Theodore D. Drivas – Lecture 1 Room: Aula Dini We will discuss some foundational aspects of three-dimensional incompressible fluid turbulence, including guiding experimental observations, Kolmogorov’s 1941 theory on the structure of a turbulent flow, Onsager’s 1949 conjecture on anomalous dissipation and weak Euler solutions, and Landau’s Kazan remark concerning intermittency. Mathematical constraints on, as well as constructions that exhibit features of turbulent behavior will be discussed.

Contributed Talks 1: Alexander Migdal & Jeremie Bec Room: Aula Dini Alexander Migdal. Duality of Decaying Turbulence to a Solvable String Theory. We propose a novel analytical framework for incompressible Navier-Stokes (NS) turbulence, revealing a duality between classical fluid dynamics and one-dimensional nonlinear dynamics in loop space. This duality is an exact mathematical equivalence without any model assumptions or approximations. This reformulation of NS dynamics leads to a universal momentum loop equation, which excludes finite-time blow-ups, establishing a No Explosion Theorem for turbulent flows with finite initial noise. Decaying turbulence emerges as a solution to this equation and is interpreted as a solvable string theory with a discrete target space composed of regular star polygons. The derived decay spectrum exhibits excellent agreement with experimental data and direct numerical simulations (DNS), replacing classical Kolmogorov scaling laws with universal functions derived from number theory. These results suggest a deeper mathematical structure underlying turbulence, uniting fluid dynamics, quantum mechanics, and number theory. Jeremie Bec. Eulerian vs. Lagrangian intermittency in turbulence. Bridging multifractal descriptions Intermittency in turbulence manifests as intense fluctuations in energy dissipation and anomalous scaling laws, which can be described within the multifractal cascade framework. While the Eulerian approach characterises these fluctuations through spatial fields, the Lagrangian perspective captures them along tracer particle trajectories. These complementary viewpoints require a unified theoretical framework. In this talk, we examine the statistical signatures of Eulerian and Lagrangian intermittency, emphasising the role of bridge relations in linking their respective fluctuations and exploring how causality constraints influence the temporal evolution of intermittency. By comparing theoretical predictions with numerical data, we highlight open challenges in reconciling these two descriptions of turbulence.

Alexei A. Mailybaev – Lecture 1 Room: Aula Dini Lecture 1: We introduce the phenomenon of spontaneous stochasticity and demonstrate it in different fluid models, descending from the fluctuating Navier-Stokes equations to toy models.

Contributed Talks 2: Erika Ortiz, Victor Valadao & Dipankar Roy, Room: Aula Dini Erika Ortiz. We implement numerically a model representing the 3D incompressible Navier-Stokes system with small-scale white noise on logarithmic lattices, i.e., 3D space lattices with logarithmi- cally spaced nodes. Our goal is the numerical analysis of spontaneous stochasticity in this system. For this, we consider decreasing sequences of viscosities and noise parameters and analyze the convergence of the corresponding probability densities, thereby, verifying numerically weak con- vergence in the inviscid and zero-noise limit. We will report the numerical results obtained in this direction. This is a join work with Ciro S. Campolina and Alexei A. Mailybaev. Victor Valadao. The Surface Quasi-Geostrophic (SQG) equation is a two-dimensional model for flow dynamics at the surface of a rotating and stratified fluid. It is applied to geophysical con- texts such as the Earth’s atmosphere at the tropopause, ocean surface dynamics, and Jupiter’s atmosphere. Despite these applications, when SQG develops turbulence through forcing and dissipation, it shares statistical properties with three-dimensional (3D) turbulence, including an energy cascade with a Kolmogorov spectrum E(k) ∝ k^−5/3. The similarities between SQG and 3D turbulence motivate its exploration as a simplified model for understanding 3D turbulence. Our study focuses on the scaling properties of SQG turbulence, particularly the energy flux and spectrum, and Eulerian predictability through finite-time Lyapunov exponents (FTLE) in extensive numerical simulations over a wide range of Reynolds numbers (Re). We observe that Kolmogorov scaling emerges only at high Re, while at lower Re, the energy spectrum’s exponent ξ(Re) is steeper than −5/3, indicating a non-constant energy flux. On the predictability side, we found that the Lyapunov exponent λ(Re) scales approximately as λ ∝ Re^0.7, deviating from the expected λ ∝ Re^1/2 scaling. It is worth emphasizing that this scaling holds even when ξ < −5/3. Similar deviations were observed in 3D turbulence, where λ ∝ Re^0.64, suggesting a fundamental mechanism common to both SQG and 3D turbulence that governs the separation of solutions in these systems despite their differences. Dipankar Roy. We study one-dimensional stochastic models with two conservation laws. One of the models is the coupled continuum stochastic Burgers equations. In this model, each current is a sum of quadratic non-linearities, linear diffusion, and spacetime white noise. The second model is a two-lane stochastic lattice gas. The two conserved densities are tuned so that the flux Jacobian, a 2 × 2 matrix, has the same eigenvalues. In the steady state, we investigate spacetime correlations of the conserved fields and the time-integrated currents at the origin. For a certain choice of couplings, we observe the dynamical exponent of 3/2. Moreover, at these couplings, we demonstrate that the coupled continuum stochastic Burgers equations and the lattice gas are in the same universality class. This presentation is based on the work reported in Dipankar Roy et al J. Stat. Mech. (2024) 033209.

Alexei A. Mailybaev – Lecture 2 Room: Aula Dini Lecture 2: We show how the Feigenbaum-style renormalization-group (RG) is introduced, characterizing the inviscid limit, its stability and bifurcations.

