Local existence for the stochastic Euler equations with Lévy noise

Abstract

We consider the stochastic Euler equations driven by the Lévy noise. We construct a unique local pathwise solution in the Sobolev space $W^{s, p} (R^{d})$ where $s > d / p + 1$ .

Keywords

Stochastic Euler equations Lévy noise existence

1. Introduction

In this paper, we consider the local existence and uniqueness of solutions to the Euler equation forced by a multiplicative Lévy noise, $\begin{array}{l} (1.1) & \begin{matrix} d u (t, x) & = - P ((u (t, x) \cdot \nabla) u (t, x)) d t + σ (u (t, x)) d W (t) \\ + \int_{R^{d} ∖ {0}} G (u (t_{-}, x), z) \tilde{N} (d t, d z), \end{matrix} \\ (1.2) & \nabla \cdot u = 0, (t, x) \in (0, T) \times R^{d}, \\ (1.3) & u (0, x) = u_{0} (x), x \in R^{d}, \end{array}$ on the full Euclidean space $R^{d}$ for $d = 2, 3$ . Here, u is the velocity field of a stochastic flow, $P$ stands for the Leray projection, $W$ represents the cylindrical Wiener process, and $\tilde{N} (d t, d z)$ is a compensated Poisson measure on $[0, T] \times R^{d} / {0}$ (cf. Section 3 for details). Note that in the formulation (1.1) the pressure has been eliminated by utilization of the projector $P$ . The second term $σ (u) d W (t)$ on the right-hand side of (1.1) can be formally written as $\sum_{k ⩾ 1} σ_{k} (u) d W_{k} (t)$ where ${W_{k} : k \in N}$ is a collection of mutually independent 1-D Brownian motions.

The deterministic Euler equations are the classical model for the motion of an inviscid, incompressible, and homogeneous fluid [22], and the noise terms are added to capture the empirical, numerical, experimental, and physical uncertainties. While the Wiener process has been pervasively adopted for random phenomena that evolve continuously in time, Poisson-type noise arises naturally to describe randomness that is impulsive in nature. For example, the shocks that an airplane experiences when it flies through a turbulent air zone (cf. [8]) may be modeled by such noise.

The local existence of solutions for the stochastic Euler equations has been investigated in many works. While [1,2,5–7,13] address the 2D case, the references [14,17], and [18] consider the local well-posedness in dimension three with additive white noise. Finally, the paper [11] provides the local existence theory for the stochastic Euler equations with multiplicative white noise in a bounded domain. The Euler equations with Lévy noise in $H^{s} (R^{d})$ was initially considered in [19] by Mohan and Sritharan, including both additive and multiplicative versions, while Cyr, Tang, and Temam addressed in [8] the problem in $H^{s}$ on a torus and a bounded domain with more natural noise assumptions, by not assuming that the coefficient of large jumps is necessarily Lipschitz or of linear growth. There are also many results on the stochastic Navier-Stokes equations with white noise [12,16] or the version including jumps [3,4,9,10,20,21].

In this paper, we establish the local existence and uniqueness with initial datum in the general Sobolev space $W^{s, p}$ , where $s > n / p + 1$ and $p > 2$ . As mentioned above, this has previously been obtained in [19] when $p = 2$ and $s > n / 2 + 1$ and in [8] when $p = 2$ and $s > n / 2 + 2$ for the case of the torus and a bounded domain. In our proof, we do not approximate solutions to (1.1)–(1.3) by those to the Navier-Stokes equations but instead regularize the initial data and utilize the results from [8] and [19] to establish the existence of approximating pathwise solutions. Then we prove that this sequence of solutions converges in $L^{p} (Ω; L^{\infty} (0, T \land τ; W^{s, p} (R^{d})))$ for $s > d / p + 1$ and $p > 2$ , and the limit is a solution to (1.1)–(1.3). As pointed out in [11], since the a priori estimates hold only up to a stopping time that may depend on the approximating step n, we need to show these stopping times have a uniform lower bound. This difficulty is typically overcome by using continuity of trajectories, which is not the case here. Instead, we introduce a modified argument (cf. Lemma 5.3 below) that is based on the fact that the probability for stopping times not exceeding T is small when the time interval is sufficiently small.

The paper is organized as follows. In Section 2, we introduce the basic notation, while Section 3 contains the preliminaries on stochastic analysis and the statement of the main result. Next, in Section 4, we derive the energy estimates for approximating solutions. In the last section, we show that this sequence of solutions converges on a uniform random time interval in the desired function spaces.

2. Notation

For a scalar function $u = u (t, x)$ on $[0, T] \times R^{d}$ , we denote its partial derivatives by $\partial_{t} u = \partial u / \partial t$ , $\partial_{i} u = \partial u / \partial x_{i}$ , and $\partial_{i j}^{2} u = \partial^{2} u / \partial x_{i} x_{j}$ . Also, denote its gradient with respect to x by $\nabla u = (\partial_{1} u, \dots, \partial_{d} u)$ , while $D^{γ} u = \partial^{| γ |} u / \partial x_{1}^{γ_{1}} \dots \partial x_{d}^{γ_{d}}$ , where $γ = (γ_{1}, \dots, γ_{d}) \in N_{0}^{d}$ is a multi-index. (We write $N_{0} = {0, 1, 2, 3, \dots,}$ and $N = N_{0} ∖ {0}$ .)

Vector-valued functions are indicated by boldface letters, and $u_{k}$ , for $k = 1, \dots, d$ , represents the k-th component of the vector function u.

Denote by $S (R^{d})$ the Schwartz space and by $S^{'} (R^{d})$ the space of tempered distributions. For $s \in R$ , we write $\begin{array}{rcl} J^{s} f = F^{- 1} ({(1 + | ξ |^{2})}^{s / 2} F f) \end{array}$ and $\begin{array}{rcl} \partial^{s} f = F^{- 1} (| ξ |^{s} F f), \end{array}$ for $f \in S^{'} (R^{d})$ .

The usual $L^{p}$ norms are denoted by $‖ \cdot ‖_{p}$ , while $W^{s, p} (R^{d})$ , where $s ⩾ 0$ and $p ⩾ 1$ , is the class of tempered distributions $f \in S^{'} (R^{d})$ such that $\begin{array}{rcl} ‖ f ‖_{s, p} : = {‖ J^{s} f ‖}_{p} < \infty . \end{array}$ For the $L^{2}$ based spaces, we abbreviate $H^{s} (R^{d}) = W^{s, 2} (R^{d})$ . It is well-known that $\begin{array}{rcl} \frac{1}{C} ‖ f ‖_{s, p} ⩽ ‖ f ‖_{0} + {‖ \partial^{s} f ‖}_{p} ⩽ C ‖ f ‖_{s, p}, s ⩾ 0, 1 ⩽ p < \infty \end{array}$ for some positive constant $C > 0$ independent of f. We use $W^{1, \infty} (R^{d})$ to denote the collection of functions whose $‖ \cdot ‖_{1, \infty}$ norm is finite, where $\begin{array}{rcl} ‖ f ‖_{1, \infty} : = ‖ f ‖_{\infty} + ‖ \nabla f ‖_{\infty} . \end{array}$ The Leray projection $P$ is defined by $\begin{matrix} {\hat{(P u)}}_{j} (ξ) = \sum_{k = 1}^{d} (δ_{j k} - \frac{ξ_{j} ξ_{k}}{| ξ |^{2}}) \hat{u_{k}} (ξ), j = 1, 2, \dots, d . \end{matrix}$ Using the Riesz transforms $\begin{array}{rcl} R_{j} = - \frac{\partial}{\partial x_{j}} {(- Δ)}^{- \frac{1}{2}}, j = 1, 2, \dots, d, \end{array}$ the projector $P$ may be rewritten as $\begin{array}{rcl} {(P u)}_{j} (x) = \sum_{k = 1}^{d} (δ_{j k} + R_{j} R_{k}) u_{k} (x), j = 1, 2, \dots, d, \end{array}$ from where $\begin{array}{rcl} (2.1) & {((I - P) u)}_{j} (x) = - \sum_{k = 1}^{d} R_{j} R_{k} u_{k} (x), j = 1, 2, \dots, d . \end{array}$ As usual, C denotes a generic positive constant, whose value may increase from line to line, with explicit dependence indicated when necessary.

3. Preliminaries, assumptions, and main results

Let $(Ω, F, {(F_{t})}_{t ⩾ 0}, P)$ be a complete filtered probability space, where ${(F_{t})}_{t ⩾ 0}$ is the augmented natural filtration generated by the cylindrical Brownian motion $W$ . Suppose that $W$ is an $H$ -valued process for some separable real Hilbert space $H$ which may be infinite dimensional. Choosing a complete orthogonal basis ${e_{k}}_{k ⩾ 1}$ for $H$ , we may formally write $W (t, ω) = \sum_{k ⩾ 1} e_{k} W_{k} (t, ω)$ , where ${W_{k} : k \in N}$ is a collection of mutually independent 1-D Brownian motions.

Denote by $l^{2} (H, Y)$ the set of all bounded operators from $H$ to some Banach space $Y$ such that $‖ f ‖_{l^{2} (H, Y)}^{2} : = \sum_{k} | f \circ e_{k} |_{Y}^{2} < \infty$ , and write $l^{2} (H, R^{d}) = l^{2} (H)$ . For $p ⩾ 2$ and $s ⩾ 0$ , we define $\begin{array}{rcl} W^{s, p} = {f : R^{d} \to l^{2} (H) : J^{s} f \in l^{2} (H) and \int_{R^{d}} {‖ J^{s} f ‖}_{l^{2} (H)}^{p} d x < \infty}, \end{array}$ identifying functions equal a.e., which is a Banach space endowed with the norm $\begin{array}{rcl} ‖ f ‖_{W^{s, p}} = {(\int_{R^{d}} {‖ J^{s} f ‖}_{l^{2} (H)}^{p} d x)}^{1 / p} . \end{array}$ Letting $(P f) \circ e_{k} = P (f \circ e_{k})$ , where $P$ is the Leray projector, we have $P f \in W^{s, p}$ if $f \in W^{s, p}$ . Define $\begin{array}{rcl} X^{s, p} = {P f : f \in W^{s, p}} . \end{array}$ When $H$ reduces to a one-dimensional space, this definition is consistent with that of the divergence-free space $\begin{matrix} X^{s, p} = {f \in W^{s, p} (R^{d}) : \nabla \cdot f = 0} \end{matrix}$ in the deterministic setting. In this paper, we assume for (1.1) that σ maps $X^{s, p}$ into $X^{s, p}$ .

Suppose that $N (d t, d z)$ is a Poisson random measure on $[0, T] \times R^{d} ∖ {0}$ corresponding to an $F_{t}$ -adapted Poisson process on $(Ω, F, P)$ . Denote $\begin{matrix} E N (d t, d z) = ν (d z) d t, \end{matrix}$ where ν is a Lévy measure on $R^{d} ∖ {0}$ . The compensated Poisson measure is defined as $\begin{array}{rcl} \tilde{N} (d t, d z) = N (d t, d z) - ν (d z) d t . \end{array}$ Since $N (d t, d z)$ is a counting measure, the stochastic integration with respect to $N (d t, d z)$ can be seen as an infinite sum. We assume in (1.1) that G maps $X^{s, p} \times R^{d}$ into $X^{s, p}$ .

