Probabilistically Strong Solutions to Stochastic Euler Equations

In this paper, we establish the existence of probabilistically strong, measure-valued solutions for the stochastic incompressible Navier–Stokes equations and prove their convergence, in the vanishing viscosity limit, to probabilistically strong solutions for the stochastic incompressible Euler equations. In particular, this solves the open problem of constructing probabilistically strong solutions for the stochastic Euler equations that satisfy the energy inequality for general $L^2$ initial data. We introduce the concept of energy-variational solutions in the stochastic context in order to treat the nonlinearities without changing the probability space. Furthermore, we extend these results to fluids driven by transport noise.

💡 Research Summary

This paper addresses a long‑standing open problem in stochastic fluid dynamics: the construction of probabilistically strong solutions to the stochastic incompressible Euler equations that satisfy the natural energy inequality for arbitrary (L^{2}) initial data, and that arise as the vanishing‑viscosity limit of stochastic Navier–Stokes (N‑S) solutions. The authors achieve this by introducing a new analytical framework—energy‑variational solutions—in the stochastic setting and by exploiting Young measures to capture oscillations without changing the underlying probability space.

The first major result is the existence of probabilistically strong, measure‑valued solutions to the three‑dimensional stochastic incompressible Navier–Stokes equations with both additive and transport noise. Classical approaches rely on the Skorokhod–Jakubowski representation theorem, which forces a change of probability space and yields only probabilistically weak (martingale) solutions. By contrast, the authors apply a combination of Helmholtz projection, Itô‑Stratonovich conversion, and a careful Hilbert‑Schmidt interpretation of the transport term ((\sigma_{2}!\cdot!\nabla)u). This allows them to obtain uniform (L^{2})‑energy bounds and a BV‑type time regularity for the kinetic energy, which in turn yields a càdlàg representative for the energy process. The energy‑variational formulation encodes the energy inequality as a variational inequality, turning the nonlinear convection term into a term that can be handled via Young measures.

With these uniform estimates in hand, the authors perform a vanishing‑viscosity analysis. They let the viscosity (\nu\to0) and use compactness tools (Komogorov continuity, Aldous’ tightness criterion, and Jakubowski’s generalized Skorokhod theorem) to extract a subsequence that converges strongly in probability to a limit. Crucially, the associated Young measures collapse to Dirac measures, which means that the limit does not retain any hidden oscillations. Consequently, the limit satisfies the stochastic Euler equations in the strong sense, retains the energy inequality, and is defined on the original probability space—hence a probabilistically strong solution. This result resolves the previously open question of constructing such Euler solutions directly from Navier–Stokes approximations.

The third contribution is the systematic development of the energy‑variational solution concept for stochastic PDEs. Inspired by deterministic variational approaches, the authors define a solution not by the usual distributional identity but by a variational inequality that reflects the dissipation of kinetic energy. They prove that any energy‑variational solution automatically yields a dissipative measure‑valued solution in the sense of recent works on stochastic fluid dynamics, and they establish a weak‑strong uniqueness principle: if a strong (classical) solution exists, any energy‑variational solution coincides with it. This provides a robust framework for proving convergence results without having to identify delicate nonlinear limits directly.

Finally, the paper extends all previous results to the case of transport noise, where the stochastic forcing appears as ((\sigma_{2}!\cdot!\nabla)u\circ dW). Transport noise models unresolved small‑scale fluctuations and has been shown to possess regularizing effects. The authors demonstrate that the energy‑variational structure is preserved under transport noise, and the same vanishing‑viscosity limit holds: stochastic Navier–Stokes equations driven by transport noise converge to probabilistically strong stochastic Euler solutions that satisfy the energy inequality for any (L^{2}) initial datum.

In summary, the paper delivers (i) the existence of probabilistically strong, measure‑valued Navier–Stokes solutions; (ii) a novel energy‑variational framework that bridges strong and weak stochastic solution concepts; (iii) a rigorous vanishing‑viscosity limit yielding probabilistically strong Euler solutions with the energy inequality; and (iv) a generalization to transport noise. These advances not only solve a prominent open problem but also provide powerful new tools for future investigations of stochastic fluid equations, their long‑time behavior, regularization by noise, and numerical approximation schemes.