arXiv subjects

Antonio Agresti

Publications and source records attributed to Antonio Agresti.

At least 19 recordsLinked to original sources

arXiv2026

On the absence of blow-up in the 3D Navier-Stokes equations with transport noise

Establishing the global-in-time smoothness of solutions to the 3D Navier-Stokes equations (NSEs) with large initial data remains a long-standing open problem. In this paper, we prove that the 3D NSEs driven by a suitably chosen transport noise -- a physically motivated stochastic perturbation -- admit global-in-time smooth solutions with high probability. This regularizing effect holds uniformly for initial data in arbitrarily large balls of subcritical function spaces with positive smoothness.

math.PR↗

arXiv2026

Global classical solutions by transport noise for reaction-diffusion systems with entropy dissipation

The existence of global classical solutions for reaction-diffusion systems arising from chemical reaction networks remains a major open problem in the deterministic setting, especially for reactions with high polynomial growth. We prove that a suitably chosen, physically motivated transport noise yields unique strong solutions for complex balanced chemical reaction networks that are global in time with arbitrarily high probability. These solutions possess paths in $C^θ_t C^{\infty}_x$ for all $θ<1/2$ and are, in particular, classical in space. Furthermore, we show that a suitable transport noise can enhance the dissipation of spatial fluctuations at an arbitrarily prescribed exponential rate. Our proofs rely on a combination of scaling-limit arguments, maximal $L^p(L^q)$-regularity, and entropy-entropy dissipation estimates.

math.AP↗

arXiv2026

Global smooth solutions by transport noise of 3D Navier-Stokes equations with small hyperviscosity

The existence of global smooth solutions to the Navier-Stokes equations (NSEs) with hyperviscosity $(-Δ)^γ$ is open unless $γ$ is close to the J.-L. Lions exponent $ \frac{5}{4}$ at which the energy balance is strong enough to prevent singularity formation. If $1<γ\ll \frac{5}{4}$, then the global well-posedness of the hyperviscous NSEs is widely open as for the usual NSEs. In this paper, for all $γ>1$, we show the existence of a transport noise for which global smooth solutions to the stochastic hyperviscous NSEs on the three-dimensional torus exist with high probability. In particular, a suitable transport noise considerably improves the known well-posedness results in the deterministic setting.

math.AP↗

arXiv2026

Sharp bounds for non-trace class noise and applications to SPDEs

In the study of stochastic PDEs with colored, non-trace class space-time noise, one frequently encounters Gaussian series of the form $$g \sum_{n\geq 1} γ_n μ_n f_n, $$ where $(γ_n)_{n}$ is a sequence of standard independent Gaussian variables, $g$ is an $L^η(\mathcal{O})$ function, $(μ_n)_{n}$ is a sequence of scalars, and $(f_n)_n$ is an orthonormal system in $L^2(\mathcal{O})$ where $\mathcal{O} \subseteq \mathbb{R}^d$ is an open set. In this manuscript, we establish necessary and sufficient conditions for the above sum to converge in Bessel potential spaces $H^{-s,q}(\mathcal{O})$. The latter can be interpreted as a Sobolev embedding for Gaussian series. Our main theorem is formulated using weighted sequence spaces that encode the $L^\infty$-growth of the orthonormal system $(f_n)_{n}$, a feature that is crucial for obtaining sharp estimates. We apply our results to the stochastic heat equation with additive non-trace class noise. In this case, our conditions capture the scaling relationship between the heat operator and the coloring of the noise.

math.PR↗

arXiv2026

An optimal local theory for reaction-diffusion equations driven by non-trace-class noise

We study local well-posedness for a class of stochastic reaction-diffusion equations driven by multiplicative, possibly colored, noise. The interaction between rough stochastic forcing and polynomial nonlinearities naturally leads to solutions with low spatial regularity, making the treatment of the nonlinear terms delicate. Our main contribution is a general local existence and uniqueness theory for SPDEs with rough noise and highly irregular initial data. The framework also yields new results in standard noise regimes, including trace-class noise and space-time white noise. We identify the critical initial-data spaces for a wide range of nonlinearities, and we establish instantaneous parabolic regularization, general blow-up criteria, and sufficient conditions for positivity preservation. We apply the abstract theory to several prototypical models, including the stochastic Allen-Cahn, Burgers, Fisher-KPP, and coupled Gray-Scott equations. Finally, in the one-dimensional space-time white-noise setting, we combine our local theory with existing global a priori results in a highly singular regime.

math.AP↗

arXiv2026

Fractal dimension of singular times for SPDEs: Energy bounds, criticality, and weak-strong uniqueness