For the initial datum, we assume that $u_{0} \in L^{p} (Ω; W^{s, p} (R^{d}))$ , which is divergence-free a.s.

In order to differentiate from the notion of strong solutions in the deterministic PDE setting, we replace the term “strong solution” in the probabilistic framework by “pathwise solution”.

Definition 3.1 (Local pathwise solution).

We say that a pair $(u, τ)$ is a local pathwise solution of the stochastic Euler equations (1.1)–(1.3) if

the stopping time τ is almost surely positive, i.e., $P (τ > 0) = 1$ ,

the vector-valued function u is a $F_{t}$ -measurable stochastic process so that

it has càdlàg paths componentwise and $u \in L^{p} (Ω; L^{\infty} (0, T \land τ; W^{s, p} (R^{d})))$ for $s > d / p + 1$ and $p > 2$ ,

for all $t \in [0, T]$ , u satisfies $\begin{array}{l} (3.1) & \begin{matrix} u (t \land τ, x) & = u_{0} - \int_{0}^{t \land τ} P (u (r, x) \cdot \nabla) u (r, x) d r + \int_{0}^{t \land τ} σ (u (r, x)) d W (r) \\ + \int_{0}^{t \land τ} \int_{R^{d} ∖ {0}} G (u (r_{-}, x), z) \tilde{N} (d r, d z) . \end{matrix} \end{array}$

Note that in [19] the equations (1.1)–(1.3) were studied by assuming $u_{0} \in L^{2} (Ω, H^{s^{'}} (R^{d}))$ when $s^{'} > d / 2 + 1$ and $d = 2, 3$ . As a result, their local pathwise solution belongs to the space $L^{2} (Ω; L^{\infty} (0, T \land τ_{N}; H^{s^{'}} (R^{d})))$ . In the paper [19], the authors assumed the following.

Hypothesis A.
Fix $s^{'} > d / 2 + 1$ .
For all $u \in H^{s^{'}} (R^{d})$ , there exists a positive constant K such that $\begin{matrix} {‖ σ (u) ‖}_{l^{2} (H, H^{s^{'}})}^{2} + \int_{R^{d} ∖ {0}} {‖ G (u, z) ‖}_{s^{'}, 2}^{2} ν (d z) ⩽ K (1 + ‖ u ‖_{s^{'}, 2}^{2}) . \end{matrix}$

For all $u_{1}, u_{2} \in H^{s^{'}} (R^{d})$ , there exists a positive constant L such that $\begin{matrix} {‖ σ (u_{1}) - σ (u_{2}) ‖}_{l^{2} (H, H^{s^{'}})}^{2} + \int_{R^{d} ∖ {0}} {‖ G (u_{1}, z) - G (u_{2}, z) ‖}_{s^{'}, 2}^{2} ν (d z) ⩽ L ‖ u_{1} - u_{2} ‖_{s^{'}, 2}^{2} . \end{matrix}$

We assume throughout the paper that $u_{0} \in L^{p} (Ω, W^{s, p} (R^{d}))$ , where $s > d / p + 1$ and $p > 2$ , and that $u_{0}$ is divergence-free a.s. Also, $d \in {2, 3}$ .

On the noise, we impose the following assumption.
Hypothesis $\tilde{A}$ .
Fix $s > d / p + 1$ and $p > 2$ .
Hypothesis A is satisfied for some $s^{'} > d / 2 + 1$ .

Denote by ${Lip}_{x} (X, Y)$ the class of operators from $X$ to $Y$ that are Lipschitz in the variable x, and by ${Lin}_{x} (X, Y)$ those which are of sublinear growth in x. We assume $\begin{matrix} σ \in {Lip}_{u} (W^{s - 1, p}, W^{s - 1, p}) \cap {Lip}_{u} (W^{s, p}, W^{s, p}) \end{matrix}$ and $\begin{array}{rcl} σ \in {Lin}_{u} (W^{s, p}, W^{s, p}) \cap {Lin}_{u} (W^{s + 1, p}, W^{s + 1, p}) . \end{array}$

There exists $K > 0$ such that $\begin{matrix} {(\int_{R^{d} ∖ {0}} {‖ G (u, z) ‖}_{i, p} ν (d z))}^{p} + \int_{R^{d} ∖ {0}} {‖ G (u, z) ‖}_{i, p}^{p} ν (d z) \\ ⩽ K {(1 + ‖ u ‖_{i, p})}^{p}, i = s, s + 1 \end{matrix}$ for $u \in W^{s + 1, p} (R^{d})$ .

There exists $L > 0$ such that $\begin{matrix} \int_{R^{d} ∖ {0}} {‖ G (u_{1}, z) - G (u_{2}, z) ‖}_{i, p}^{p} ν (d z) + {(\int_{R^{d} ∖ {0}} {‖ G (u_{1}, z) - G (u_{2}, z) ‖}_{i, p} ν (d z))}^{p} \\ ⩽ L ‖ u_{1} - u_{2} ‖_{i, p}^{p}, i = s - 1, s \end{matrix}$ for all $u_{1}, u_{2} \in W^{s, p} (R^{d})$ . Also, $\begin{matrix} \int_{R^{d} ∖ {0}} {‖ G (u_{1}, z) - G (u_{2}, z) ‖}_{s - 1, p}^{i} {‖ G (u_{1}, z) ‖}_{s + 1, p}^{j} ν (d z) \\ ⩽ L ‖ u_{1} - u_{2} ‖_{s - 1, p}^{i} {(1 + ‖ u_{1} ‖_{s + 1, p})}^{j} \end{matrix}$ for $u_{1}, u_{2} \in W^{s, p} (R^{d})$ and all pairs of indices $(i, j) = (1, 1), (1, p), (p, 1), (p, p)$ .

The following is the main result of this paper. Theorem 3.1.
Assume that Hypothesis $\tilde{A}$ holds, in particular $s > d / p + 1$ and $p > 2$ , and that $u_{0} \in L^{p} (Ω, W^{s, p} (R^{d}))$ is $F_{0}$ -measurable and divergence-free a.s. Then there exists a local strong solution $(u, τ)$ to ( 1.1 )–( 1.3 ) such that
$(u, τ \land T)$ is pathwise unique,

the $F_{t}$ -adapted paths of $(u, τ \land T)$ are càdlàg,

$u \in L^{p} (Ω; L^{\infty} (0, T \land τ; W^{s, p} (R^{d})))$ ,

we have $\begin{array}{l} (3.2) & \begin{matrix} E [{‖ u (t \land τ, \cdot) ‖}_{s, p}^{p}] ⩽ e^{C_{p} N t} (E [{‖ u (0, \cdot) ‖}_{s, p}^{p}] + C_{p} T) \end{matrix} \end{array}$ and $\begin{array}{rcl} (3.3) & E [sup_{0 ⩽ r ⩽ t \land τ} {‖ u (r, \cdot) ‖}_{s, p}^{p}] ⩽ e^{C_{p} N t} (2 E [{‖ u (0, \cdot) ‖}_{s, p}^{p}] + C_{p} T) \end{array}$ for all $t \in [0, T]$ .

4. Smoothly truncated data

In this section, we smoothen and localize the initial datum $u_{0} \in L^{p} (Ω, W^{s, p} (R^{d}))$ , where we always assume $s > d / p + 1$ and $p > 2$ , so that it belongs to $L^{2} (Ω, H^{s^{'}} (R^{d}))$ for some $s^{'} > d / 2 + 1$ .

Let ${η_{ϵ}}_{ϵ > 0}$ , be the standard mollifier, and let ϕ be a function in $C_{0}^{\infty} (R^{d})$ such that $0 ⩽ ϕ (x) ⩽ 1$ for $x \in R^{d}$ , with $ϕ (x) = 1$ if $| x | ⩽ 1$ , and $supp ϕ \subseteq {x \in R^{d} : | x | ⩽ 2}$ . For $m \in N$ , denote $\begin{matrix} (4.1) & v^{m} = (v ϕ (\frac{\cdot}{m})) * η_{1 / m}, v \in W^{s, p} (R^{d}) . \end{matrix}$

Lemma 4.1.
Let $v \in W^{\bar{s}, p} (R^{d})$ , where $\bar{s} > 0$ and $p ⩾ 2$ . For $v^{m}$ defined above, we have $‖ v^{m} - v ‖_{\bar{s}, p} \to 0$ as $m \to \infty$ .
Proof of Lemma 4.1.
Indeed, we have $\begin{matrix} {‖ v^{m} - v ‖}_{\bar{s}, p} & ⩽ {‖ v^{m} - v * η_{1 / m} ‖}_{\bar{s}, p} + ‖ v * η_{1 / m} - v ‖_{\bar{s}, p} \\ ⩽ {‖ v ϕ (\frac{\cdot}{m}) - v ‖}_{\bar{s}, p} ‖ η ‖_{1} + \int {‖ v (\cdot - \frac{y}{m}) - v (\cdot) ‖}_{\bar{s}, p} | η (y) | d y . \end{matrix}$ The Dominated Convergence Theorem then implies $‖ v^{m} - v ‖_{\bar{s}, p} \to 0$ as $m \to \infty$ . □
Lemma 4.2.
Let $v \in W^{\bar{s}, p} (R^{d})$ , where $\bar{s} > 0$ and $p ⩾ 2$ . Then for any $s^{'} > 0$ , we have $‖ v^{m} ‖_{s^{'}, 2} ⩽ C (\bar{s}, s^{'}, d, m) ‖ v ‖_{\bar{s}, p}$ , for $m \in N$ .
Proof of Lemma 4.2.
A direct computation shows that $\begin{matrix} {‖ v^{m} ‖}_{s^{'}, 2} & ⩽ {‖ v ϕ (\frac{\cdot}{m}) ‖}_{2} {‖ J^{s^{'}} η_{1 / m} ‖}_{1} ⩽ ‖ v ‖_{p} {‖ ϕ (\frac{\cdot}{m}) ‖}_{2 p / (p - 2)} {‖ J^{s^{'}} η_{1 / m} ‖}_{1} \\ ⩽ C (s^{'}, d, m) ‖ v ‖_{\bar{s}, p}, \end{matrix}$ and the statement follows. □
Lemma 4.3.
Let $v \in W^{s, p} (R^{d})$ , where $s > d / p + 1$ and $p ⩾ 2$ . Then $\begin{matrix} {‖ v^{m} ‖}_{s + i, p} & ⩽ C m^{i} ‖ v ‖_{s, p}, i = - 1, 0, 1, \end{matrix}$ for some $C > 0$ independent of m.
Proof of Lemma 4.3.
Since the statement is well-known, we only sketch the proofs for $i = 1$ and $i = - 1$ . When $i = 1$ , we have $\begin{matrix} {‖ v^{m} ‖}_{s + 1, p} & ⩽ {‖ J^{s} [v ϕ (\frac{\cdot}{m})] ‖}_{p} (‖ \nabla η_{1 / m} ‖_{1} + ‖ η_{1 / m} ‖_{1}) ⩽ C (s, d) m ‖ v ‖_{s, p}, \end{matrix}$ while by the Minkowski inequality, $\begin{matrix} {‖ v^{m} ‖}_{s - 1, p} & = {‖ \int J^{s - 1} [v (x - y) ϕ (\frac{x - y}{m}) - v (x) ϕ (\frac{x}{m})] η_{1 / m} (y) d y ‖}_{p} \\ = {‖ \int J^{s - 1} [v (x - \frac{y}{m}) ϕ (\frac{x - y / m}{m}) - v (x) ϕ (\frac{x}{m})] η (y) d y ‖}_{p} \\ ⩽ C (s, d) m^{- 1} {‖ \nabla [v ϕ (\frac{\cdot}{m})] ‖}_{s - 1, p} ⩽ C (s, d) m^{- 1} ‖ v ‖_{s, p}, \end{matrix}$ and we obtain the statement for $i = - 1$ . □

