Antonio Agresti (Sapienza University of Rome)
Title: On partial regularity for stochastic 3D Navier-Stokes equations
Abstract: It is known that global weak solutions to the stochastic 3D Navier-Stokes equations with various types of multiplicative noise exist. Capturing their regularity is one of the most fundamental open problems in fluid dynamics, even in the absence of noise. The set of singularities can be of a very intricate nature, even of fractal type. Partial regularity aims to provide sharp estimates on the fractal dimension (e.g., Hausdorff) for such sets. In this talk, we will discuss a way to estimate the set of singular times, that is, times at which a weak solution is not regular. In particular, we extend to the stochastic setting the well-known $1/2$-bound on the fractal dimension of singular times, which goes back to the works by Leray and Scheffer. Interestingly, the bound is independent of the roughness of the noise. Our viewpoint is new even in the deterministic case, and can be applied to a wide class of stochastic PDEs, e.g., reaction-diffusion equations.
Alexandra Blessing (University of Konstanz)
Title: Pathwise solutions for parabolic stochastic evolution equations and their long-time behavior
Abstract: We provide a pathwise construction of the solutions of stochastic partial di fferential equations with nonlinear multiplicative boundary noise and quasilinear systems. To this aim we combine functional analytic tools with rough path theory. We also discuss the long-time behavior of such systems using a random dynamical system approach. This talk is based on joint works with Mazyar Ghani Varzaneh, Antoine Hocquet and Tim Seitz.
Charles-Edouard Bréhier (Université de Pau et des Pays de l'Adour)
Title: Structure preserving schemes for some SPDEs
Abstract: I will present examples of situations where the discretization of (parabolic semilinear) stochastic partial differential equations requires some attention if one is interested in the preservation of qualitative properties of the solutions. I will first provide examples of positivity and domain preserving schemes for some SPDEs driven by multiplicative noise. I will also present a regularity-preserving scheme. Finally, I will explain the construction of asymptotic preserving schemes for some slow-fast SPDE systems in the averaging principle. All examples will be illustrated with numerical experiments and will be associated with strong and weak error estimates.
Zdzislaw Brzezniak (University of York)
Title: Weak martingale solutions to stochastic Navier-Stokes-Cahn-Hilliard system with transport noise
Abstract: We consider a diffuse interface model for the mixture of two incompressible fluids driven by transport noise. Under suitable abstract assumptions, we prove the existence of a global weak martingale solution as well as the pathwise uniqueness of a global strong solution in the two dimensional case. This is the first result addressing a unified framework derived from the works by Brze{\'z}niak and Motyl and Mikulevicius and Rozovskii, on the study of stochastic partial differential equations. This talk is based on joint research with A. Ndongmo Ngana (York).
Petr Čoupek (Charles University)
Title: SPDEs driven by fractional processes
Abstract: The talk will be devoted to linear SPDEs with additive but possibly non-Gaussian fractional noise. In the first part, we will briefly discuss stochastic integration and give necessary and sufficient conditions for the existence of the mild solution as well as sufficient conditions for its continuity in a broad class of possibly non-UMD Banach spaces. We will then shift focus to parameter estimation in such models. In particular, we will discuss the sensitivity of drift estimators to noise misspecification - while for the fractional Brownian motion, consistent drift estimation from a discretely observed trajectory with a fixed time horizon under in-fill asymptotics is impossible, this is, surprisingly, not the case for the Rosenblatt process. As a result, a Girsanov-type theorem cannot hold for Rosenblatt processes in the usual form.
Fernanda Cipriano (NOVA University Lisbon)
Title: Random dynamics and invariant measures for a class of non-Newtonian fluids of differential type on 2D and 3D Poincar\'e domains
Abstract:
In this talk, we consider a class of incompressible stochastic third-grade fluids (non-Newtonian fluids) equations on two- as well as three-dimensional Poincar\'e domains $\mathcal{O}$ (which may be bounded or unbounded). Our aims are to study the well-posedness and asymptotic analysis for the solutions of the underlying system. More precisely, we prove that the underlying system defined on $\mathcal{O}$ has a unique weak solution under Dirichlet boundary condition, and generates a random dynamical system $\Psi$. We then consider the system on bounded domains. Using the compact Sobolev embedding $\mathbb{H}^1(\mathcal{O}) \hookrightarrow \mathbb{L}^2(\mathcal{O})$, we prove the existence of a unique random attractor for the underlying system on bounded domains with external forcing in $\mathbb{H}^{-1}(\mathcal{O})$. Next, we consider the underlying system on unbounded Poincar\'e domains with external forcing in $\mathbb{L}^{2}(\mathcal{O})$ and show the existence of a unique random attractor. In order to obtain the existence of a unique random attractor on unbounded domains, due to the lack of compact Sobolev embedding $\mathbb{H}^1(\mathcal{O}) \hookrightarrow\L^2(\mathcal{O})$, we use the uniform-tail estimates method which helps us to demonstrate the asymptotic compactness of $\Psi$. These results are based on joint work with K. Kinra. Finally, we discuss the results on the existence of invariant measures established in joint works with K. Kinra and Y. Tahraoui.