For several physically relevant SPDEs, it is known that global weak solutions coexist with local strong ones. Typically, weak-strong uniqueness results are known, and ensure that the global and strong solutions coincide as long as the latter exist. Times at which a weak solution does not coincide with a strong one are called singular times. Determining their fractal dimension is fundamental to capturing the regularity of weak solutions. We define singular times for a wide class of semilinear SPDEs. We show that sets of singular times have fractal dimension (i.e., Hausdorff and/or Minkowski) at most $ 1-\ell\, \mathsf{Exc}$, where $\ell$ and $\mathsf{Exc}$ are the time integrability and the excess of spatial regularity compared to the critical regularity of the energy bound associated with weak solutions, respectively. Moreover, their corresponding $(1-\ell\,\mathsf{Exc} )$-dimensional measure is zero. We formulate and apply our theory to quenched strong Leray-Hopf solutions of 3D Navier-Stokes equations (NSEs) with physically relevant noises, including rough Kraichnan and Lie transport. In particular, we extend the fundamental $1/2$-dimensional bound of Leray and Scheffer on singular times for 3D NSEs to the stochastic setting, and we prove new conditional results under supercritical Serrin's conditions, irrespective of the roughness of the noise. Our framework is new even in the deterministic case, and provides the first partial regularity results for weak solutions to SPDEs with multiplicative noise.

math.PR↗

arXiv2026

Global smooth solutions by high mode Lie-Transport noise for Logarithmically Hyperdissipative Navier-Stokes equations

We study a logarithmically hyperviscous Navier-Stokes model on the three-dimensional torus with Lie-transport noise, which includes both transport and stretching. We prove that, for noise of sufficiently large intensity and high frequency, the system admits a unique global smooth solution with probability arbitrarily close to one. Unlike previous works, this physically motivated noise does not preserve energy or enstrophy, but rather circulation. Global well-posedness is established through a probabilistic mechanism that produces effective dissipation via a scaling limit. Crucially, this approach bypasses the lack of conserved quantities and tames the singular nature of stochastic stretching.

math.PR↗

arXiv2026

A Note on a threshold for temporal regularity of stochastic PDEs

We consider solutions to linear parabolic SPDEs of the form \[ \mathrm{d} u(t) + A u(t)\, \mathrm{d} t = g(t)\, \mathrm{d} β, \qquad u(0)=0, \] where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$, $β$ is a one-dimensional Brownian motion, and $g(t)\equiv x\in X$. We show that, for all $α\in [0,\frac{1}{2}),$ \[ u\in L^2(Ω;W^{α,2}(0,T;\mathsf{D}(A^{1/2}))) \quad \text{ if and only if }\quad x\in \mathsf{D}(A^α). \] In particular, there is a lack of persistence of temporal regularity from the diffusion coefficient $g$ to the solution, and additional spatial regularity is required to improve time regularity. In particular, this provides a counterexample to a conjectured time-regularity property for monotone stochastic evolution equations posed by D. Breit and M. Hofmanová in [C. R. Math. Acad. Sci. Paris 354 (2016), 33-37].

math.PR↗

arXiv2026

Well-posedness of the stochastic thin-film equation with an interface potential

We consider strictly positive solutions to a class of fourth-order conservative quasilinear SPDEs on the $d$-dimensional torus modeled after the stochastic thin-film equation. We prove local Lipschitz estimates in Bessel potential spaces under minimal assumptions on the parameters and corresponding stochastic maximal $L^p$-regularity estimates for thin-film type operators with measurable in-time coefficients. As a result, we deduce local well-posedness of the stochastic thin-film equation as well as blow-up criteria and instantaneous regularization for the solution. In dimension one, we additionally close $α$-entropy estimates and subsequently an energy estimate for the stochastic thin-film equation with an interface potential so that global well-posedness follows. We allow for a wide range of mobility functions including the power laws $u^n$ for $n\in [0,6)$ as long as the interface potential is sufficiently repulsive.

math.AP↗

arXiv2025

The large deviation principle for the stochastic 3D primitive equations with transport noise

We prove the small-noise large deviation principle for the three-dimensional primitive equations with transport noise and turbulent pressure. Transport noise is important for geophysical fluid dynamics applications, as it takes into account the effect of small scales on the large scale dynamics. The main mathematical challenge is that we allow for the transport noise to act on the full horizontal velocity, therefore leading to a non-trivial turbulent pressure, which requires an involved analysis to obtain the necessary energy bounds. Both Stratonovich and Itô noise are treated.

math.PR↗

arXiv2025

A stochastic flow approach to De Giorgi-Nash-Moser estimates for SPDEs with smooth transport noise

The celebrated De Giorgi-Nash-Moser theory ensures that solutions to uniformly elliptic or parabolic PDEs are bounded and Hölder continuous, even with merely bounded measurable coefficients. For parabolic SPDEs with transport noise, boundedness has recently been established, but Hölder continuity remains a key open problem in the regularity theory of parabolic SPDEs. In this work, we resolve this question under the assumption that the noise coefficients are sufficiently regular in space. Our approach relies on Kunita's stochastic method of characteristics, which allows us to transform the original SPDE-via a stochastic flow of diffeomorphisms-into a random PDE to which the classical De Giorgi-Nash-Moser estimates apply. This program is accomplished through new a-priori estimates for the inverse of stochastic flows of diffeomorphisms, and a novel version of the Itô-Wentzell formula adapted to rough random fields. To demonstrate the applicability of our results, we establish the existence of global, regular solutions to quasilinear SPDEs with transport noise.