With $u_{0}$ the initial data, let, as in (4.1), $\begin{matrix} v_{0}^{m} = (u_{0} ϕ (\frac{\cdot}{m})) * η_{1 / m}, \end{matrix}$ and $\begin{matrix} (4.2) & u_{0}^{m} = P v_{0}^{m} \end{matrix}$ By Lemmas 4.1–4.3 and continuity of the projector $P$ on $W^{s, p} (R^{d})$ and $H^{s^{'}} (R^{d})$ , we have $u_{0}^{m} \in L^{2} (Ω, H^{s^{'}} (R^{d}))$ for $s^{'} > d / 2 + 1$ if $u_{0} \in L^{p} (Ω, W^{s, p} (R^{d}))$ , where $s > d / p + 1$ . Moreover, since $u_{0}$ is divergence-free, $u_{0}^{m}$ converges to $u_{0}$ in $L^{p} (Ω, W^{s, p} (R^{d}))$ as $m \to \infty$ . Applying [19, Theorem 4.4], we obtain the following statement.
Lemma 4.4.
Assume that Hypothesis $\tilde{A}$ holds and suppose $u_{0} \in L^{p} (Ω, W^{s, p} (R^{d}))$ , where $s > d / p + 1$ and $p > 2$ , is divergence-free a.s. Then for all $m \in N$ and $u_{0}^{m}$ defined as ( 4.2 ), there is a pathwise unique solution $u^{m} \in L^{2} (Ω; L^{\infty} (0, T \land τ_{N}; H^{s^{'}} (R^{d})))$ to $\begin{array}{l} (4.3) & \begin{matrix} d u^{m} (t, x) & = - P (u^{m} (t, x) \cdot \nabla) u^{m} (t, x) d t + σ (u^{m} (t, x)) d W (t) \\ + \int_{R^{d} ∖ {0}} G (u^{m} (t_{-}, x), z) \tilde{N} (d t, d z), (t, x) \in [0, T] \times R^{d}, \end{matrix} \end{array}$ and $\begin{matrix} (4.4) & u^{m} (0, x) = u_{0}^{m} (x), ω -a.s., x \in R^{d}, \end{matrix}$ where $\begin{matrix} (4.5) & τ_{N} : = inf_{t ⩾ 0} {t : {‖ \nabla u^{m} ‖}_{\infty} + {‖ u^{m} ‖}_{\infty} ⩾ N} . \end{matrix}$
Remark 4.5.
The results in [19] were established for a stopping time using the $H^{s}$ norm, but it can be shown without essential changes that the nonlinear term $P ((u^{m} \cdot \nabla) u^{m})$ becomes sublinear when $‖ u^{m} ‖_{1, \infty}$ is bounded and then the existence of a strong solution follows for (4.3).

