arXiv subjects

Martina Hofmanová

Publications and source records attributed to Martina Hofmanová.

At least 19 recordsLinked to original sources

arXiv2026

Fluctuation dynamics in randomly advected Navier-Stokes equations below critical scaling

We study randomly advected incompressible Navier-Stokes equations, where the advecting field is a mean-zero, divergence-free, space-time stationary velocity field with smooth order-one correlations. We introduce a two-parameter family of models in which the advection is accelerated on a fast temporal scale $\varepsilon^2$ and has spatial correlation length $δ$; the critical regime $\varepsilon = δ$ corresponds to the natural parabolic scaling of the Navier-Stokes equation. In the full subcritical regime $\varepsilon = o (δ)$, we prove a law of large numbers in dimensions $d = 2, 3$: the solutions converge to a deterministic Navier--Stokes system with an enhanced diffusion coefficient given by a Green-Kubo formula. In two space dimensions, under the slightly stronger assumption $\varepsilon = o (δ^{1 + ι})$ for some $ι> 0$, we identify the leading-order fluctuations: after subtracting deterministic macroscopic corrections satisfying a nonlinear system of Navier-Stokes type, the rescaled fluctuations converge to a Gaussian field solving a linearized Navier-Stokes equation driven by multiplicative space-time white noise.

math.AP↗

arXiv2025

Self-similar blowup from arbitrary data for supercritical wave maps with additive noise

We consider stochastically perturbed wave maps from $\mathbb{R}^{1+d}$ into $\mathbb{S}^d$, in all energy-supercritical dimensions $d \geq 3$. We show that corotational non-degenerate Gaussian additive noise leads to self-similar blowup with positive probability for any corotational initial data. The same result without noise is conjectured, but unknown, for large data.

math.AP↗

arXiv2025

Non-uniqueness of mild solutions to supercritical heat equations

We consider the focusing power nonlinearity heat equation \begin{equation}\label{Eq:Heat_abstract}\tag{NLH} \partial_t u -Δu = |u|^{p-1}u, \quad p>1, \end{equation} in dimensions $d \geq 3$. It is well-known that if $p$ is large enough then \eqref{Eq:Heat_abstract} is unconditionally locally well-posed in $L^q(\mathbb{R}^d)$ for $q \geq d(p-1)/2$. We prove that this result is optimal in the sense that uniqueness of local solutions fails when $q < d(p-1)/2$ as long as $p < p_{JL}$, where $p_{JL}$ stands for the Joseph-Lundgren exponent. Our proof is based on the method that Jia-Šverák proposed in \cite{JiaSve15} to show non-uniqueness of Leray solutions to incompressible 3d Navier-Stokes equations. In particular, we rigorously verify for \eqref{Eq:Heat_abstract} the (analogue of the) spectral assumption made in \cite{JiaSve15}. To our knowledge, this is the first rigorous implementation of the Jia-Šverák method to a nonlinear parabolic equation without forcing.

math.AP↗

arXiv2025

Statistical solutions to the Schrödinger map equation in 1D, via the randomly forced Landau-Lifschitz-Gilbert equation

We prove the existence of statistically stationary solutions to the Schrödinger map equation on a one-dimensional domain, with null Neumann boundary conditions. We deal directly with the equation in its real-valued formulation, without using any transform. To approximate the Schrödinger map equation, we employ the stochastic Landau-Lifschitz-Gilbert equation. By a limiting procedure à la Kuksin, we establish existence of a random initial datum, whose distribution is preserved under the dynamics of the deterministic equation. Among other properties, the corresponding statistically stationary solution is proved to exhibit non-trivial dynamics in space and time and to be genuinely random. With an analogous argument, we prove the existence of stationary solutions to a stochastic Schrödinger map equation. We discuss the relationship between the statistically stationary solutions to the Schrödinger map equation, the binormal curvature flow and the cubic non-linear Schrödinger equation. Additionally, we prove the existence of statistically stationary solutions to the binormal curvature flow.

math.AP↗

arXiv2024

Rough analysis of two scale systems

We address a slow-fast system of coupled three dimensional Navier--Stokes equations where the fast component is perturbed by an additive Brownian noise. By means of the rough path theory, we establish the convergence in law of the slow component towards a Navier--Stokes system with an It{ô}--Stokes drift and a rough path driven transport noise. This gives an alternative, more general and direct proof to \cite{DP22}. Notably, the limiting rough path is identified as a geometric rough path, which does not necessarily coincide with the Stratonovich lift of the Brownian motion.