math.PR↗

arXiv2025

On anomalous dissipation induced by transport noise

In this paper, we show that suitable transport noises produce anomalous dissipation of both enstrophy of solutions to 2D Navier-Stokes equations and of energy of solutions to diffusion equations in all dimensions. The key ingredients are Meyers' type estimates for SPDEs with transport noise, which are combined with recent scaling limits for such SPDEs. The former enables us to establish, for the first time, uniform-in-time convergence in a space of positive smoothness for such scaling limits. Compared to previous work, one of the main novelties is that anomalous dissipation might take place even in the presence of a transport noise of arbitrarily small intensity. Physical interpretations of our results are also discussed.

math.AP↗

arXiv2025

Nonlinear SPDEs and Maximal Regularity: An Extended Survey

In this survey, we provide an in-depth exposition of our recent results on the well-posedness theory for stochastic evolution equations, employing maximal regularity techniques. The core of our approach is an abstract notion of critical spaces, which, when applied to nonlinear SPDEs, coincides with the concept of scaling-invariant spaces. This framework leads to several sharp blow-up criteria and enables one to obtain instantaneous regularization results. Additionally, we refine and unify our previous results, while also presenting several new contributions. In the second part of the survey, we apply the abstract results to several concrete SPDEs. In particular, we give applications to stochastic perturbations of quasi-geostrophic equations, Navier-Stokes equations, and reaction-diffusion systems (including Allen--Cahn, Cahn--Hilliard and Lotka--Volterra models). Moreover, for the Navier--Stokes equations, we establish new Serrin-type blow-up criteria. While some applications are addressed using $L^2$-theory, many require a more general $L^p(L^q)$-framework. In the final section, we outline several open problems, covering both abstract aspects of stochastic evolution equations, and concrete questions in the study of linear and nonlinear SPDEs.

math.PR↗

arXiv2025

Global well-posedness of 2D Navier-Stokes with Dirichlet boundary fractional noise

In this paper, we prove the global well-posedness and interior regularity for the 2D Navier-Stokes equations driven by a fractional noise acting as an inhomogeneous Dirichlet-type boundary condition. The model describes a vertical slice of the ocean with a relative motion between the two surfaces and can be thought of as a stochastic variant of the Couette flow. The relative motion of the surfaces is modeled by a Gaussian noise which is coloured in space and fractional in time with Hurst parameter greater than 3/4.

math.AP↗

arXiv2025

Lagrangian chaos and unique ergodicity for stochastic primitive equations

We show that the Lagrangian flow associated with the stochastic 3D primitive equations (PEs) with non-degenerate noise is chaotic, i.e., the corresponding top Lyapunov exponent is strictly positive almost surely. This result builds on the landmark work by Bedrossian, Blumenthal, and Punshon-Smith on Lagrangian chaos in stochastic fluid mechanics. Our primary contribution is establishing an instance where Lagrangian chaos can be proven for a fluid flow with supercritical energy, a key characteristic of 3D fluid dynamics. For the 3D PEs, establishing the existence of the top Lyapunov exponent is already a challenging task. We address this difficulty by deriving new estimates for the invariant measures of the 3D PEs, which capture the anisotropic smoothing in the dynamics of the PEs. As a by-product of our results, we also obtain the first uniqueness result for invariant measures of stochastic PEs.

math.PR↗

arXiv2024

The stochastic primitive equations with non-isothermal turbulent pressure

In this paper, we introduce and study the primitive equations with $\textit{non}$-isothermal turbulent pressure and transport noise. They are derived from the Navier-Stokes equations by employing stochastic versions of the Boussinesq and the hydrostatic approximations. The temperature dependence of the turbulent pressure can be seen as a consequence of an additive noise acting on the small vertical dynamics. For such a model we prove global well-posedness in $H^1$ where the noise is considered in both the Itô and Stratonovich formulations. Compared to previous variants of the primitive equations, the one considered here presents a more intricate coupling between the velocity field and the temperature. The corresponding analysis is seriously more involved than in the deterministic setting. Finally, the continuous dependence on the initial data and the energy estimates proven here are new, even in the case of isothermal turbulent pressure.

math.AP↗

arXiv2024

Global well-Posedness and Interior Regularity of 2D Navier-Stokes Equations with Stochastic Boundary Conditions

The paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier-Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere-ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.

math.AP↗

arXiv2024

Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems

In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the $d$-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g. the Allen-Cahn equation), and dissipative systems (e.g. equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka-Volterra equations for $d\in\{1, 2, 3, 4\}$ and the Brusselator for $d\in \{1, 2, 3\}$. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on Itô's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new $L^ζ$-coercivity/dissipativity conditions and the development of an $L^p(L^q)$-framework for systems of reaction-diffusion equations, which are needed when treating dimensions $d\in \{2, 3\}$ in the case of cubic or higher order nonlinearities.

math.AP↗