Denote by $u^{m}$ the solution to (4.3)–(4.4). In the rest of this section, for $M, N ⩾ 1$ let $\begin{matrix} τ_{N} : = inf {t ⩾ 0 : {‖ u^{m} ‖}_{\infty} + {‖ \nabla u^{m} ‖}_{\infty} ⩾ N} \end{matrix}$ and $\begin{matrix} {\tilde{τ}}_{M} : = inf {t ⩾ 0 : {‖ u^{m} ‖}_{s, p} ⩾ M} . \end{matrix}$ Theorem 4.6.
Assume that Hypothesis $\tilde{A}$ holds and let $u^{m}$ be the solution to ( 4.3 )–( 4.4 ). If $u_{0} \in L^{p} (Ω, W^{s, p} (R^{d}))$ where $s > d / p + 1$ and $p > 2$ , then $\begin{matrix} E [{‖ u^{m} (t \land τ_{N}, \cdot) ‖}_{s, p}^{p}] ⩽ e^{C_{p} N t} (E [{‖ u (0, \cdot) ‖}_{s, p}^{p}] + C_{p} T) \end{matrix}$ and $\begin{matrix} E [sup_{0 ⩽ r ⩽ t \land τ_{N}} {‖ u^{m} (r, \cdot) ‖}_{s, p}^{p}] ⩽ e^{C_{p} N t} (2 E [{‖ u (0, \cdot) ‖}_{s, p}^{p}] + C_{p} T) \end{matrix}$ for all $t \in [0, T]$ .
Proof of Theorem 4.6.
Applying $J^{s}$ , where $s > d / p + 1$ , to (4.3)–(4.4), we get $\begin{matrix} d J^{s} u^{m} = - J^{s} (P (u^{m} \cdot \nabla) u^{m}) d t + J^{s} σ (u^{m}) d W (t) \\ + \int_{R^{d} ∖ {0}} J^{s} G (u^{m} (t_{-}, x), z) \tilde{N} (d t, d z), \\ J^{s} u^{m} (0) = J^{s} u_{0}^{m} . \end{matrix}$ Set $f (y) = | y |^{p} = {(y \cdot y)}^{p / 2}$ where $p > 2$ , and let $τ = τ_{N} \land {\tilde{τ}}_{M}$ . We apply the Itô formula to $f (J^{s} u^{m} (t, x))$ for each $x \in R^{d}$ and $t \in [0, T \land τ]$ , and then integrate both sides with respect to x obtaining $\begin{matrix} {‖ J^{s} u^{m} ‖}^{p} & = {‖ J^{s} u_{0}^{m} ‖}^{p} - p \int \int_{0}^{t} {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot J^{s} [(u^{m} \cdot \nabla) u^{m}] d r d x \\ + p \int \int_{0}^{t} {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot (I - P) J^{s} [(u^{m} \cdot \nabla) u^{m}] d r d x \\ + p \int \int_{0}^{t} \int_{R^{d} ∖ {0}} {| J^{s} u^{m} (r_{-}) |}^{p - 2} J^{s} u^{m} (r_{-}) \cdot J^{s} (G (u^{m} (r_{-}), z)) \tilde{N} (d z, d r) d x \\ + \frac{p (p - 2)}{2} \int \int_{0}^{t} {‖ J^{s} u^{m} \cdot J^{s} σ (u^{m}) ‖}_{l^{2} (H)}^{2} {| J^{s} u^{m} |}^{p - 4} d r d x \\ + \frac{p}{2} \int \int_{0}^{t} {| J^{s} u^{m} |}^{p - 2} {‖ J^{s} σ (u^{m}) ‖}_{l^{2} (H)}^{2} d r d x \\ + \int \int_{0}^{t} \int_{R^{d} ∖ {0}} ({| J^{s} u^{m} (r_{-}) + J^{s} G (u^{m} (r_{-}), z) |}^{p} - {| J^{s} u^{m} (r_{-}) |}^{p} \\ - p {| J^{s} u^{m} (r_{-}) |}^{p - 2} J^{s} u^{m} (r_{-}) \cdot J^{s} (G (u^{m} (r_{-}), z))) N (d r, d z) d x \\ + p \int \int_{0}^{t} {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot J^{s} σ (u^{m}) d W (r) d x . \end{matrix}$ Note the $N (d r, d z)$ -integral above is an infinite sum that is $P$ -a.s. absolutely convergent up to the stopping time τ, and the usual Fubini theorem applies. Also observe that the stochastic Fubini theorem is justified by the Burkholder–Davis–Gundy (BDG) inequality for the local martingales $\begin{matrix} \int_{0}^{t} \int_{R^{d} ∖ {0}} {| J^{s} u^{m} (r_{-}) |}^{p - 2} J^{s} u^{m} (r_{-}) \cdot J^{s} (G (u^{m} (r_{-}), z)) \tilde{N} (d z, d r) \end{matrix}$ and $\begin{matrix} \int_{0}^{t} {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot J^{s} σ (u^{m}) d W (r) . \end{matrix}$ After switching the order of integration and combining finite $N (d r, d z)$ -integrals, we arrive at $\begin{matrix} d {‖ J^{s} u^{m} ‖}_{p}^{p} & = - p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot J^{s} [(u^{m} \cdot \nabla) u^{m}] d x d r \\ + p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot (I - P) J^{s} [(u^{m} \cdot \nabla) u^{m}] d x d r \\ - p \int_{R^{d} ∖ {0}} \int {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot J^{s} [G (u^{m}, z)] d x ν (d z) d r \\ + \frac{p (p - 2)}{2} \int {‖ J^{s} u^{m} \cdot J^{s} σ (u^{m}) ‖}_{l^{2} (H)}^{2} {| J^{s} u^{m} |}^{p - 4} d x d r \\ + \frac{p}{2} \int {| J^{s} u^{m} |}^{p - 2} {‖ J^{s} σ (u^{m}) ‖}_{l^{2} (H)}^{2} d x d r \\ + \int_{R^{d} ∖ {0}} \int [{| J^{s} u^{m} (r_{-}) + J^{s} G (u^{m} (r_{-}), z) |}^{p} - {| J^{s} u^{m} (r_{-}) |}^{p}] d x N (d r, d z) \\ + p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot J^{s} σ (u^{m}) d x d W (r) \\ : = \sum_{k = 1}^{5} I_{s; k} d r + \int_{R^{d} ∖ {0}} I_{s; 6} N (d r, d z) + I_{s; 7} d W (r) . \end{matrix}$ First, we have $\begin{matrix} I_{s; 1} & = - p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot (u^{m} \cdot \nabla) J^{s} u^{m} d x \\ - p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot (J^{s} ((u^{m} \cdot \nabla) u^{m}) - (u^{m} \cdot \nabla) J^{s} u^{m}) d x \\ : = I_{s; 11} + I_{s; 12} . \end{matrix}$ By the divergence-free condition, $\begin{array}{rcl} I_{s; 11} = \int {| J^{s} u^{m} |}^{p} div u^{m} d x = 0, \end{array}$ while, using the commutator estimates, $\begin{matrix} | I_{s; 12} | & ⩽ {‖ J^{s} u^{m} ‖}_{p}^{p - 1} {‖ J^{s} ((u^{m} \cdot \nabla) u^{m}) - (u^{m} \cdot \nabla) J^{s} u^{m} ‖}_{p} ⩽ C_{p} {‖ J^{s} u^{m} ‖}_{p}^{p} {‖ \nabla u^{m} ‖}_{\infty} . \end{matrix}$ Thus we obtain $\begin{matrix} | I_{s; 1} | & ⩽ C_{p} {‖ J^{s} u^{m} ‖}_{p}^{p} {‖ \nabla u^{m} ‖}_{\infty} ⩽ C_{p} {‖ u^{m} ‖}_{s, p}^{p} {‖ \nabla u^{m} ‖}_{\infty} . \end{matrix}$ Next, by (2.1), $\begin{matrix} I_{s; 2} & = p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u^{m} \cdot (I - P) J^{s} ((u^{m} \cdot \nabla) u^{m}) d x \\ = - p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u_{j}^{m} R_{j} R_{k} (u_{i}^{m} \partial_{i} J^{s} u_{k}^{m}) d x \\ - p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u_{j}^{m} R_{j} R_{k} [J^{s} (u_{i}^{m} \partial_{i} u_{k}^{m}) - (u_{i}^{m} J^{s} \partial_{i} u_{k}^{m})] d x \\ : = I_{s; 21} + I_{s; 22} . \end{matrix}$ Similarly, the divergence-free assumption implies $\begin{matrix} I_{s; 21} & = - p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u_{j}^{m} R_{j} R_{k} \partial_{i} (u_{i}^{m} J^{s} u_{k}^{m}) d x \\ = - p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u_{j}^{m} R_{j} R_{i} \partial_{k} (u_{i}^{m} J^{s} u_{k}^{m}) d x \\ = - p \int {| J^{s} u^{m} |}^{p - 2} J^{s} u_{j}^{m} R_{j} R_{i} (\partial_{k} u_{i}^{m} J^{s} u_{k}^{m}) d x \end{matrix}$ where we also used $R_{k} \partial_{i} = R_{i} \partial_{k}$ . By the Calderón–Zygmund theorem, we get $\begin{matrix} | I_{s; 21} | ⩽ C_{p} {‖ u^{m} ‖}_{s, p}^{p} {‖ \nabla u^{m} ‖}_{\infty} . \end{matrix}$ For the same reason, $\begin{matrix} | I_{s; 22} | & ⩽ C_{p} {‖ J^{s} u^{m} ‖}_{p}^{p - 1} ({‖ \nabla u^{m} ‖}_{\infty} {‖ J^{s - 1} \nabla u^{m} ‖}_{p} + {‖ J^{s} u^{m} ‖}_{p} {‖ \nabla u^{m} ‖}_{\infty}) \\ ⩽ C_{p} {‖ u^{m} ‖}_{s, p}^{p} {‖ \nabla u^{m} ‖}_{\infty}, \end{matrix}$ which implies $\begin{array}{rcl} | I_{s; 2} | ⩽ C_{p} {‖ u^{m} ‖}_{s, p}^{p} {‖ \nabla u^{m} ‖}_{\infty} . \end{array}$ Since G and σ are of linear growth, we get $\begin{matrix} | I_{s, 3} | ⩽ C_{p} {‖ u^{m} ‖}_{s, p}^{p - 1} (1 + {‖ u^{m} ‖}_{s, p}) \end{matrix}$ and $\begin{matrix} | I_{s; 4} |, | I_{s; 5} | ⩽ C_{p} {‖ u^{m} ‖}_{s, p}^{p - 2} {(1 + {‖ u^{m} ‖}_{s, p})}^{2} . \end{matrix}$ Summarizing, we obtain $\begin{matrix} E [| \int_{0}^{t \land τ} \sum_{k = 1}^{5} I_{s; k} d r |] ⩽ C_{p} N \int_{0}^{t \land τ} E [{‖ u^{m} ‖}_{s, p}^{p}] d r + C_{p} T \end{matrix}$ and $\begin{matrix} E [sup_{0 ⩽ t^{'} ⩽ t \land τ} | \int_{0}^{t^{'}} \sum_{k = 1}^{5} I_{s; k} d r |] ⩽ C_{p} N \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ} {‖ u^{m} ‖}_{s, p}^{p}] d r + C_{p} T, \end{matrix}$ for all $t \in (0, T]$ . For the term $I_{s; 6}$ , we write $\begin{matrix} | I_{s; 6} | & ⩽ p \int \int_{0}^{1} {| J^{s} u^{m} + θ J^{s} G (u^{m}, z) |}^{p - 1} | J^{s} G (u^{m}, z) | d θ d x \\ ⩽ C_{p} \int {| J^{s} u^{m} |}^{p - 1} | J^{s} G (u^{m}, z) | + {| J^{s} G (u^{m}, z) |}^{p} d x \\ ⩽ C_{p} ({‖ u^{m} ‖}_{s, p}^{p - 1} {‖ G (u^{m}, z) ‖}_{s, p} + {‖ G (u^{m}, z) ‖}_{s, p}^{p}) . \end{matrix}$ By the linear growth assumption on G and the fact that $N (d r, d z)$ is a counting measure, we have $\begin{matrix} E [| \int_{0}^{t \land τ} \int_{R^{d} ∖ {0}} I_{s; 6} N (d r, d z) |] \\ ⩽ E [\int_{0}^{t \land τ} \int_{R^{d} ∖ {0}} | I_{s; 6} | ν (d z) d r] \\ ⩽ C_{p} E [\int_{0}^{t \land τ} \int_{R^{d} ∖ {0}} ({‖ u^{m} ‖}_{s, p}^{p - 1} {‖ G (u^{m}, z) ‖}_{s, p} + {‖ G (u^{m}, z) ‖}_{s, p}^{p}) ν (d z) d r] \\ ⩽ C_{p} (\int_{0}^{t \land τ} E [{‖ u^{m} ‖}_{s, p}^{p}] d r + T) . \end{matrix}$ Moreover, $\begin{matrix} E [sup_{0 ⩽ t^{'} ⩽ t \land τ} | \int_{0}^{t^{'}} \int_{R^{d} ∖ {0}} I_{s; 6} N (d r, d z) |] & ⩽ C_{p} E [\int_{0}^{t \land τ} \int_{R^{d} ∖ {0}} | I_{s; 6} | ν (d z) d r] \\ ⩽ C_{p} (\int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ} {‖ u^{m} ‖}_{s, p}^{p}] d r + T) . \end{matrix}$ Using the linear growth of σ again, we have $\begin{array}{rcl} ‖ I_{s; 7} ‖_{l^{2} (H)} ⩽ C {‖ J^{s} u^{m} ‖}_{p}^{p - 1} (1 + {‖ u^{m} ‖}_{s, p}) ⩽ C ({‖ u^{m} ‖}_{s, p}^{p} + 1), \end{array}$ which is bounded up to the stopping time ${\tilde{τ}}_{M}$ . Then $\int_{0}^{t \land τ} I_{s; 7} d W_{r}$ , for $t \in [0, T]$ is a well-defined local martingale and thus $\begin{matrix} E [\int_{0}^{t \land τ} I_{s; 7} d W (r)] = 0, t \in [0, T] . \end{matrix}$ By the BDG inequality, $\begin{matrix} E [sup_{0 ⩽ t^{'} ⩽ t \land τ} | \int_{0}^{t^{'}} I_{s; 7} d W (r) |] & ⩽ C_{p} E [{(\int_{0}^{t \land τ} ‖ I_{s; 7} ‖_{l^{2} (H)}^{2} d r)}^{1 / 2}] \\ ⩽ C_{p} E [{(\int_{0}^{t \land τ} {‖ u^{m} ‖}_{s, p}^{2 p - 2} {(1 + {‖ u^{m} ‖}_{s, p})}^{2} d r)}^{1 / 2}] \\ ⩽ C_{p} E [sup_{0 ⩽ r ⩽ t \land τ} {‖ u^{m} ‖}_{s, p}^{p / 2} {(\int_{0}^{t \land τ} {‖ u^{m} ‖}_{s, p}^{p - 2} {(1 + {‖ u^{m} ‖}_{s, p})}^{2} d r)}^{1 / 2}] \end{matrix}$ for all $t \in (0, T]$ . Applying Young’s inequality, we get $\begin{matrix} E [sup_{0 ⩽ t^{'} ⩽ t \land τ} | \int_{0}^{t^{'}} I_{s; 7} d W (r) |] \\ ⩽ \frac{1}{2} E [sup_{0 ⩽ t^{'} ⩽ t \land τ} {‖ u^{m} ‖}_{s, p}^{p}] + C_{p} E [\int_{0}^{t \land τ} {‖ u^{m} ‖}_{s, p}^{p - 2} {(1 + {‖ u^{m} ‖}_{s, p})}^{2} d r] \\ ⩽ \frac{1}{2} E [sup_{0 ⩽ t^{'} ⩽ t \land τ} {‖ u^{m} ‖}_{s, p}^{p}] + C_{p} \int_{0}^{t} E [sup_{o ⩽ t^{'} ⩽ r \land τ} {‖ u^{m} ‖}_{s, p}^{p}] d r + C_{p} T . \end{matrix}$ Combining all the estimates, $\begin{matrix} E [{‖ u^{m} (t \land τ, \cdot) ‖}_{s, p}^{p}] - E [{‖ u^{m} (0, \cdot) ‖}_{s, p}^{p}] ⩽ C_{p} N \int_{0}^{t \land τ} E [{‖ u^{m} ‖}_{s, p}^{p}] d r + C_{p} T, \end{matrix}$ which by the Grönwall inequality implies $\begin{array}{rcl} E [{‖ u^{m} (t \land τ, \cdot) ‖}_{s, p}^{p}] ⩽ e^{C_{p} N t} (E [{‖ u^{m} (0, \cdot) ‖}_{s, p}^{p}] + C_{p} T) . \end{array}$ Since the right-hand side does not depend on M, we have $τ = τ_{N} \land {\tilde{τ}}_{M} \to τ_{N}$ as $M \to \infty$ . Therefore, $\begin{matrix} E [{‖ u^{m} (t \land τ_{N}, \cdot) ‖}_{s, p}^{p}] & ⩽ e^{C_{p} N t} (E [{‖ u^{m} (0, \cdot) ‖}_{s, p}^{p}] + C_{p} T) \\ ⩽ e^{C_{p} N t} (E [{‖ u (0, \cdot) ‖}_{s, p}^{p}] + C_{p} T) . \end{matrix}$ Similarly, we have $\begin{matrix} E [sup_{0 ⩽ r ⩽ t \land τ} {‖ u^{m} (r, \cdot) ‖}_{s, p}^{p}] - 2 E [{‖ u^{m} (0, \cdot) ‖}_{p}^{p}] \\ ⩽ C_{p} N \int_{0}^{t} E [sup_{0 ⩽ r ⩽ t^{'} \land τ} {‖ u^{m} (r, \cdot) ‖}_{s, p}^{p}] d t^{'} + C_{p} T . \end{matrix}$ Applying the Grönwall inequality and passing to the limit $M \to \infty$ , we get $\begin{array}{rcl} E [sup_{0 ⩽ r ⩽ t \land τ_{N}} {‖ u^{m} (r, \cdot) ‖}_{s, p}^{p}] ⩽ e^{C_{p} N t} (2 E [{‖ u (0, \cdot) ‖}_{s, p}^{p}] + C_{p} T), \end{array}$ for all $t \in [0, T]$ . □

