescaled functions \(\eta _R (\cdot ) = \eta (\frac{.}{R})\). Multiplying (3.11) by \(\eta _R\) and integrating over \( \mathbb R^d \) we have

$$\begin{aligned} \int _{\mathbb R^d} u_{\epsilon }^2(t,x)\eta _R(x) \mathrm{d}x =&-\int _0^t\int _{\mathbb R^d} b^{i}(s,x)\cdot \partial _i u_{\epsilon }^2(s,x) \eta _R(x) \mathrm{d}x \mathrm{d}s \\&- \int _{0}^{t} \int _{\mathbb R^d} \partial _{i} u_{\epsilon }^2(s,x)\eta _R(x) \ \mathrm{d}x \,{\mathrm{d}B^i_s} \nonumber \\&+\frac{1}{2}\int _0^t\int _{\mathbb R^d} \triangle u^2_{\epsilon }(s,x) \eta _R(x)\mathrm{d}x \,\mathrm{d}s \\&+ \int _0^t \int _{\mathbb R^d} 2u_{\epsilon }(s,x)\mathcal {R}_{\epsilon } (b,u)(s,x) \eta _R(x) \mathrm{d}s, \end{aligned}$$

which we rewrite as

$$\begin{aligned} \int _{\mathbb R^d} u_{\epsilon }^2(t,x)\eta _R(x) \mathrm{d}x =&\int _0^t\int _{\mathbb R^d} u_{\epsilon }^2(s,x) b^{i}(s,x)\cdot \partial _i \eta _R(x) \mathrm{d}x \mathrm{d}s \nonumber \\&+ \int _{0}^{t} \bigg (\int _{\mathbb R^d} u_{\epsilon }^2(s,x) \partial _{i}\eta _R(x) \ \mathrm{d}x \bigg )\,{\mathrm{d}B^i_s} \nonumber \\&+\frac{1}{2}\int _0^t\int _{\mathbb R^d} u^2_{\epsilon }(s,x) \triangle \eta _R(x)\mathrm{d}x \,\mathrm{d}s\nonumber \\&+ \int _0^t \int _{\mathbb R^d} 2u_{\epsilon }(s,x)\mathcal {R}_{\epsilon } (b,u)(s,x) \eta _R(x) \mathrm{d}s. \end{aligned}$$
(3.12)

Step 4: Passage to the limit Finally, in this step we shall pass to the limit in \(\varepsilon \) and R to obtain uniqueness. We first take the limit \(\varepsilon \rightarrow 0\) in above equation (3.12). By standard properties of mollifiers \(u_\varepsilon \rightarrow u\) strongly in \(L^{2}\big ([0,T]; H^1(\mathbb R^d)\big ) \). Then using Lemma 3.4, we obtain

$$\begin{aligned} \int _{\mathbb R^d} u^2(t,x)\eta _R(x) \mathrm{d}x =&\int _0^t\int _{\mathbb R^d} u^2(s,x) b^{i}(s,x)\cdot \partial _i\eta _R(x) \mathrm{d}x \mathrm{d}s \nonumber \\&+ \int _{0}^{t} \bigg (\int _{\mathbb R^d} u^2(s,x) \partial _{i}\eta _R(x) \ \mathrm{d}x \bigg )\,{\mathrm{d}B^i_s} \nonumber \\&+\frac{1}{2}\int _0^t\int _{\mathbb R^d} u^2(s,x) \triangle \eta _R(x)\mathrm{d}x \,\mathrm{d}s. \end{aligned}$$
(3.13)

Now, we observe that

$$\begin{aligned} \bigg |\int _{0}^{t} \int _{\mathbb R^d} u^{2}(s,x) b^{i}(s,x) \cdot \partial _i \eta _R(x) \, \mathrm{d}x \mathrm{d}s \bigg | \le C \bigg |\int _{0}^{t} \bigg (\int _{ R<|x|< 2R} |b(s,x) |^{p} \, \mathrm{d}x \bigg )^{q/p} \mathrm{d}s\bigg |^{1/q}. \end{aligned}$$

Thus, taking the limit in (3.13) as \(R\rightarrow \infty \) we have

$$\begin{aligned} \begin{aligned} \int _{\mathbb R^d} u^{2}(t,x) \, \mathrm{d}x = 0 \end{aligned} \end{aligned}$$

Taking expectation and integrating on [0, T] we have

$$\begin{aligned} \begin{aligned} \int _{\Omega } \int _0^T \int _{\mathbb R^d} u^{2}(t,x) \, \mathrm{d}x \ \mathrm{d}t \ \mathbb {P}(\mathrm{d}\omega )=0. \end{aligned} \end{aligned}$$

Therefore, we conclude that \(u=0\) almost everywhere on \(\Omega \times [0,T] \times \mathbb R^d\). \(\square \)

References

  1. 1.

