Convex Integration Fails to Demonstrate Non-Uniqueness for Singular Stochastic Partial Differential Equations

Singular stochastic partial differential equations pose a significant challenge to mathematicians, as the random forces within these equations often create ill-defined nonlinear terms. Hongjie Dong of the NSF and Kazuo Yamazaki, also of the NSF, have been investigating the potential of convex integration as a method for constructing solutions to these complex equations. Their research reviews recent progress in applying this technique, offering a novel perspective on tackling singularity problems. Importantly, the study suggests that proving non-uniqueness using convex integration may be difficult for certain models, including those arising in field theory, thereby refining the scope of this powerful analytical tool.

Convex Integration Solves Singular Stochastic Equations

Scientists demonstrate a novel approach to understanding and solving singular stochastic partial differential equations (SPDEs), equations that arise in modelling physical phenomena with inherent randomness. These equations, commonly used to describe hydrodynamic fluctuations, incorporate a rough random force that creates mathematical difficulties in defining solutions due to ill-defined nonlinear terms. This work builds upon established theories of regularity structures and paracontrolled distributions, offering a complementary pathway to address the challenges posed by singular SPDEs. The study rigorously examines the convex integration technique, revealing its potential and limitations when applied to these complex equations.

Researchers employ fractional derivative operators and Hölder-Besov spaces to analyse the roughness of the random force and its impact on solution regularity, establishing a framework for assessing the viability of convex integration. Experiments show that the technique often yields global-in-time solutions, a significant advantage over some existing methods limited to short-term predictions, and are generally more amenable to higher dimensional problems. This breakthrough reveals a critical constraint on the applicability of convex integration to a specific SPDE: the Φ4 model, originating from quantum field theory. Detailed analysis proves that any weak solution to a related heat equation with specific damping characteristics is unique, suggesting that demonstrating non-uniqueness, a key feature expected from solutions constructed via convex integration, is improbable for the Φ4 model unless the random force introduces such behaviour.

The team establishes this result through Theorem 4.1, solidifying the theoretical understanding of the technique’s boundaries. The research establishes a rigorous mathematical foundation for evaluating the effectiveness of convex integration in tackling singular SPDEs. By focusing on the Φ4 model, the study provides a concrete example where the technique appears unlikely to succeed in proving non-uniqueness, despite its promise in other contexts. The findings have implications for modelling a wide range of physical systems, including fluid dynamics, magnetism, and materials science, where stochastic partial differential equations play a crucial role.

Convex Integration for Rough Stochastic PDEs

The research addresses singular stochastic partial differential equations, employing convex integration as a primary solution method. Scientists developed a rigorous approach to tackle equations containing rough random forces, which traditionally create ill-defined nonlinear terms, necessitating innovative methods to construct viable solutions where standard techniques fail. The study examines the limitations of convex integration in proving non-uniqueness for a particular model originating from field theory. Experiments employ iterative procedures designed to improve solution regularity, with an observed gain at each step.

However, the research reveals a critical limitation in four dimensions, where solution regularity does not improve, leading to “infinite trees” as the system approaches criticality. A crucial element of the methodology involves subtracting renormalization constants to accurately solve equations, resulting in a solution comprising products of a stochastic time-varying white noise. The team meticulously computes these products using Wick products, ensuring appropriate renormalization constant choices, referencing established methods. Recent investigations extend this approach to the stochastic Yang-Mills equation, building upon deterministic Navier-Stokes equations, defining weak solutions and leveraging the convex integration technique, rooted in Nash’s isometric embedding theorem, to demonstrate non-uniqueness. This involved constructing solutions to the Navier-Stokes-Reynolds system with inductive hypotheses ensuring convergence towards solutions of the deterministic Navier-Stokes equations with prescribed energy.

Convex Integration Resolves Singular SPDEs and Onsager’s Conjecture

Scientists have achieved a significant breakthrough in understanding singular stochastic partial differential equations, demonstrating the construction of solutions where traditional methods fail. The research team successfully applied the convex integration technique to a specific model in d ≥ 2 dimensions, proving non-uniqueness in comparison to a zero solution with compact support in space-time. This result establishes a foundation for exploring solutions to equations with rough random forces that previously presented mathematical challenges, and enables the resolution of Onsager’s conjecture regarding the C-threshold for deterministic Euler equations, confirming energy conservation. Subsequent work demonstrated the non-uniqueness of weak solutions to the 3D deterministic Navier-Stokes equations, constructing a weak solution with a prescribed energy profile.

This capability was then extended to the L2(T) -supercritical threshold by Buckmaster, Colombo, and Vicol, and by Luo and Titi, showcasing the technique’s versatility. The core of the method involves constructing a sequence of solutions, satisfying inductive hypotheses, and cancelling out previous errors to reduce the overall error. This iterative process allows for the creation of solutions that approach the deterministic Navier-Stokes equations with a prescribed energy. In the stochastic realm, the convex integration technique was initially applied to compressible Euler equations, demonstrating path-wise non-uniqueness in two and three dimensions, resulting in probabilistically strong solutions, a characteristic previously unachieved in solutions forced by random noise. The team’s findings effectively eliminate the possibility of proving path-wise uniqueness for these equations using Cherney’s theorem.

Global Solutions for Singular Stochastic Equations

Recent work has advanced the application of convex integration techniques to singular stochastic partial differential equations, a class of equations complicated by rough random forcing terms. Researchers have demonstrated the potential of this approach to extend local solutions into global-in-time solutions, even when those solutions are not unique, and, notably, to construct global solutions where no local solution was previously known. This represents a significant step forward in addressing previously intractable problems. Findings suggest that, for certain equations with specific damping characteristics, uniqueness of solutions is guaranteed, diminishing the likelihood of successfully applying convex integration to induce non-uniqueness. Authors acknowledge limitations stemming from the specific equations examined and the reliance on the sign of nonlinearities within those equations. Future research could explore the technique’s efficacy with different stochastic forcing terms or alternative nonlinear structures, potentially broadening its scope and revealing further insights into the behaviour of singular stochastic partial differential equations.