5. Existence and uniqueness

5.1. Uniqueness

For the next lemma, fix $k \in {1, \dots, d}$ , and consider the commutator $\begin{matrix} [J^{s - 1}, f] \partial_{k} g = J^{s - 1} (f \partial_{k} g) - f (J^{s - 1} \partial_{k} g) . \end{matrix}$

Lemma 5.1.
Let $d = 2$ or 3 and $f, g \in W^{s, p} (R^{d})$ , where $s > d / p + 1$ . If $2 < p < \infty$ , then $\begin{matrix} (5.1) & {‖ [J^{s - 1}, f] \partial_{k} g ‖}_{p} ⩽ C_{s, p, d} ‖ f ‖_{s - 1, p} ‖ g ‖_{s, p} \end{matrix}$ and $\begin{matrix} (5.2) & {‖ [J^{s - 1}, f] \partial_{k} g ‖}_{p} ⩽ C_{s, p, d} ‖ f ‖_{s, p} ‖ g ‖_{s - 1, p}, p ⩽ d or p > d, s > 2 . \end{matrix}$ If, in addition, $\nabla \cdot f = 0$ , then $\begin{matrix} {‖ J^{s - 1} (f_{k} \partial_{k} g) - f_{k} (J^{s - 1} \partial_{k} g) ‖}_{p} ⩽ C_{s, p, d} ‖ f ‖_{s, p} ‖ g ‖_{s - 1, p}, p > d, s ⩽ 2 . \end{matrix}$
Proof of Lemma 5.1.
We consider several possibilities.

Case 1: Suppose $2 < p = d = 3$ and $s > d / p + 1 = 2$ . By Theorem A.1, we have $\begin{matrix} (5.3) & {‖ [J^{s - 1}, f] \partial_{k} g ‖}_{p} ⩽ C_{s, p, p_{1}, p_{2}, p_{3}, p_{4}, d} ({‖ J^{s - 1} f ‖}_{p_{1}} ‖ \partial_{k} g ‖_{p_{2}} + ‖ \nabla f ‖_{p_{3}} {‖ J^{s - 2} \partial_{k} g ‖}_{p_{4}}) \end{matrix}$ with $1 / p_{1} + 1 / p_{2} = 1 / p_{3} + 1 / p_{4} = 1 / p$ to be determined.

If $s ⩾ 3$ , by taking $p_{1} = p_{2} = 2 p$ , $p_{3} = \infty$ and $p_{4} = p$ , we have $\begin{array}{l} (5.4) & \begin{matrix} {‖ [J^{s - 1}, f] \partial_{k} g ‖}_{p} & ⩽ C_{s, p, d} ({‖ J^{s - 1} f ‖}_{1, p} ‖ \partial_{k} g ‖_{s - 2, p} + ‖ \nabla f ‖_{\infty} {‖ J^{s - 2} \partial_{k} g ‖}_{p}) \\ ⩽ C_{s, p, d} ‖ f ‖_{s, p} ‖ g ‖_{s - 1, p} . \end{matrix} \end{array}$ Letting $p_{1} = p$ , $p_{2} = \infty$ and $p_{3} = p_{4} = 2 p$ , we obtain $\begin{array}{l} (5.5) & \begin{matrix} {‖ [J^{s - 1}, f] \partial_{k} g ‖}_{p} & ⩽ C_{s, p, d} ({‖ J^{s - 1} f ‖}_{p} ‖ \partial_{k} g ‖_{\infty} + ‖ \nabla f ‖_{s - 2, p} {‖ J^{s - 2} \partial_{k} g ‖}_{1, p}) \\ ⩽ C_{s, p, d} ‖ f ‖_{s - 1, p} ‖ g ‖_{s, p} . \end{matrix} \end{array}$ If $2 < s < 3$ , we select $p_{2} = d / (d / p - (s - 2))$ , $p_{1} = p p_{2} / (p_{2} - p)$ , $p_{3} = \infty$ , and $p_{4} = p$ to obtain (5.2). On the other hand, if we set $p_{1} = p$ , $p_{2} = \infty$ , $p_{3} = d / (d / p - (s - 2))$ , and $p_{4} = p p_{3} / (p_{3} - p)$ , we arrive at (5.1).

Case 2: Suppose that $2 < p < d = 3$ and $s > d / p + 1 > 2$ . The inequality (5.3) still applies. If $s ⩾ d / p + 2$ , we have (5.2) for $p < p_{1} < d / (d / p - 1)$ , $p_{2} = p p_{1} / (p_{1} - p)$ , $p_{3} = \infty$ , and $p_{4} = p$ . Choosing $p_{1} = p$ , $p_{2} = \infty$ , $p_{4} = d p / (d - p)$ , and $p_{3} = p p_{4} / (p_{4} - p)$ , we obtain (5.1).

If $d / p + 1 < s < d / p + 2$ , we first take $p_{1} = d p / (d - p)$ , $p_{2} = p p_{1} / (p_{1} - p)$ , $p_{3} = \infty$ and $p_{4} = p$ , leading to (5.2). Choosing $p_{1} = p$ , $p_{2} = \infty$ , $p_{4} = d p / (d - p)$ , and $p_{3} = p p_{4} / (p_{4} - p)$ instead, we get (5.1).

Case 3: Suppose that $p > d$ and $s > 2$ . Then $\begin{matrix} {‖ [J^{s - 1}, f] \partial_{k} g ‖}_{p} & ⩽ C_{s, p, d} ({‖ J^{s - 1} f ‖}_{\infty} ‖ \partial_{k} g ‖_{p} + ‖ \nabla f ‖_{\infty} {‖ J^{s - 2} \partial_{k} g ‖}_{p}) \\ ⩽ C_{s, p, d} ({‖ J^{s - 1} f ‖}_{1, p} ‖ \partial_{k} g ‖_{p} + ‖ \nabla f ‖_{1, p} {‖ J^{s - 2} \partial_{k} g ‖}_{p}) \\ ⩽ C_{s, p, d} ‖ f ‖_{s, p} ‖ g ‖_{s - 1, p} \end{matrix}$ and $\begin{matrix} {‖ [J^{s - 1}, f] \partial_{k} g ‖}_{p} & ⩽ C_{s, p, d} ({‖ J^{s - 1} f ‖}_{p} ‖ \partial_{k} g ‖_{\infty} + ‖ \nabla f ‖_{p} {‖ J^{s - 2} \partial_{k} g ‖}_{\infty}) \\ ⩽ C_{s, p, d} ({‖ J^{s - 1} f ‖}_{p} ‖ \partial_{k} g ‖_{1, p} + ‖ \partial f ‖_{p} {‖ J^{s - 2} \partial_{k} g ‖}_{1, p}) \\ ⩽ C_{s, p, d} ‖ f ‖_{s - 1, p} ‖ g ‖_{s, p} . \end{matrix}$

Case 4: Suppose that $p > d$ and $d / p + 1 < s ⩽ 2$ . Then by Theorem A.1, $\begin{matrix} {‖ [J^{s - 1}, f] \partial_{k} g ‖}_{p} & ⩽ C_{s, p, d} {‖ J^{s - 1} f ‖}_{p} ‖ \partial_{k} g ‖_{\infty} ⩽ C_{s, p, d} ‖ f ‖_{s - 1, p} ‖ g ‖_{s, p} . \end{matrix}$ If in addition $\nabla \cdot f = 0$ , then $\begin{matrix} {‖ J^{s - 1} (f_{k} \partial_{k} g) - f_{k} (J^{s - 1} \partial_{k} g) ‖}_{p} & ⩽ {‖ J^{s - 1} \partial_{k} (f_{k} g) - f_{k} (J^{s - 1} \partial_{k} g) ‖}_{p} \\ ⩽ C_{s, p, d} ‖ f ‖_{s, p} ‖ g ‖_{s - 1, p}, \end{matrix}$ and this case is concluded as well. □