    Ambrosio, L.: Transport equation and Cauchy problem for \(BV\) vector fields. Invent. Math. 158, 227–260 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Ambrosio, L., Crippa, G.: Continuity equations and ODE flows with non-smooth velocity, Lecture Notes of a course given at Heriot Watt University, Edinburgh. In: Proceeding of the Royal Society of Edinburgh, Section A: Mathematics, vol. 144, pp 1191–1244 (2014)

  3. 3.

    Beck, L., Flandoli, F., Gubinelli, M., Maurelli, M.: Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness, PP 1401–1530 (2014)

  4. 4.

    Bouchut, F., Desvillettes, L.: On two-dimensional Hamiltonian transport equations with continuous coefficients. Diff. Integral Equ. 14, 1015–1024 (2001)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, 3 edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325. Springer (2010)

  6. 6.

    De Lellis, C.: Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio, Bourbaki Seminar, Preprint, pp 1–26 (2007)

  7. 7.

    DiPerna, R., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Fedrizzi, E., Flandoli, F.: Noise prevents singularities in linear transport equations. J. Funct. Anal. 264, 1329–1354 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Fedrizzi, E., Neves, W., Olivera, C.: On a class of stochastic transport equations for \(L_{loc}^{2}\) vector fields, to appears in the Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. arXiv:1410.6631v2 (2014)

  10. 10.

    Figalli, A.: Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254, 109–153 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Flandoli, F., Gubinelli, M., Priola, E.: Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180, 1–53 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Hauray, M.: On two-dimensional Hamiltonian transport equations with \(L_{loc}^p\) coefficients. Ann. I. H. Poincaré (C) 20, 625–644 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Krylov, N.V., Röckner, M.: Strong solutions to stochastic equations with singular time dependent drift. Probab. Theory Relat. Fields 131, 154–196 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  15. 15.

    Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. Lectures Notes in Mathematics, vol. 1097, pp. 143–303. Springer, Berlin (1982)

  16. 16.

    Kunita, H.: First order stochastic partial differential equations. In: Stochastic Analysis, Katata Kyoto, North-Holland Math. Library, vol. 32, pp. 249–269 (1984)

  17. 17.

    Le Bris, C., Lions, P.L.: Existence and uniqueness of solutions to Fokker–Planck type equations with irregular coefficients. Comm. Partial Diff. Equ. 33, 1272–1317 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Lions, P.L.: Mathematical topics in fluid mechanics: incompressible models. Oxford Lecture Series in Mathematics and its applications, vol. 1, 3rd edn. Oxford University Press, London (1996)

    Google Scholar 

  19. 19.

    Lions, P.L.: Mathematical Topics in Fluid Mechanics, Vol. II: Compressible Models. Oxford Lecture Series in Mathematics and Its Applications, vol. 10. Oxford University Press, London (1996)

    Google Scholar 

  20. 20.

    Maurelli, M.: Wiener chaos and uniqueness for stochastic transport equation. Comptes Rendus Mathématique 349, 669–672 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Mohammed, S.A., Nilssen, T.K., Proske, F.N.: Sobolev differentiable stochastic flows for SDE’s with singular coefficients: applications to the transport equation. Annals Prob. 43, 1535–1576 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Neves, W., Olivera, C.: Wellposedness for stochastic continuity equations with Ladyzhenskaya–Prodi–Serrin condition. Nonlinear Diff. Equ. Appl. 22, 1247–1258 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Neves, W., Olivera, C.: Stochastic continuity equations-a general uniqueness result. Braz. Math. Soc. (N.S.) 47, 631–639 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Nilssen, T.: Regularity of strong solutions of one-dimensional SDE’s with discontinuous and unbounded drift. Stoch. Int. J. Prob. Stoch. Processes 88(5), 779–802 (2016)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Pardoux, E.: Equations aux dérivées partielles stochastiques non linéaires monotones. Etude de solutions fortes de type Itô. PhD Thesis, Universite Paris Sud(1975)

  26. 26.