Contributed Talks 3: Augusto Del Zotto & Marc Nualart Batalla Room: Aula Dini Augusto Del Zotto. In this talk, we investigate the effect of rotation on shear flows, focusing on the simple case of a three-dimensional Couette flow. The dispersive behavior induced by the Coriolis force manifests at both the linear and nonlinear levels. In the linear case, stability depends on the strength and direction of rotation, with both stable and unstable regimes observed. In the nonlinear case, rotation positively influences the nonlinear stability threshold. Marc Nualart Batalla. Streamline Geometries of Steady Euler Flows. Steady states of the two- dimensional Euler equations generally come in infinite-dimensional families and play an important role in the long-time dynamics of generic initial data. In this talk we will introduce several classes of steady states, we will discuss recent results on their structures and we will provide a geometric characterization of them in the periodic channel and annulus.

Theodore D. Drivas – Lecture 2 Room: Aula Dini We will discuss some foundational aspects of three-dimensional incompressible fluid turbulence, including guiding experimental observations, Kolmogorov’s 1941 theory on the structure of a turbulent flow, Onsager’s 1949 conjecture on anomalous dissipation and weak Euler solutions, and Landau’s Kazan remark concerning intermittency. Mathematical constraints on, as well as constructions that exhibit features of turbulent behavior will be discussed.

Poster Session Room: Aula Dini

Theodore D. Drivas – Lecture 3 Room: Aula Dini We will discuss some foundational aspects of three-dimensional incompressible fluid turbulence, including guiding experimental observations, Kolmogorov’s 1941 theory on the structure of a turbulent flow, Onsager’s 1949 conjecture on anomalous dissipation and weak Euler solutions, and Landau’s Kazan remark concerning intermittency. Mathematical constraints on, as well as constructions that exhibit features of turbulent behavior will be discussed.

Contributed Talks 4: Giorgio Cialdea & Lars Eric Hientzsch Room: Aula Dini Giorgio Cialdea. We show finite-time vorticity blowup for smooth solutions of the 2D compressible Euler equations with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity, and is accompanied by an axisymmetric implosion in which the swirl velocity enjoys full stability, as opposed to finite co-dimension stability. This is a joint work with Jiajie Chen, Steve Shkoller and Vlad Vicol. Lars Eric Hientzsch. On the ill-posedness of the 2D Boussinesq equations in the class of bounded initial data The Boussinesq equations describe the evolution of a stratified fluid under the influence of gravity. We investigate the system in vorticity form and with a stable continuous background stratification increasing with depth (spectrally stable density profile). We prove that the system is strongly ill-posed in the class of data with bounded initial vorticity and density gradient. The mechanism that allows us to exhibit the norm-inflation in infinitesimal time is purely nonlinear - in contrast to previous results on mild ill-posedness of the system. Time permitting, we discuss applications to the 3D axisymmetric Euler equations with small bounded initial vorticity are discussed. Based on joint work with R. Bianchini (IAC Rome) and F. Iandoli (Università della Calabria)

Alexei A. Mailybaev – Lecture 3 Room: Aula Dini Lecture 3: We extend the RG approach to the stochastic framework, demonstrating how it describes the joint limit of zero viscosity and noise.

Contributed Talks 5: Daniel Boutros, Daniel Goodair & Victor Navarro Fernandez Room: Aula Dini Daniel Boutros. We consider the analogue of Onsager’s conjecture for the hydrodynamic helicity, which is a topological conserved quantity of the incompressible Euler equations. We establish a local helicity balance with a defect term capturing the local helicity flux at small scales. We find a sufficient criterion for helicity conservation which is more general than the previous criteria in the literature. Using these results we can establish a Kolmogorov-type scaling law for the helicity. We also make some remarks on the non-barotropic compressible Euler equations, where the helicity is no longer conserved but bounds on the evolution of a related topological quantity can still be deduced. We also prove new results for the conservation of magnetic helicity for the ideal MHD equations. We observe that the conservation laws of the MHD system only hold under the assumption that the magnetic field remains divergence-free, which is not guaranteed at the level of weak solutions. We therefore show that this property is preserved for Leray-Hopf weak solutions of the viscous MHD equations and their vanishing viscosity limits. Finally, if time allows we will show new results on convex integration for a wide class of geophysical models. These results are joint works with John D. Gibbon, Simon Markfelder and Edriss S. Titi. Daniel Goodair. Fluids under Transport Noise and Boundary Conditions: Energy Estimates, Well- Posedness and Inviscid Limits. The seemingly random nature of turbulence has long suggested the use of noise in fluid equations, inviting the study from numerous perspectives of physically motivated forms of noise in such models. Backed by variational principles, model reduction and regularisation phenomena, there is a strong trend towards the use of transport noise. Introducing a stochastic term dependent on the gradient of the solution necessitates precise cancellation in the energy estimates at the core of our solution theory. I will explore the muddied waters of boundary conditions and their interplay with transport noise and energy estimates, discussing what does or does not play out like the torus, with implications for well-posedness and inviscid limit results concerning the Stochastic Navier-Stokes and Euler Equations. Victor Navarro Fernandez. Exponential mixing and enhanced dissipation with random cellular flows via hypocoercivity. In this work we study the evolution of a passive scalar advected by a cellular flow on a two-dimensional periodic box, where the center of the flow undergoes a random walk. We prove exponential decay of correlations for mean-free H^s functions, uniformly in diffusivity, leading to almost sure exponential mixing and optimal enhanced dissipation rates. Despite the stochastic nature of the flow, our approach is entirely analytical. We introduce a hypoelliptic PDE for the expectation of the two-point process, and establish its hypocoercivity in a weighted H^1 norm that degenerates at the diagonal. Hypocoercivity of the two-point PDE can be understood as an L^2-analogue of geometric ergodicity of the two-point chain in the random dynamical systems framework. This is a joint work with Christian Seis, Univeristat Munster.