arXiv subjects

Anne Bronzi

Publications and source records attributed to Anne Bronzi.

5 recordsLinked to original sources

arXiv2026

Optimal minimax formula for bounds on ensemble averages of statistically stationary three-dimensional Navier-Stokes flows

We establish an optimal upper bound formula for ensemble averages of flow quantities associated with the three-dimensional incompressible Navier-Stokes equations. The formula takes the form of a minimax problem, extending the framework developed by Tobasco, Goluskin, and Doering (2018) for finite-dimensional systems and by Rosa and Temam (2022) for the two-dimensional Navier-Stokes equations. The lack of global well-posedness for the 3D case presents a significant challenge, which we overcome by working within the space of Foias-Prodi stationary statistical solutions. The minimax formula is derived by exploiting suitable compactness and continuity properties of specific subspaces of probability measures under the weak topology of the phase space. A distinguishing feature of our result is the characterization of the maximizing measures: unlike the previous cases, the optimal bounds in 3D are achieved on extreme points that are specific convex combinations of at most two Dirac delta measures, instead of exactly one, a structure that naturally appears from the constraints given by the mean energy dissipation inequalities in the characterization of the Foias-Prodi stationary statistical solutions.

math.AP↗

arXiv2024

On the locally self-similar blowup for the generalized SQG equation

We analyze finite-time blowup scenarios of locally self-similar type for the inviscid generalized surface quasi-geostrophic equation (gSQG) in $\mathbb{R}^2$. Under an $L^r$ growth assumption on the self-similar profile and its gradient, we identify appropriate ranges of the self-similar parameter where the profile is either identically zero, and hence blowup cannot occur, or its $L^p$ asymptotic behavior can be characterized, for suitable $r, p$. Our results extend the work [Xue; Journal of Differential Equations, 2016] regarding the SQG equation, and also partially recover the results proved in [Cannone, Xue; Proceedings of the American Mathematical Society, 2015] concerning globally self-similar solutions of the gSQG equation.

math.AP↗

arXiv2014

On the energy behavior of locally self-similar blowup for the Euler equation

In this note we study locally self-similar blow up for the Euler equation. The main result states that under a mild $L^p$-growth assumption on the profile $v$, namely, $\int_{|y| \sim L} |v|^p dy \lesssim L^{\g}$ for some $\g <p-2$, the self-similar solution carries a positive amount of energy up to the time of blow-up $T$, namely, $\int_{|y| \sim L} |v|^2 dy \sim L^{N-2\a}$. The result implies and extends several previously known exclusion criteria. It also supports a general conjecture relating fractal local dimensions of the energy measure with the rate of velocity growth at the time of possible blowup.

math.AP↗

arXiv2014

Wild solutions for 2D incompressible ideal flow with passive tracer

In Ann. Math., 170 (2009), 1417-1436, C. De Lellis and L. Székelyhidi Jr. constructed wild solutions of the incompressible Euler equations using a reformulation of the Euler equations as a differential inclusion together with convex integration. In this article we adapt their construction to the system consisting of adding the transport of a passive scalar to the two-dimensional incompressible Euler equations.

math.AP↗

arXiv2013

Trajectory Statistical Solutions for three-dimensional Navier-Stokes-like systems

A general framework for the theory of statistical solutions on trajectory spaces is constructed for a wide range of equations involving incompressible viscous flows. This framework is constructed with a general Hausdorff topological space as the phase space of the system, and with the corresponding set of trajectories belonging to the space of continuous paths in that phase space. A trajectory statistical solution is a Borel probability measure defined on the space of continuous paths and carried by a certain subset which is interpreted, in the applications, as the set of solutions of a given problem. The main hypotheses for the existence of a trajectory statistical solution concern the topology of that subset of "solutions", along with conditions that characterize those solutions within a certain larger subset (a condition related to the assumption of strong continuity at the origin for the Leray-Hopf weak solutions in the case of the Navier-Stokes and related equations). The aim here is to raise the current theory of statistical solutions to an abstract level that applies to other evolution equations with properties similar to those of the three-dimensional Navier-Stokes equations. The applicability of the theory is illustrated with the Bénard problem of convection in fluids.

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