Franco Flandoli (Scuola Normale Superiore)
Title: SPDEs for turbulent transport
Abstract: Modern fusion plasma devices called Tokamak aim to work in the so-called H-mode regime (High confinement), opposite to the L-mode one (Low confinement). Both of them are turbulent fluid dynamic regimes, for a plasma subject to a very strong magnetic field. After a brief introduction to the topic, the talk will focus on the turbulent heat transport, showing how SPDEs suitably model the problem and allow one to compute the energy confinement time, the quantity of major interest. SPDEs may be crucial also for the understanding of the L-H transition and for the development of anomalous heat transport models; some ideas in this direction will be also mentioned. The research has been performed in the framework of the ERC AdG project NoisyFluid, no. 101053472.
Lucio Galeati (Università degli Studi dell'Aquila)
Title: Well-posedness by rough Kraichnan noise for active scalar equations
Abstract: Motivated by turbulence modelling, we consider a class of 2D stochastic PDEs, characterized by the presence of a Gaussian, rough, solenoidal velocity field acting as a transport term. The linear version of such an equation (modelling passive scalars) was proposed in the 60s by Kraichnan as a synthetic model for passive scalar turbulence; it is by now well-understood that it displays numerous interesting features, like anomalous dissipation of energy and Lagrangian spontaneous stochasticity. Here we consider more complicated cases of active scalar equations, like 2D Euler in vorticity form and the generalized Surface Quasi Geostrophic (gSQG) equations. When the Gaussian field is white in time and $\alpha$-Holder in space, for some $\alpha\in (0,1)$, the associated transport term has a dissipative nature that helps stabilize the dynamics. As a consequence, for any $\alpha\in (0,1)$ and suitable $p\in (1,\infty)$, we can show strong existence and pathwise uniqueness of global solutions in $L^p$ spaces, which are recovered as the unique limit of vanishing viscosity approximations. This is in sharp contrast with the deterministic system, where such a result is open or even false in the presence of suitable forcing. The proofs are based on suitable refinements of the techniques first introduced in Coghi, Maurelli (2023). Based on a joint work with M. Bagnara (arXiv:2601.05982).
Ludovic Goudenège (Université d'Évry-Paris Saclay)
Title: Finite volume scheme for stochastic heat equation with transport noise in Stratonovich sense
Abstract: We prove that a finite volume approximation of a stochastic heat equation on a polygonal two-dimensional domain $\Lambda$, with multiplicative transport noise in the Stratonovich sense, satisfies a pathwise discrete energy estimate. Coupled with two possible strategies for the time discretization, this yields, almost surely, an $L^\infty((0,T);L^2(\Lambda))$ bound on the discrete solution $u$, as well as an $L^2((0,T);L^2(\Lambda))$ bound on $\nabla u$. These estimates are expected to be sufficient to pass to the limit in the numerical scheme and to prove convergence of the approximation toward the unique solution of the limiting equation. This is not a completely standard result since it is expected that there is a residual term in the limit equation which has the form of an additional second-order viscous term. Finally, I will illustrate the performance of the finite volume method with numerical simulations based on a two-point flux approximation. This is a joint work with Anne de Bouard and Flore Nabet from CMAP at Polytechnique, Institut Polytechnique de Paris, France.
Erika Hausenblas (Montanuniversitaet)
Title: Existence of a weak solution to a stochastic elliptic-parabolic system with multiplicative noise
Abstract: In this talk, we discuss a stochastic model for single-phase miscible displacement in porous media, motivated by enhanced oil recovery. The model describes the interaction between an injected solvent and resident oil, wherethe solvent concentration influences both the viscosity and the density of the mixture. Mathematically, the system is formulated on a bounded Lipschitz domain \(\U\subset\mathbb R^2\) as a coupled elliptic--parabolic problem consisting of an incompressibility equation for the Darcy velocity and a transport--diffusion equation for the solvent concentration. The main result is the existence of a probabilistic weak, or martingale, solution under natural structural assumptions on the physical coefficients and data, including porosity, permeability, viscosity, density, and the diffusion--dispersion tensor. The proof is based on a compactness method combined with suitable a priori energy estimates for the coupled system. We also discuss a positivity and boundedness property for the concentration: in the case of a degenerate diffusion of the form \(c(1-c)\), the concentration remains in the physically relevant interval \([0,1]\).