    Zhang, T.: Stochastic flows of SDEs with irregular coefficients and stochastic transport equations. Bull. Sci. Math. 134, 340–378 (2010)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

Christian Olivera is partially supported by CNPq through the Grant 460713/2014-0 and FAPESP by the Grants 2015/04723-2 and 2015/07278-0.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Christian Olivera.

Appendix

Appendix

Lemma 3.6

Assume \(b\in C_c^{\infty }(\mathbb R)\) and that satisfies hypothesis 2.1. Then for \(T>0\) there are constants \(k_1=k_1(k,T)\) and \(k_2=k_2(k,T)\) such that

$$\begin{aligned} \mathbb {E}\bigg [\bigg |\frac{\mathrm{d}}{\mathrm{d}x}X_t(x)\bigg |^{-2}\bigg ] \le k_1 t^{-3/8} e^{k_2 x^2} \,, \end{aligned}$$
(3.14)

where \(k_1= \sqrt{c_1}\root 4 \of {c_2}e^{35 T k^2} \) and \(k_2=2(k+99 T k^2)\) (\(c_1\) and \(c_2\) are defined below in the proof).

Proof

We consider the SDE associated with the vector field b :

$$\begin{aligned} \mathrm{d} X_t = b (X_t) \, \mathrm{d}t + \mathrm{d} B_t, X_0 = x . \end{aligned}$$

We denote

$$\begin{aligned} \mathcal {E}\bigg (\int _0^t b(X_u)\mathrm{d}B_u\bigg )=\exp \bigg \{\int _0^t b(X_u)\mathrm{d}B_u-\frac{1}{2} \int _0^{t} b^2(X_u)\mathrm{d}u \bigg \}, \end{aligned}$$

and

$$\begin{aligned} d Q(\omega )= \mathcal {E}\bigg (\int _0^t b(X_u)\mathrm{d}B_u\bigg ) d\mathbb {P}(\omega ). \end{aligned}$$

Using Girsanov’s theorem we obtain that

$$\begin{aligned} \mathbb E\bigg [\bigg |\frac{\mathrm{d} X_t}{\mathrm{d}x}(x)\bigg |^{-2}\bigg ]&= \mathbb E_{Q}\bigg [\bigg |\frac{d Y_t}{\mathrm{d}x}(x)\bigg |^{-2}\bigg ] \\&= \mathbb E\bigg [\exp \bigg \{-2\int _0^t b'(x+B_s) \mathrm{d}s \bigg \} \mathcal {E}\bigg (\int _0^t b(x+B_s)\mathrm{d}B_s\bigg )\bigg ]. \end{aligned}$$

Now, we proceed as in the proof of Lemma 3.6 of [24]. Let \( b_1=-b\), then we have

$$\begin{aligned} \mathbb E\bigg [\bigg |\frac{\mathrm{d} X_t}{\mathrm{d}x}(x)\bigg |^{-2}\bigg ] = \mathbb E\bigg [\exp \bigg \{2\int _0^t b_1'(x+B_s) \mathrm{d}s \bigg \} \mathcal {E}\bigg (\int _0^t b(x+B_s)\mathrm{d}B_s\bigg )\bigg ]. \end{aligned}$$

Applying the Itô’s formula to \(\tilde{b}(z)=\int _{\infty }^z b_1(y)\mathrm{d}y\) we get

$$\begin{aligned} \tilde{b}(x+B_t)=\tilde{b}(x)+\int _0^t b_1(x+B_s) \mathrm{d}B_s+\frac{1}{2} \int _0^t b_1'(x+B_s) \mathrm{d}s. \end{aligned}$$

By Hölder inequality we get

$$\begin{aligned}&\mathbb E\bigg [\bigg |\frac{\mathrm{d} X_t}{\mathrm{d}x}(x)\bigg |^{-2}\bigg ] \nonumber \\&\quad = \mathbb E\bigg [\exp \bigg \{4(\tilde{b}(x+B_t) -\tilde{b}(x) -\int _0^t b_1(x+B_s) \mathrm{d}B_s )\bigg \} \mathcal {E}\bigg (\int _0^t b(x+B_s)\mathrm{d}B_s\bigg )\bigg ] \nonumber \\&\quad \le \Vert \exp \{4(\tilde{b}(x+B_t) -\tilde{b}(x) )\} \Vert _{L^2(\Omega )} \ \Vert \exp \bigg \{-4\int _0^t b_1(x+B_s) \mathrm{d}B_s \bigg \}\nonumber \\&\quad \quad \times \mathcal {E}\bigg (\int _0^t b(x+B_s)\mathrm{d}B_s\bigg ) \Vert _{L^2(\Omega )}. \end{aligned}$$
(3.15)