Assume that $u^{(1)}$ , $u^{(2)}$ are two solutions to (1.1)–(1.3). Set $v = u^{(1)} - u^{(2)}$ and $\begin{array}{l} τ^{(1)} : = inf {t ⩾ 0 : {‖ u^{(1)} ‖}_{s, p} ⩾ N}, \\ τ^{(2)} : = inf {t ⩾ 0 : {‖ u^{(2)} ‖}_{s, p} ⩾ N} . \end{array}$ Theorem 5.2.
Assume that Hypothesis $\tilde{A}$ holds and that $u_{0} \in L^{p} (Ω, W^{s, p} (R^{d}))$ , where $s > d / p + 1$ and $p > 2$ . Suppose $(u^{(1)}, τ^{(1)})$ , $(u^{(2)}, τ^{(2)})$ are two local pathwise solutions to ( 1.1 )–( 1.3 ). Then $\begin{array}{rcl} P (u^{(1)} (t) = u^{(2)} (t); \forall t \in [0, τ^{(1)} \land τ^{(2)} \land T]) = 1 . \end{array}$
Proof of Theorem 5.2.
Denote $τ = τ^{(1)} \land τ^{(2)}$ . Then on the interval $[0, τ \land T]$ , we have $\begin{array}{l} \begin{matrix} d J^{s - 1} v & = - P J^{s - 1} [(u^{(1)} \cdot \nabla) u^{(1)} - (u^{(2)} \cdot \nabla) u^{(2)}] d t + J^{s - 1} [σ (u^{(1)}) - σ (u^{(2)})] d W (t) \\ + \int_{R^{d} ∖ {0}} J^{s - 1} [G (u^{(1)} (t_{-}), z) - G (u^{(2)} (t_{-}), z)] \tilde{N} (d t, d z), \end{matrix} \\ J^{s - 1} v (0, x) = J^{s - 1} v_{0} (x) . \end{array}$ Applying the Itô formula to $| J^{s - 1} v |^{p}$ , where $p > 2$ , on $[0, τ \land T]$ and then integrating with respect to x, we get $\begin{matrix} d {‖ v (t, \cdot) ‖}_{s - 1, p}^{p} \\ = - p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v \cdot J^{s - 1} [(u^{(1)} \cdot \nabla) u^{(1)} - (u^{(2)} \cdot \nabla) u^{(2)}] d x d t \\ + p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v \cdot (I - P) J^{s - 1} [(u^{(1)} \cdot \nabla) u^{(1)} - (u^{(2)} \cdot \nabla) u^{(2)}] d x d t \\ - p \int_{R^{d} ∖ {0}} \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v \cdot J^{s - 1} [G (u^{(1)}, z) - G (u^{(2)}, z)] d x ν (d z) d t \\ + \frac{p (p - 2)}{2} \int {‖ J^{s - 1} v \cdot (J^{s - 1} σ (u^{(1)}) - J^{s - 1} σ (u^{(2)})) ‖}_{l^{2} (H)}^{2} {| J^{s - 1} v |}^{p - 4} d x d t \\ + \frac{p}{2} \int {| J^{s - 1} v |}^{p - 2} {‖ J^{s - 1} σ (u^{(1)}) - J^{s - 1} σ (u^{(2)}) ‖}_{l^{2} (H)}^{2} d x d t \\ + \int_{R^{d} ∖ {0}} \int [{| J^{s - 1} v + J^{s - 1} [G (u^{(1)}, z) - G (u^{(2)}, z)] |}^{p} - {| J^{s - 1} v |}^{p}] d x N (d t, d z) \\ + p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v \cdot J^{s - 1} [σ (u^{(1)}) - σ (u^{(2)})] d x d W (t) \\ : = \sum_{k = 1}^{5} A_{s - 1; k} d t + \int_{R^{d} ∖ {0}} A_{s - 1; 6} N (d t, d z) + A_{s - 1; 7} d W (t) . \end{matrix}$ Similarly to Section 3, we write $A_{s - 1; 1} = \sum_{k = 1}^{4} A_{s - 1; 1 k}$ , where $\begin{array}{l} (5.6) & \begin{matrix} A_{s - 1; 11} & = - p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v \cdot [(u^{(1)} \cdot \nabla) J^{s - 1} v] d x = - \int {| J^{s - 1} v |}^{p} div u^{(1)} d x = 0 \end{matrix} \end{array}$ and $\begin{array}{l} A_{s - 1; 12} = p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v \cdot [(u^{(1)} \cdot \nabla) J^{s - 1} v - J^{s - 1} ((u^{(1)} \cdot \nabla) v)] d x, \\ A_{s - 1; 13} = p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v \cdot [(v \cdot \nabla) J^{s - 1} u^{(2)} - J^{s - 1} ((v \cdot \nabla) u^{(2)})] d x, \\ A_{s - 1; 14} = - p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v \cdot [(v \cdot \nabla) J^{s - 1} u^{(2)}] d x . \end{array}$ Lemma 5.1 shows that $\begin{matrix} | A_{s - 1; 12} |, | A_{s - 1; 13} | & ⩽ C_{p} ‖ v ‖_{s - 1, p}^{p} ({‖ u^{(1)} ‖}_{s, p} + {‖ u^{(2)} ‖}_{s, p}), \end{matrix}$ while $\begin{matrix} | A_{s - 1; 14} | ⩽ C_{p} ‖ v ‖_{s - 1, p}^{p - 1} {‖ u^{(2)} ‖}_{s, p} ‖ v ‖_{\infty} . \end{matrix}$ Summarizing, $\begin{matrix} | A_{s - 1; 1} | & ⩽ C_{p} ‖ v ‖_{s - 1, p}^{p} ({‖ u^{(1)} ‖}_{s, p} + {‖ u^{(2)} ‖}_{s, p}) . \end{matrix}$ In the same vein, denote $A_{s; 2} = \sum_{k = 1}^{4} A_{s; 2 k}$ , where $\begin{array}{l} (5.7) & \begin{array}{l} A_{s - 1; 21} = p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v_{j} R_{j} R_{k} (u_{i}^{(1)} \partial_{i} J^{s - 1} v_{k}) d x, \\ A_{s - 1; 22} = p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v_{j} R_{j} R_{k} [J^{s - 1} (u_{i}^{(1)} \partial_{i} v_{k}) - u_{i}^{(1)} \partial_{i} J^{s - 1} v_{k}] d x, \\ A_{s - 1; 23} = p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v_{j} R_{j} R_{k} [J^{s - 1} (v_{i} \partial_{i} u_{k}^{(2)}) - v_{i} \partial_{i} J^{s - 1} u_{k}^{(2)}] d x, \\ A_{s - 1; 24} = p \int {| J^{s - 1} v |}^{p - 2} J^{s - 1} v_{j} R_{j} R_{k} [v_{i} \partial_{i} J^{s - 1} u_{k}^{(2)}] d x . \end{array} \end{array}$ Then $\begin{matrix} | A_{s - 1; 21} | & ⩽ p \int {| J^{s - 1} v |}^{p - 1} | R_{j} R_{i} [\partial_{k} u_{i}^{(1)} J^{s - 1} v_{k}] | d x ⩽ C_{p} ‖ v ‖_{s - 1, p}^{p} {‖ \nabla u^{(1)} ‖}_{\infty}, \\ | A_{s - 1; 2 i} | & ⩽ C_{p} ‖ v ‖_{s - 1, p}^{p} ({‖ u^{(1)} ‖}_{s, p} + {‖ u^{(2)} ‖}_{s, p}), i = 2, 3, 4, \end{matrix}$ and we conclude $\begin{matrix} | A_{s - 1; 2} | & ⩽ C_{p} ‖ v ‖_{s - 1, p}^{p} ({‖ u^{(1)} ‖}_{s, p} + {‖ u^{(2)} ‖}_{s, p}) . \end{matrix}$ Using the Lipschitz condition on σ and G, we obtain $\begin{matrix} (5.8) & | A_{s - 1; i} | ⩽ C_{p} ‖ v ‖_{s - 1, p}^{p}, i = 3, 4, 5 . \end{matrix}$ Then for all $t \in [0, T]$ , $\begin{matrix} | \sum_{k = 1}^{5} A_{s - 1; k} | & ⩽ C_{p} ‖ v ‖_{s - 1, p}^{p} ({‖ u^{(1)} ‖}_{s, p} + {‖ u^{(2)} ‖}_{s, p} + 1) ⩽ C_{p} N ‖ v ‖_{s - 1, p}^{p}, \end{matrix}$ and therefore, $\begin{matrix} (5.9) & E [sup_{0 ⩽ t^{'} ⩽ t \land τ} | \int_{0}^{t^{'}} \sum_{k = 1}^{5} A_{s - 1; k} d r |] ⩽ C_{p} N \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ} ‖ v ‖_{s - 1, p}^{p}] d r, t \in [0, T] . \end{matrix}$ Similarly to the energy estimate, we have $\begin{matrix} | A_{s - 1; 6} | ⩽ C_{p} (‖ v ‖_{s - 1, p}^{p - 1} {‖ G (u^{(1)}, z) - G (u^{(2)}, z) ‖}_{s - 1, p} + {‖ G (u^{(1)}, z) - G (u^{(2)}, z) ‖}_{s - 1, p}^{p}) . \end{matrix}$ Then by the Lipschitz condition on G, $\begin{array}{l} (5.10) & \begin{matrix} E [sup_{0 ⩽ t^{'} ⩽ t \land τ} | \int_{0}^{t^{'}} \int_{R^{d} ∖ {0}} A_{s - 1; 6} N (d r, d z) |] \\ ⩽ E [\int_{0}^{t \land τ} \int_{R^{d}} | A_{s - 1; 6} | ν (d z) d r] ⩽ C_{p} \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ} ‖ v ‖_{s - 1, p}^{p}] d r \end{matrix} \end{array}$ for all $t \in [0, T]$ . Clearly, $\int_{0}^{t} A_{s - 1; 7} d W_{r}$ is a well-defined local martingale up to the stopping time $τ \land T$ . Note that $\begin{matrix} ‖ A_{s - 1; 7} ‖_{l^{2} (H)} ⩽ C_{p} ‖ v ‖_{s - 1, p}^{p} . \end{matrix}$ Utilizing the BDG and Young’s inequalities, we get $\begin{array}{l} (5.11) & \begin{matrix} E [sup_{0 ⩽ t^{'} ⩽ t \land τ} | \int_{0}^{t^{'}} A_{s - 1; 7} d W (r) |] \\ ⩽ C_{p} E [{(\int_{0}^{t \land τ} ‖ A_{s - 1; 7} ‖_{l^{2} (H)}^{2} d r)}^{1 / 2}] \\ ⩽ C_{p} E [sup_{0 ⩽ r ⩽ t \land τ} ‖ v ‖_{s - 1, p}^{p / 2} {(\int_{0}^{t \land τ} ‖ v ‖_{s - 1, p}^{p} d r)}^{1 / 2}] \\ ⩽ \frac{1}{3} E [sup_{0 ⩽ r ⩽ t \land τ} ‖ v ‖_{s - 1, p}^{p}] + C_{p} \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ} ‖ v ‖_{s - 1, p}^{p}] d r, \end{matrix} \end{array}$ for all $t \in [0, T]$ . Combining (5.9), (5.10), and (5.11), we obtain $\begin{matrix} E [sup_{0 ⩽ r ⩽ t \land τ} ‖ v ‖_{s - 1, p}^{p}] ⩽ C_{p} N \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ} ‖ v ‖_{s - 1, p}^{p}] d r + 3 E [‖ v_{0} ‖_{s - 1, p}^{p}], \end{matrix}$ which by the Grönwall lemma and Lemma 4.1 implies $E [{sup}_{0 ⩽ r ⩽ t \land τ} ‖ v ‖_{s - 1, p}^{p}] = 0$ . Hence, $\begin{array}{rcl} P (u^{(1)} (t) = u^{(2)} (t); \forall t \in [0, τ^{(1)} \land τ^{(2)} \land T]) = 1, \end{array}$ and the uniqueness is established. □

5.2. Existence

When passing to the limit, it is important to show that the time interval on which the approximating solutions are Cauchy does not vanish. If the noise has continuous trajectories, then one could apply [12, Lemma 5.1] to obtain the existence of a positive stopping time (cf. [11]). However, the solutions we are considering have discontinuities in general due to Lévy noise. Next lemma, which is of independent interest, provides a modification of the statement providing the existence of a sufficiently small but almost surely positive stopping time.

First note that since $W^{s, p} (R^{d})$ is continuously embedded in $W^{1, \infty} (R^{d})$ if $s > d / p + 1$ , there exists $C_{0} > 1$ such that $\begin{matrix} (5.12) & ‖ u ‖_{1, \infty} ⩽ C_{0} ‖ u ‖_{s, p}, u \in W^{s, p} (R^{d}) . \end{matrix}$

Lemma 5.3.