Hannelore-Inge Lisei (Babes-Bolyai University)
Title: Approximation and Optimal Control Theory for Stochastic Schrödinger Equations
Abstract: In this presentation, we examine stochastic nonlinear Schrödinger equations driven by fractional Brownian motion. We introduce a linearization method for approximating variational solutions and study the corresponding stochastic control problems, focusing particularly on the existence and approximation of optimal controls.
Christian Horacio Olivera (UNICAMP)
Title: An Eulerian-Lagrangian Formulation of the Compressible Euler Equations with Vacuum
Abstract: In this paper, we present a novel Eulerian-Lagrangian formulation for the compressible isentropic Euler equations with vaccum. Using the developed Lagrangian flow map formulation, we show a short-time solution for a general pressure law. A particularly appealing feature of the approach used, it is well defined in the presence of vacuum, namely for compactly supported initial data which constitute an important problem in gas dynamics. Moreover, it does so without relying on any special symmetrization. While analogous results are well understood for incompressible fluids, the compressible setting, particularly in the presence of vacuum, remained open. This work was support by FAPESP-ANR by the grant Stochastic and Deterministic Analysis for Irregular Models.
Paul Razafimandimby (Dublin City University)
Title: The Stochastic Ericksen–Leslie System: From Ginzburg-Landau Approximations to the Constrained Dynamics
Abstract: Motivated by the importance of noise in the dynamics of nematic liquid crystals, we consider a nonlinear and constrained stochastic PDEs modelling the dynamics of 2-dimensional nematic liquid crystals under random perturbation. This system is a coupling of the Navier-Stokes and the Heat Flow of Harmonic maps and is known as the stochastic Ericksen-Leslie equations (SELEs). In this talk we aim to present recent results on mathematical analysis of this stochastic nematic liquid crystal model. In the first part, we consider a penalised system modelling the dynamics of nematic liquid crystals in bounded domains of \(\mathbb{R}^2\) or \(\mathbb{R}^3\), driven by multiplicative Gaussian noise, where the constraint \(|\mathbf{n}| = 1\) is relaxed through a Ginzburg-Landau penalisation. Using a fixed-point argument combined with a cut-off technique, we prove the existence and uniqueness of a maximal local strong solution. In the two-dimensional case, we further establish that this solution is global by employing Khashminskii's test for non-explosion together with careful a priori estimates. In the second part, we investigate the asymptotic behaviour of the penalised model as the penalisation parameter \(\varepsilon \to 0\), aiming to recover the original stochastic Ericksen-Leslie equations with the strict constraint \(|\mathbf{n}| = 1\). Working on the two-dimensional torus, we establish uniform estimates and, using tightness arguments and the Jakubowski-Skorokhod representation theorem, prove the existence of a martingale local solution to the stochastic Ericksen-Leslie system for initial data in \(\mathbf{H}^1 \times \mathbf{H}^2\), improving upon previous results requiring higher regularity. The talk is based on several joint works with E. Hausenblas, Z. Brze\'zniak and G. Deugou\'e.
Francesco Russo (ENSTA Paris, Institut Polytechnique de Paris)
Title: About a Fokker-Planck partial differential equation with terminal condition
Abstract: Stochastic differential equations (SDEs) in the sense of McKean are stochastic differential equations, whose coefficients do not only depend on time and on the position of the solution process, but also on its marginal laws. Often they constitute probabilistic representation of conservative PDEs, called Fokker-Planck equations. In general Fokker-Planck PDEs are well-posed if the initial condition is specified. Here, alternatively, we consider the inverse problemwhich consists in prescribing the final data: in particular we give sufficient conditions for existence and uniqueness. We also provide a probabilistic representation of those PDEs in the form a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process. The research is motivated by some application consisting in representing some semilinear PDEs (typically Hamilton-Jacobi-Bellman in stochastic control) fully backwardly.
Jonas Tölle (AAlto University)
Title: Quantitative mixing for locally monotone stochastic evolution equations
Abstract: We establish general quantitative conditions for stochastic evolution equations with locally monotone drift and degenerate additive Wiener noise in variational formulation resulting in the existence of a unique invariant probability measure for the associated ergodic Markovian Feller semigroup. We prove improved moment estimates for the solutions and the $e$-property of the semigroup. Furthermore, we provide quantitative upper bounds for the Wasserstein $\varepsilon$-mixing times. Examples on possibly unbounded domains include the stochastic incompressible 2D Navier-Stokes equations, shear thickening stochastic power-law fluid equations, the stochastic $p$-Laplace equation, the stochastic heat equation, as well as, stochastic semilinear equations such as the 1D stochastic Burgers equation. The main novelties are the quantitative nature of the mixing rate and the assumptions, as well as the existence of invariant probability measures without the need for compact energy embeddings or conditions on the dimension of the range of the covariance operator. Joint works with Gerardo Barrera (IST Lisboa), \\https://arxiv.org/abs/2412.01381 and https://arxiv.org/abs/2602.00853