For the first term, we have

$$\begin{aligned} |\tilde{b}(x+B_t) -\tilde{b}(x)|&=|\int _0^1 b_1(x+\theta (B_t)) \mathrm{d}\theta | \ |B_t| \\&\le \int _0^1 (k+k|x+\theta B_t|) \mathrm{d} \theta |B_t| \\&\le k |B_t|+k|x||B_t|+\frac{k}{2}(B_t)^2 \\&\le \frac{k}{2}x^2+k |B_t|+k(B_t)^2 . \end{aligned}$$

Thus we obtain

$$\begin{aligned} \mathbb E[\exp \{8(\tilde{b}(x+B_t) -\tilde{b}(x) )\}]&\le \mathbb E[\exp \{8(\frac{k}{2}x^2+k |B_t|+k(B_t)^2)\}] \\&= e^{4kx^2} \mathbb E\left[ \exp \{8(k |B_t|+k(B_t)^2)\}\right] \\&= e^{4kx^2} \frac{1}{\sqrt{2\pi t}}\int _{\mathbb R} \ \exp \left\{ 8k(|z|+z^2)-\frac{z^2}{2 t}\right\} \ \mathrm{d}z. \end{aligned}$$

Then , we conclude that

$$\begin{aligned} \Vert \exp \{4(\tilde{b}(x+B_t) -\tilde{b}(x) )\} \Vert _{L^2(\Omega )} \le e^{2kx^2} t^{-1/4} \sqrt{c_1}, \end{aligned}$$
(3.16)

where

$$\begin{aligned} c_1=\frac{1}{\sqrt{2\pi }}\int _{\mathbb R} \exp \left\{ 8k(|z|+z^2)-\frac{z^2}{2 T}\right\} \mathrm{d}z . \end{aligned}$$

For the second term of (3.15) we have

$$\begin{aligned}&\mathbb E\bigg [\exp \bigg \{-8\int _0^t b_1(x+B_s) \mathrm{d}B_s \bigg \} \mathcal {E}\bigg (\int _0^t b(x+B_s)\mathrm{d}B_s\bigg )^2\bigg ] \nonumber \\&\mathbb E\bigg [\exp \bigg \{-8\int _0^t b_1(x+B_s) \mathrm{d}B_s \bigg \} \exp \bigg \{2\int _0^t b(x+B_s)\mathrm{d}B_s-\int _0^t b^2(x+B_s)\mathrm{d}s\bigg \}\bigg ] \nonumber \\&\quad =\mathbb E\bigg [\exp \bigg \{-10\int _0^t b_1(x+B_s) \mathrm{d}B_s -\int _0^t b^2_1(x+B_s) \mathrm{d}s\bigg \}\bigg ] \nonumber \\&\quad = \mathbb E\bigg [\exp \bigg \{-10\int _0^t b_1(x+B_s) \mathrm{d}B_s -\alpha \int _0^t b^2_1(x+B_s) \mathrm{d}s\bigg \}\exp \bigg \{(\alpha -1)\int _0^t b^2_1(x+B_s) \mathrm{d}s\bigg \}\bigg ] \nonumber \\&\quad \le \Vert \exp \bigg \{-10\int _0^t b_1(x+B_s) \mathrm{d}B_s -\alpha \int _0^t b^2_1(x+B_s) \mathrm{d}s\bigg \}\Vert _{L^2(\Omega )} \times \nonumber \\&\quad \quad \times \Vert \exp \bigg \{(\alpha -1)\int _0^t b^2_1(x+B_s) \mathrm{d}s\bigg \}\Vert _{L^2(\Omega )}. \end{aligned}$$
(3.17)

Now, we choose \(\alpha =100\) because \(\frac{1}{2}(-20 b_1(x+B_s))^2=2\alpha b_1^2(x+B_s)\). Then the process \(\exp \{-20\int _0^t b_1(x+B_s) \mathrm{d}B_s-200\int _0^t b_1^2(x+B_s)\mathrm{d}s\}=\mathcal {E}\bigg (\int _0^t (-20b_1(x+B_s) \mathrm{d}B_s)\bigg )\) is a martingale with expectation equal to one. Then

$$\begin{aligned} \left\| \exp \bigg \{-10\int _0^t b_1(x+B_s) \mathrm{d}B_s -100 \int _0^t b^2_1(x+B_s) \mathrm{d}s\bigg \}\right\| _{L^2(\Omega )}=1 \end{aligned}$$