Assume that Hypothesis $\tilde{A}$ holds, and let $u^{m}$ be a solution to ( 4.3 )–( 4.4 ). Suppose that ${sup}_{m} ‖ u_{0}^{m} ‖_{s, p} ⩽ ‖ u_{0} ‖_{s, p} ⩽ K$ a.s., for some deterministic $K > 1$ . Fix any $N ⩾ 2 K$ . Set $\begin{matrix} τ_{N}^{m} & = inf {t ⩾ 0 : {‖ u^{m} (t, \cdot) ‖}_{\infty} + {‖ \nabla u^{m} (t, \cdot) ‖}_{\infty} ⩾ C_{0} N} \end{matrix}$ and $\begin{matrix} λ^{m} & = inf {t ⩾ 0 : {‖ u^{m} (t, \cdot) ‖}_{s, p} ⩾ {‖ u_{0}^{m} ‖}_{s, p} + N} . \end{matrix}$ Then $\begin{matrix} P [λ > 0] = 1 \end{matrix}$ where $λ = {lim^{‾}}_{m \to \infty} λ^{m}$ .

Proof of Lemma 5.3.

In the proof of Theorem 4.6, we have shown that $\begin{matrix} E [sup_{0 ⩽ r ⩽ t \land τ_{N}^{m}} {‖ u^{m} (r, \cdot) ‖}_{s, p}^{p}] - 2 E [{‖ u_{0}^{m} ‖}_{p}^{p}] ⩽ C_{p} N \int_{0}^{t} E [sup_{0 ⩽ r ⩽ t^{'} \land τ_{N}^{m}} {‖ u^{m} (r, \cdot) ‖}_{s, p}^{p}] d t^{'} + C_{p} T, \end{matrix}$ for all $t \in [0, T]$ . This implies $\begin{matrix} E [sup_{0 ⩽ r ⩽ t \land τ_{N}^{m}} {({‖ u^{m} (r, \cdot) ‖}_{s, p}^{p} - 2 {‖ u_{0}^{m} ‖}_{s, p}^{p})}_{+}] \\ ⩽ C_{p} N \int_{0}^{t} E [sup_{0 ⩽ r ⩽ t \land τ_{N}^{m}} {({‖ u^{m} (r, \cdot) ‖}_{s, p}^{p} - 2 {‖ u_{0}^{m} ‖}_{s, p}^{p})}_{+}] d r + C_{p} T (N K^{p} + 1), \end{matrix}$ which by Grönwall lemma leads to $\begin{matrix} E [sup_{0 ⩽ r ⩽ T \land τ_{N}^{m}} {({‖ u^{m} (r, \cdot) ‖}_{s, p}^{p} - 2 {‖ u_{0}^{m} ‖}_{s, p}^{p})}_{+}] ⩽ C_{p} T (N K^{p} + 1) e^{C_{p} N T} . \end{matrix}$ Then, for this fixed N and any $ϵ > 0$ , there is a sufficiently small $T_{ϵ} > 0$ so that $\begin{matrix} E [sup_{0 ⩽ r ⩽ T_{ϵ} \land τ_{N}^{m}} {({‖ u^{m} (r, \cdot) ‖}_{s, p}^{p} - 2 {‖ u_{0}^{m} ‖}_{s, p}^{p})}_{+}] ⩽ ϵ, m \in N . \end{matrix}$ Denote $\begin{matrix} Ω_{N}^{m} = {ω : sup_{0 ⩽ r ⩽ τ_{N}^{m} \land T_{ϵ}} {‖ u^{m} (r, ω, \cdot) ‖}_{s, p}^{p} ⩽ 2 {‖ u_{0}^{m} ‖}_{s, p}^{p} + N^{p} 2^{- p}} . \end{matrix}$ Since $‖ u^{m} ‖_{1, \infty} ⩽ \sqrt[p]{3} C_{0} N / 2$ on $Ω_{N}^{m}$ , with exception of a null set, we in fact have $\begin{matrix} Ω_{N}^{m} = {ω : sup_{0 ⩽ r ⩽ T_{ϵ}} {‖ u^{m} (r, ω, \cdot) ‖}_{s, p}^{p} ⩽ 2 {‖ u_{0}^{m} ‖}_{s, p}^{p} + N^{p} 2^{- p}} . \end{matrix}$ By Chebyshev’s inequality, $\begin{matrix} P [Ω_{N}^{m}] ⩾ 1 - ϵ N^{- p} 2^{p}, \end{matrix}$ and thus $\begin{matrix} P [⋂_{i = 1}^{\infty} ⋃_{m = i}^{\infty} Ω_{N}^{m}] ⩾ 1 - ϵ N^{- p} 2^{p}, \end{matrix}$ due to $\begin{matrix} P [⋂_{i = 1}^{\infty} ⋃_{m = i}^{\infty} Ω_{N}^{m}] = lim_{i \to \infty} P [⋃_{m = i}^{\infty} Ω_{N}^{m}] ⩾ \underset{i \to \infty}{lim inf} P [Ω_{N}^{i}] ⩾ 1 - ϵ N^{- p} 2^{p} . \end{matrix}$ Note that $\begin{matrix} λ (ω) = {lim^{‾}}_{m \to \infty} λ^{m} (ω) ⩾ T_{ϵ} > 0, ω \in ⋂_{i = 1}^{\infty} ⋃_{m = i}^{\infty} Ω_{N}^{m}, \end{matrix}$ from where $\begin{array}{rcl} P [{ω : λ (ω) = {lim^{‾}}_{m \to \infty} λ^{m} (ω) > 0}] ⩾ 1 - ϵ N^{- p} 2^{p} . \end{array}$ Since ϵ can be arbitrarily small, we conclude $\begin{array}{rcl} P [{ω : λ (ω) > 0}] = 1, \end{array}$ and the lemma is established. □

By Lemma 4.4, we have that $(u^{j}, τ_{N}^{j})$ is a local pathwise solution to (4.3) given initial data $u_{0}^{j}$ . Because of the continuous embedding relation between $W^{s, p} (R^{d})$ , where $s > d / p + 1$ , and $W^{1, \infty} (R^{d})$ , we may set $\begin{matrix} (5.13) & τ_{j} = inf {t ⩾ 0 : {‖ u^{j} ‖}_{s, p} ⩾ 3 K + {‖ u_{0}^{j} ‖}_{s, p}} \land T \end{matrix}$ under the assumption $\begin{matrix} sup_{j} {‖ u_{0}^{j} ‖}_{s, p} ⩽ ‖ u_{0} ‖_{s, p} ⩽ K \end{matrix}$ for some deterministic $K > 0$ . Then $\begin{matrix} τ_{j} ⩽ inf {t ⩾ 0 : {‖ u^{j} ‖}_{1, \infty} ⩾ 4 C_{0} K} \land T, \end{matrix}$ where $C_{0}$ could be chosen as the smallest constant such that (5.12) holds.

We aim to show in this section that ${(u^{j}, t \land τ_{j}); j \in N}$ is a Cauchy sequence in $L^{p} (Ω; L^{\infty} (0, T \land τ; W^{s, p} (R^{d})))$ for some positive stopping time τ.

Lemma 5.4.

Let $s > d / p + 1$ and suppose that $u^{j}$ is a solution to ( 4.3 )–( 4.4 ). Assume that Hypothesis $\tilde{A}$ holds and ${sup}_{j} ‖ u_{0}^{j} ‖_{s, p} ⩽ ‖ u_{0} ‖_{s, p} ⩽ K$ a.s., for some deterministic $K > 0$ . Then for ${τ_{j}}_{j ⩾ 1}$ defined in ( 5.13 ), we have $\begin{matrix} lim_{i \to \infty} sup_{j ⩾ i} E [sup_{t \in [0, τ_{j} \land τ_{i}]} {‖ u^{j} (t) - u^{i} (t) ‖}_{s, p}^{p}] = 0 . \end{matrix}$

Proof of Lemma 5.4.