math.PR↗

arXiv2024

Non-uniqueness of Leray-Hopf solutions for stochastic forced Navier-Stokes equations

We consider stochastic forced Navier--Stokes equations on $\mathbb{R}^{3}$ starting from zero initial condition. The noise is linear multiplicative and the equations are perturbed by an additional body force. Based on the ideas of Albritton, Brué and Colombo \cite{ABC22}, we prove non-uniqueness of local-in-time Leray--Hopf solutions as well as joint non-uniqueness in law for solutions on $\mathbb{R}^{+}$. In the deterministic setting, we show that the set of forces, for which Leray--Hopf solutions are non-unique, is dense in $L^{1}_{t}L^{2}_{x}$. In addition, by a simple controllability argument we show that for every divergence-free initial condition in $L^{2}_{x}$ there is a force so that non-uniqueness of Leray--Hopf solutions holds.

math.PR↗

arXiv2024

Non-unique ergodicity for deterministic and stochastic 3D Navier--Stokes and Euler equations

We establish the existence of infinitely many stationary solutions, as well as ergodic stationary solutions, to the three dimensional Navier--Stokes and Euler equations in both deterministic and stochastic settings, driven by additive noise. These solutions belong to the regularity class $C(\mathbb{R};H^{\vartheta})\cap C^{\vartheta}(\mathbb{R};L^{2})$ for some $\vartheta>0$ and satisfy the equations in an analytically weak sense. The solutions to the Euler equations are obtained as vanishing viscosity limits of stationary solutions to the Navier--Stokes equations. Furthermore, regardless of their construction, every stationary solution to the Euler equations within this regularity class, which satisfies a suitable moment bound, is a limit in law of stationary analytically weak solutions to Navier--Stokes equations with vanishing viscosities. Our results are based on a novel stochastic version of the convex integration method, which provides uniform moment bounds locally in the aforementioned function spaces.

math.PR↗

arXiv2024

Kolmogorov $4/5$ law for the forced 3D Navier-Stokes equations

We identify a sufficient condition under which solutions to the 3D forced Navier--Stokes equations satisfy an $L^p$-in-time version of the Kolmogorov 4/5 law for the behavior of the averaged third order longitudinal structure function along the vanishing viscosity limit. The result has a natural probabilistic interpretation: the predicted behavior is observed on average after waiting for some sufficiently generic random time. The sufficient condition is satisfied e.g. by the solutions constructed by Bruè, Colombo, Crippa, De~Lellis, and Sorella. In this particular case, our results can be applied to derive a bound for the exponent of the third order absolute structure function in accordance with the Kolmogorov turbulence theory.

math.AP↗

arXiv2024

Decay of correlations in stochastic quantization: the exponential Euclidean field in two dimensions

We present two approaches to establish the exponential decay of correlation functions of Euclidean quantum field theories (EQFTs) via stochastic quantization (SQ). In particular we consider the elliptic stochastic quantization of the Høegh--Krohn (or $\exp (αϕ)_2$) EQFT in two dimensions. The first method is based on a path-wise coupling argument and PDE apriori estimates while the second on estimates of the Malliavin derivative of the solution to the SQ equation.

math-ph↗

arXiv2024

Anomalous and total dissipation due to advection by solutions of randomly forced Navier-Stokes equations

We propose a novel approach to induce anomalous dissipation through advection driven by turbulent fluid flows. Specifically, we establish the existence of a velocity field $v$ satisfying randomly forced Navier-Stokes equations, leading to total dissipation of kinetic energy in finite time when advecting a passive scalar. This dissipation phenomenon is uniform across viscosity parameters and initial conditions, representing a case of anomalous dissipation. We further explore dissipation induced by individual realizations of $v$. Our results extend to scenarios where the passive scalar is replaced by solutions to two or three-dimensional deterministic Navier-Stokes equations advected by $v$.

