On uniqueness of solutions to stochastic Navier--Stokes equations

Technical Summary: On Uniqueness of Solutions to Stochastic Navier-Stokes Equations

Problem Statement
The paper addresses the problem of uniqueness and continuous dependence on initial conditions for solutions to the stochastic Navier-Stokes equations (SNSE). While classical results by Leray and Hopf establish the existence of weak solutions for the deterministic equations, and uniqueness is known for the two-dimensional case, the three-dimensional deterministic problem remains open regarding the uniqueness of weak solutions. Recent work has demonstrated non-uniqueness for specific 3D deterministic cases, motivating the search for "conditional uniqueness" theorems.

The authors extend this investigation to the stochastic setting, specifically considering equations driven simultaneously by multiplicative Wiener processes and compensated Poisson random measures (jump noise). The study focuses on establishing uniqueness within a restricted class of solutions (admissible solutions) on arbitrary Lipschitz domains, moving beyond the whole space $\mathbb{R}^d$.

Methodology
The authors employ a functional analytic approach grounded in the theory of stochastic evolution equations in Gelfand triples ($V \hookrightarrow H \hookrightarrow V^*$). The core methodology involves:

1. Reformulation of Noise: The authors utilize a technique (referencing [29]) to rewrite stochastic Itô integrals with respect to martingale measures (Poisson noise) as stochastic integrals with respect to Hilbert space-valued martingales. This allows the system driven by both Wiener and Poisson noise to be treated as a unified system driven by a sequence of martingales.
2. Admissible Solutions and Morrey Spaces: A new class of solutions, termed "admissible solutions," is introduced. These solutions are decomposed into a "Morrey component" ($u_M$) and a "bounded component" ($u_B$). The Morrey component satisfies a specific local integrability condition related to Morrey spaces ($L^{r, \lambda}$), while the bounded component is essentially bounded. This decomposition allows for weaker assumptions than the classical Ladyzhenskaya-Prodi-Serrin (LPS) condition.
3. Stochastic Energy Equality: A critical technical tool is a specialized version of Itô's formula for Banach space-valued semimartingales. This formula is adapted to handle drift components that are sums of $V^*$-valued and $H$-valued processes, where the $H$-valued part is only locally integrable in time. This is essential for deriving energy estimates for the difference of two solutions.
4. Energy Estimates and Gronwall's Lemma: By applying the stochastic energy equality to the difference of two admissible solutions ($w = u - v$), the authors derive differential inequalities. These estimates rely on the smallness of the Morrey component of the solution and the coercivity of the diffusion operator.

Key Contributions and Results

• Generalization of Noise Models: The paper extends conditional uniqueness results from SNSE driven solely by Wiener processes (as in [32]) to equations driven simultaneously by Wiener processes and compensated Poisson martingale measures.
• Extension to Lipschitz Domains: The results are established for arbitrary Lipschitz domains, generalizing previous work that was often restricted to the whole space $\mathbb{R}^d$.
• Weaker Uniqueness Conditions: The authors introduce a uniqueness criterion based on a "restricted Hopf-Leray class" defined by Morrey-type conditions.
  • Theorem 2.2: Establishes that if an admissible solution $u$ satisfies a smallness condition on its Morrey component ($\hat{u} < \delta/N$), then the solution is unique within the class of admissible solutions. Furthermore, it proves continuous dependence on initial data, yielding exponential stability for the deterministic case (Theorem 2.3).
  • Corollary 2.4: Demonstrates that the classical Ladyzhenskaya-Prodi-Serrin condition ($u \in L^q(0, T; L^p(\mathbb{R}^d))$ with $d/p + 2/q \leq 1$) implies the admissibility condition required for uniqueness. Thus, the new results generalize the classical LPS uniqueness theorems to the stochastic setting with jump noise.
• Existence of H-Solutions: Theorem 2.1 proves that any admissible solution admits a modification that is an $H$-valued càdlàg process (an "H-solution"), ensuring the solution has the necessary regularity to be well-defined pointwise in time.

Significance and Claims
The paper claims to contribute to the direction of conditional uniqueness results for three-dimensional stochastic Navier-Stokes equations, a field where considerably less is known compared to the two-dimensional case or the deterministic existence theory.

The authors state that their main result (Theorem 2.2) establishes conditional uniqueness for equations driven by both Wiener and Poisson noises on general Lipschitz domains. They emphasize that their Morrey-type condition is strictly weaker than the classical LPS condition, thereby broadening the scope of initial data and force fields for which uniqueness can be guaranteed. The work also provides a stability result (Theorem 2.3) for the deterministic case, showing exponential convergence of solutions under the derived smallness conditions.

The paper does not claim to solve the global uniqueness problem for arbitrary weak solutions in 3D, but rather provides a rigorous framework for uniqueness under specific, albeit weaker, structural assumptions on the solution's regularity. The results are presented as generalizations of previous deterministic and stochastic findings, specifically extending the work of [32] to include jump noise and more general domains.