Set $τ_{i j} = τ_{i} \land τ_{j}$ and $v^{i j} = u^{j} - u^{i}$ . We repeat the decomposition (5.6)–(5.7) in Theorem 5.2 with $s - 1$ replaced by s. Then by utilizing the regular Kato-Ponce commutator estimates and (5.8), we obtain $\begin{matrix} | \sum_{k = 1}^{5} A_{s; k} | & ⩽ C_{p} {‖ v^{i j} ‖}_{s, p}^{p - 1} {‖ u^{i} ‖}_{s + 1, p} {‖ v^{i j} ‖}_{\infty} + C_{p} {‖ v^{i j} ‖}_{s, p}^{p} ({‖ u^{i} ‖}_{s, p} + {‖ u^{j} ‖}_{s, p} + 1) \\ ⩽ C_{p} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p} + C_{p} {‖ v^{i j} ‖}_{s, p}^{p} ({‖ u^{i} ‖}_{s, p} + {‖ u^{j} ‖}_{s, p} + 1) \end{matrix}$ for all $t \in [0, τ_{i j} \land T]$ . Next, $\begin{matrix} E [sup_{0 ⩽ t^{'} ⩽ t \land τ_{i j}} | \int_{0}^{t^{'}} \int_{R^{d} ∖ {0}} A_{s; 6} N (d r, d z) |] ⩽ C_{p} \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s, p}^{p}] d r \end{matrix}$ and $\begin{array}{l} E [sup_{0 ⩽ t^{'} ⩽ t \land τ_{i j}} | \int_{0}^{t^{'}} A_{s; 7} d W (r) |] ⩽ \frac{1}{2} E [sup_{0 ⩽ r ⩽ t \land τ_{i j}} {‖ v^{i j} ‖}_{s, p}^{p}] + C_{p} \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s, p}^{p}] d r . \end{array}$ Therefore, $\begin{matrix} E [sup_{0 ⩽ r ⩽ t \land τ_{i j}} {‖ v^{i j} ‖}_{s, p}^{p}] & ⩽ C_{p, K} \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s, p}^{p}] d r \\ + C_{p} T E [sup_{0 ⩽ r ⩽ t \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p}] + 2 E [{‖ v_{0}^{i j} ‖}_{s, p}^{p}], \end{matrix}$ and using the Grönwall lemma, $\begin{matrix} E [sup_{0 ⩽ r ⩽ T \land τ_{i j}} {‖ v^{i j} ‖}_{s, p}^{p}] ⩽ C_{p, K, T} (E [sup_{0 ⩽ r ⩽ T \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p}] + 2 E [{‖ v_{0}^{i j} ‖}_{s, p}^{p}]) . \end{matrix}$ We have shown in Lemmas 4.1–4.3 that $\begin{matrix} lim_{i \to \infty} sup_{j ⩾ i} E [{‖ v_{0}^{i j} ‖}_{s, p}^{p}] = 0 . \end{matrix}$ We need to prove $\begin{matrix} lim_{i \to \infty} sup_{j ⩾ i} E [sup_{0 ⩽ r ⩽ T \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p}] = 0 . \end{matrix}$ In fact, applying the Itô formula and using notations from Theorems 4.6 and 5.2, we have $\begin{matrix} d {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p} & = ({‖ v^{i j} ‖}_{s - 1, p}^{p} \sum_{k = 1}^{5} I_{s + 1; k} + {‖ u^{i} ‖}_{s + 1, p}^{p} \sum_{k = 1}^{5} A_{s - 1; k}) d t + I_{s + 1; 7} A_{s - 1; 7} d t \\ + \int_{R^{d} ∖ {0}} (I_{s + 1; 6} A_{s - 1; 6} + {‖ v^{i j} ‖}_{s - 1, p}^{p} I_{s + 1; 6} \\ + {‖ u^{i} ‖}_{s + 1, p}^{p} A_{s - 1; 6}) N (d t, d z) \\ + ({‖ v^{i j} ‖}_{s - 1, p}^{p} I_{s + 1; 7} + {‖ u^{i} ‖}_{s + 1, p}^{p} A_{s - 1; 7}) d W (t) \\ : = B_{1} d t + \int_{R^{d} ∖ {0}} B_{2} N (d t, d z) + B_{3} d W (t), \end{matrix}$ for all $t \in [0, τ_{i j}]$ . By previous estimates, $\begin{matrix} E [sup_{0 ⩽ r ⩽ t \land τ_{i j}} \int_{0}^{r} B_{1} d t^{'}] \\ ⩽ C_{p, K} (\int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p}] d r + \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p}] d r) \end{matrix}$ and then $\begin{matrix} E [sup_{0 ⩽ r ⩽ t \land τ_{i j}} | \int_{0}^{r} \int_{R^{d} ∖ {0}} B_{2} N (d t^{'}, d z) |] \\ ⩽ C_{p} E [\int_{0}^{t \land τ_{i j}} \int_{R^{d} ∖ {0}} | B_{2} | ν (d z) d r] \\ ⩽ C_{p} E [\int_{0}^{t \land τ_{i j}} \int_{R^{d} ∖ {0}} ({‖ v^{i, j} ‖}_{s - 1, p}^{p - 1} {‖ G (u^{i}, z) - G (u^{j}, z) ‖}_{s - 1, p} \\ + {‖ G (u^{i}, z) - G (u^{j}, z) ‖}_{s - 1, p}^{p}) ({‖ u^{i} ‖}_{s + 1, p}^{p - 1} {‖ G (u^{i}, z) ‖}_{s + 1, p} + {‖ G (u^{i}, z) ‖}_{s + 1, p}^{p}) \\ + {‖ v^{i j} ‖}_{s - 1, p}^{p} ({‖ u^{i} ‖}_{s + 1, p}^{p - 1} {‖ G (u^{i}, z) ‖}_{s + 1, p} + {‖ G (u^{i}, z) ‖}_{s + 1, p}^{p}) \\ + ({‖ G (u^{i}, z) - G (u^{j}, z) ‖}_{s - 1, p}^{p} \\ + {‖ v^{i j} ‖}_{s - 1, p}^{p - 1} {‖ G (u^{i}, z) - G (u^{j}, z) ‖}_{s - 1, p}) {‖ u^{i} ‖}_{s + 1, p}^{p} ν (d z) d r], \end{matrix}$ leading to $\begin{matrix} E [sup_{0 ⩽ r ⩽ t \land τ_{i j}} | \int_{0}^{r} \int_{R^{d} ∖ {0}} B_{2} N (d t^{'}, d z) |] \\ ⩽ C_{p, K} (\int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p}] d r + \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p}] d r) . \end{matrix}$ Moreover, $\begin{array}{l} E [sup_{0 ⩽ t^{'} ⩽ t \land τ_{i j}} | \int_{0}^{t^{'}} B_{3} d W_{r} |] \\ ⩽ C_{p} E [{(\int_{0}^{t \land τ_{i j}} ‖ B_{3} ‖_{l^{2} (H)}^{2} d r)}^{1 / 2}] \\ ⩽ C_{p} E [{(\int_{0}^{t \land τ_{i j}} {‖ u^{i} ‖}_{s + 1, p}^{2 p - 2} {(1 + {‖ u^{i} ‖}_{s + 1, p})}^{2} {‖ v^{i j} ‖}_{s - 1, p}^{2 p} + {‖ u^{i} ‖}_{s + 1, p}^{2 p} {‖ v^{i j} ‖}_{s - 1, p}^{2 p} d r)}^{1 / 2}] \\ ⩽ C_{p} (\int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p}] d r + \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p}] d r) \\ + \frac{1}{3} E [sup_{0 ⩽ r ⩽ t \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p}] . \end{array}$ Summarizing, for all $t \in [0, τ_{i j}]$ , $\begin{matrix} E [sup_{0 ⩽ r ⩽ t \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p}] - 3 E [{‖ v_{0}^{i j} ‖}_{s - 1, p}^{p} {‖ u_{0}^{i} ‖}_{s + 1, p}^{p}] \\ ⩽ C_{p, K} (\int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p}] d r + \int_{0}^{t} E [sup_{0 ⩽ t^{'} ⩽ r \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p}] d r) . \end{matrix}$ Therefore, $\begin{matrix} E [sup_{0 ⩽ r ⩽ T \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p} {‖ u^{i} ‖}_{s + 1, p}^{p}] \\ ⩽ C_{p, K, T} (E [sup_{0 ⩽ t ⩽ T \land τ_{i j}} {‖ v^{i j} ‖}_{s - 1, p}^{p}] + 3 E [{‖ v_{0}^{i j} ‖}_{s - 1, p}^{p} {‖ u_{0}^{i} ‖}_{s + 1, p}^{p}]) . \end{matrix}$ By the proof of Theorem 5.2, $\begin{matrix} lim_{i \to \infty} sup_{j ⩾ i} E [sup_{0 ⩽ r ⩽ t \land τ} {‖ v^{i j} ‖}_{s - 1, p}^{p}] ⩽ C_{p, K} lim_{i \to \infty} sup_{j ⩾ i} E [{‖ v_{0}^{i j} ‖}_{s - 1, p}^{p}] = 0 . \end{matrix}$ Finally, by Lemma 4.3, $\begin{matrix} lim_{i \to \infty} sup_{j ⩾ i} E [{‖ v_{0}^{i j} ‖}_{s - 1, p}^{p} {‖ u_{0}^{i} ‖}_{s + 1, p}^{p}] ⩽ C_{p, K} lim_{i \to \infty} sup_{j ⩾ i} E [{‖ v_{0}^{i j} ‖}_{s, p}^{p} {‖ u_{0}^{i} ‖}_{s, p}^{p}] = 0 . \end{matrix}$ Therefore, $\begin{matrix} lim_{i \to \infty} sup_{j ⩾ i} E [sup_{t \in [0, τ_{j} \land τ_{i}]} {‖ u^{j} (t) - u^{i} (t) ‖}_{s, p}^{p}] = 0, \end{matrix}$ and the lemma is established. □

Proof of Theorem 3.1.

As an application of Lemmas 5.3 and 5.4, there exists an almost surely positive stopping time τ and a subsequence ${u^{n_{j}}}_{j \in N}$ such that $\begin{matrix} lim_{i \to \infty} sup_{j ⩾ i} E [sup_{t \in [0, τ]} {‖ u^{n_{j}} (t) - u^{n_{i}} (t) ‖}_{s, p}^{p}] = 0, \end{matrix}$ which implies that, passing to a further subsequence, $\begin{matrix} lim_{i \to \infty} sup_{j ⩾ i} sup_{t \in [0, τ]} {‖ u^{n_{j}} (t) - u^{n_{i}} (t) ‖}_{s, p}^{p} = 0, ω -a.s . \end{matrix}$ Then there exists an $F_{t}$ -adapted process $u (ω) \in L^{\infty} ([0, T \land τ], W^{s, p} (R^{d}))$ such that $\begin{matrix} lim_{j \to \infty} sup_{t \in [0, τ]} {‖ u^{n_{j}} (t) - u (t) ‖}_{s, p} = 0, ω -a.s., \end{matrix}$ which indicates $\begin{matrix} lim_{j \to \infty} sup_{t \in [0, τ]} {‖ u^{n_{j}} (t) - u (t) ‖}_{\infty} = 0, ω -a.s. \end{matrix}$ Since each $u^{n_{j}}$ is càdlàg, the limit process u is also càdlàg.

It remains to show that u is indeed a pathwise solution to (3.1), and that requires us to prove that each term in (3.1) converges in $L^{p} (Ω; L^{\infty} (0, T \land τ; W^{s, p} (R^{d})))$ when we pass the limit. Since the convergence of the deterministic terms is well-known, we only show this for those two stochastic integrals. By the BDG and Minkowski inequalities, $\begin{matrix} E [sup_{0 ⩽ t ⩽ T \land τ} {‖ \int_{0}^{t} (σ (u^{m}) - σ (u)) d W (r) ‖}_{s, p}^{p}] \\ ⩽ C E [\int_{0}^{t \land τ} \int_{R^{d}} {‖ J^{s} σ (u^{m}) - J^{s} σ (u) ‖}_{l^{2} (H)}^{p} d x d r] \\ ⩽ C E [\int_{0}^{t \land τ} {‖ σ (u^{m}) - σ (u) ‖}_{W^{s, p}}^{p} d r] \to 0 as m \to \infty, \end{matrix}$ and $\begin{matrix} E [sup_{0 ⩽ t ⩽ T \land τ} {‖ \int_{0}^{t} \int_{R^{d} ∖ {0}} (G (u^{m} (r_{-}), z) - G (u (r_{-}), z)) \tilde{N} (d r, d z) ‖}_{s, p}^{p}] \\ ⩽ C E [\int_{0}^{t \land τ} \int_{R^{d} ∖ {0}} \int_{R^{d}} {| J^{s} G (u^{m} (r_{-}), z) - J^{s} G (u (r_{-}), z) |}^{p} d x ν (d z) d r] \\ ⩽ C E [\int_{0}^{t \land τ} \int_{R^{d} ∖ {0}} {‖ G (u^{m} (r_{-}), z) - G (u (r_{-}), z) ‖}_{s, p}^{p} ν (d z) d r] \to 0 as m \to \infty . \end{matrix}$

For a general $u_{0}$ , define $u_{0}^{k} = u_{0} 1_{k ⩽ ‖ u_{0} ‖_{s, p} ⩽ k + 1}$ , for $k \in N_{0}$ . Then applying the same argument, we obtain a local strong solution $(u^{k}, τ_{k})$ corresponding to the initial data $u_{0}^{k}$ . Then we let $\begin{matrix} u = \sum_{k ⩾ 0} u_{k} 1_{k ⩽ ‖ u_{0} ‖_{s, p} ⩽ k + 1}, \end{matrix}$ and $\begin{matrix} τ = \sum_{k ⩾ 0} τ_{k} 1_{k ⩽ ‖ u_{0} ‖_{s, p} ⩽ k + 1} . \end{matrix}$ This gives a partition of the probability space Ω. Clearly, $(u, τ)$ is a local strong solution to (1.1)–(1.3). Note that the proof of Theorem 4.6 also applies to $(u, τ)$ and thus (3.2)–(3.3) follow. Uniqueness has been shown in Theorem 5.2. □

Footnotes

Acknowledgements

IK was supported in part by the NSF grant DMS-1615239.

Appendix

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