math.AP↗

arXiv2023

Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise

We construct Hölder continuous, global-in-time probabilistically strong solutions to 3D Euler equations perturbed by Stratonovich transport noise. Kinetic energy of the solutions can be prescribed a priori up to a stopping time, that can be chosen arbitrarily large with high probability. We also prove that there exist infinitely many Hölder continuous initial conditions leading to non-uniqueness of solutions to the Cauchy problem associated with the system. Our construction relies on a flow transformation reducing the SPDE under investigation to a random PDE, and convex integration techniques introduced in the deterministic setting by De Lellis and Székelyhidi, here adapted to consider the stochastic case. In particular, our novel approach allows to construct probabilistically strong solutions on $[0,\infty)$ directly.

math.PR↗

arXiv2023

Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise

We consider a family of singular surface quasi-geostrophic equations $$ \partial_{t}θ+u\cdot\nablaθ=-ν(-Δ)^{γ/2}θ+(-Δ)^{α/2}ξ,\qquad u=\nabla^{\perp}(-Δ)^{-1/2}θ, $$ on $[0,\infty)\times\mathbb{T}^{2}$, where $ν\geq 0$, $γ\in [0,3/2)$, $α\in [0,1/4)$ and $ξ$ is a space-time white noise. For the first time, we establish the existence of infinitely many non-Gaussian $\bullet$ probabilistically strong solutions for every initial condition in $C^η$, $η>1/2$ $\bullet$ ergodic stationary solutions The result presents a single approach applicable in the subcritical, critical as well as supercritical regime in the sense of Hairer (M. Hairer, A theory of regularity structures). It also applies in the particular setting $α=γ/2$ which formally possesses a Gaussian invariant measure. In our proof, we first introduce a modified Da Prato--Debussche trick which, on the one hand, permits to convert irregularity in time into irregularity in space and, on the other hand, increases the regularity of the linear solution. Second, we develop a convex integration iteration for the corresponding nonlinear equation which yields non-unique non-Gaussian solutions satisfying powerful global-in-time estimates and generating stationary as well as ergodic stationary solutions.

math.PR↗

arXiv2022

Can probability theory really help tame problems in mathematical hydrodynamics?

Recent years have seen spectacular progress in the mathematical study of hydrodynamic equations. Novel tools from convex integration in particular prove extremely versatile in establishing non-uniqueness results. Motivated by this 'pathological' behavior of solutions in the deterministic setting, stochastic models of fluid dynamics have enjoyed growing interest from the mathematical community. Inspired by the theory of 'regularization by noise', it is hoped for that stochasticity might help avoid 'pathologies' such as non-uniqueness of weak solutions. Current research however shows that convex integration methods can prevail even in spite of random perturbations.

math.HO↗

arXiv2022

An averaged space-time discretization of the stochastic $p$-Laplace system

We study the stochastic $p$-Laplace system in a bounded domain. We propose two new space-time discretizations based on the approximation of time-averaged values. We establish linear convergence in space and $1/2$ convergence in time. Additionally, we provide a sampling algorithm to construct the necessary random input in an efficient way. The theoretical error analysis is complemented by numerical experiments.

math.NA↗

arXiv2022

Weak solutions for singular multiplicative SDEs via regularization by noise

We study multiplicative SDEs perturbed by an additive fractional Brownian motion on another probability space. Provided the Hurst parameter is chosen in a specified regime, we establish existence of probabilistically weak solutions to the SDE if the measurable diffusion coefficient merely satisfies an integrability condition. In particular, this allows to consider certain singular diffusion coefficients.

math.PR↗

arXiv2022

Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier--Stokes equations: existence and non-uniqueness

We are concerned with the three dimensional incompressible Navier--Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in $L^{2}$ we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result in particular implies non-uniqueness in law. Second, we prove non-uniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain non-uniqueness of the associated semiflows.

math.PR↗

arXiv2021

Global existence and non-uniqueness for 3D Navier--Stokes equations with space-time white noise

We establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier--Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most $-1/2-κ$ for any $κ>0$. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions.Our result applies to any divergence free initial condition in $L^{2}\cup B^{-1+κ}_{\infty,\infty}$, $κ>0$, and implies also non-uniqueness in law.

math.AP↗

arXiv2021

Non-uniqueness in law of stochastic 3D Navier--Stokes equations

We consider the stochastic Navier--Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear multiplicative and a nonlinear noise of cylindrical type, all driven by a Wiener process. In these settings, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval $[0,\infty)$. Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, non-uniqueness in law holds on an arbitrary time interval $[0,T]$, $T>0$.

math.PR↗