From (2.1) we obtain that the second term of (3.17) is bounded by

$$\begin{aligned} \mathbb E\bigg [\exp \bigg \{2(\alpha -1)&\int _0^t b^2_1(x+B_s) \mathrm{d}s\bigg \}\bigg ] = \mathbb E\bigg [\exp \bigg \{198\int _0^t b^2_1(x+B_s) \mathrm{d}s\bigg \}\bigg ] \\&\le \mathbb E\bigg [\exp \bigg \{198\int _0^t k^2(1+|x+B_s|)^2 \mathrm{d}s\bigg \}\bigg ] \\&\le \mathbb E\bigg [\exp \bigg \{198t k^2(1+B_t^{*})^2 \bigg \}\bigg ]\,, \end{aligned}$$

where \(B_t^{*}=\displaystyle \sup _{s\le t}|x+B_s|\). We define

$$\begin{aligned} Y_s=\exp \left\{ 99 t k^2(1+|x+B_s|)^2 \right\} . \end{aligned}$$

Then, by Doob’s maximal inequality we have

$$\begin{aligned} \mathbb E\bigg [ \exp \bigg \{198t k^2(1+B_t^{*})^2 \bigg \}\bigg ]&=\mathbb E\bigg [\sup _{s\le t}Y_s^2\bigg ] \\&\le 4 \ \mathbb E[Y_t^2]=\mathbb E[\exp \{198 t k^2(1+|x+B_t|)^2\}] \\&\le 4 \ \mathbb E[\exp \{396 t k^2(1+(x+B_t)^2)\}] \\&\le 4 \ \mathbb E[\exp \{396 t k^2(1+2(x^2+B_t^2))\}] \\&= 4e^{396 t k^2}e^{792 t k^2 x^2}\mathbb E[\exp \{792k^2 t B_t^2\}] \\&= 4e^{396 t k^2}e^{792 t k^2 x^2} \frac{1}{\sqrt{2\pi t}}\int _{\mathbb R} \exp \left\{ 792 t k^2 z^2-\frac{z^2}{2 t}\right\} \mathrm{d}z. \end{aligned}$$

Substituting in (3.17) we get

$$\begin{aligned} \mathbb E\bigg [\exp \bigg \{-8\int _0^t b_1(x+B_s) \mathrm{d}B_s \bigg \} \mathcal {E}\bigg (\int _0^t b(x+B_s)\mathrm{d}B_s\bigg )^2\bigg ] \le e^{198 T k^2}e^{396 T k^2 x^2} t^{-1/4}\sqrt{c_2}\,, \end{aligned}$$
(3.18)

where

$$\begin{aligned} c_2=\frac{4}{\sqrt{2\pi }}\int _{\mathbb R} \exp \left\{ 792 T k^2 z^2-\frac{z^2}{2 T}\right\} \mathrm{d}z. \end{aligned}$$

Therefore, replacing (3.16) and (3.18) in (3.15) we conclude

$$\begin{aligned} \mathbb E\bigg [\bigg |\frac{\mathrm{d} X_t}{\mathrm{d}x}(x)\bigg |^{-2}\bigg ]&\le e^{2kx^2} t^{-1/4} \sqrt{c_1}e^{99 T k^2}e^{198 T k^2 x^2} t^{-1/8}\root 4 \of {c_2} \\&= \sqrt{c_1}\root 4 \of {c_2}e^{99 T k^2} t^{-3/8}e^{2(k+99 T k^2) x^2} \\&= k_1 t^{-3/8} e^{k_2 x^2}, \end{aligned}$$

where \(k_1= \sqrt{c_1}\root 4 \of {c_2}e^{99 T k^2} \) and \(k_2=2(k+99 T k^2)\). This proves (3.14). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mollinedo, D.A.C., Olivera, C. Stochastic continuity equation with nonsmooth velocity. Annali di Matematica 196, 1669–1684 (2017). https://doi.org/10.1007/s10231-017-0633-8

Download citation

Keywords

  • Stochastic partial differential equation
  • Continuity equation
  • Stochastic characteristic method
  • Regularization by noise
  • Commutator lemma

Mathematics Subject Classification

  • 60H15
  • 35R60
  • 35F10
  • 60H30