Special Session 128: New Trends in Mathematical Fluid Dynamics and Related Problems

Quantitative classification of potential Navier-Stokes singularities beyond the blow-up time

Tobias Barker University of Bath England Co-Author(s):

Abstract: It remains an open problem whether or not solutions to the 3D Navier-Stokes equations with smooth data remain smooth for all time. All previously known regularity criteria are formulated in times of a blow-up time (where the solution loses smoothness), which make it practically impossible to use such necessary conditions to test the viability of certain numerically computed candidates. Motivated by these issues, we give the first quantitative classification of potentially singular solutions at any given time in the region of potential blow-up times. The quantitative lower bounds prior to any potential blow-up time (and in the open vicinity of it) are in principle amenable to numerical testing.

Dynamics of oscillating water columns in the shallow water regime

Edoardo Bocchi Politecnico di Milano Italy Co-Author(s):

Abstract: An oscillating water column is a wave energy converter in which waves enter a partially closed chamber, producing variations in the air volume inside, which, in turn, activate a turbine that generates electrical energy. In this talk, I will discuss the dynamics within such a configuration in the one-dimensional horizontal shallow water regime, modeled either by congested fluid models or by a spring-mass system inside the chamber interacting with the external waves.

Rotating soluitons near monotone vortices

Angel Castro ICMAT Spain Co-Author(s):    Daniel Lear

Abstract: In this talk we will consider the existence of rotating solutions arbitrarily close (in some topology) to radial monotone decreasing vorticity for 2D Euler. In a paper by Bedrossian, Coti-Zelati and Vicol was shown that radial monotone decreasing vorticities are stable at the linear level, thus, our result shows that this phenomenon can break even for small perturbation. The problem is related with the stability of shear flows and the existence of stationary and traveling waves solution near them. We also review some results on this topic. This is a joint work with Daniel Lear.

Non-uniqueness of solutions to the Navier-Stokes and related equations

Alexey Cheskidov Westlake University Peoples Rep of China Co-Author(s):

Abstract: I will overview some resent results on non-uniqueness of solutions to the Navier-Stokes and related equations. I will describe a recent instantaneous blow-up construction, where for any smooth, divergence-free initial data, we construct a solution of the Navier--Stokes equations that exhibits Type I blow-up of the L^\infty norm, while remaining smooth. An instantaneous injection of energy from infinite wavenumber initiates a bifurcation from the classical solution, producing an infinite family of spatially smooth solutions with the same data and thereby violating uniqueness of the Cauchy problem. A key ingredient is the first known construction of a complete inverse energy cascade realized by a classical Navier-Stokes flow, which transfers energy from infinitely high to low frequencies. This is a joint work with Mimi Dai and Stan Palasek.

Instantaneous blowup for NSE

Mimi Dai University of Illinois at Chicago USA Co-Author(s):    Alexey Cheskidov, Stan Palasek

Abstract: The quest of blowup solutions to the NSE remains a challenging topic. We will discuss a recent construction that illustrates a blowup phenomenon unexplored previously for the NSE. In particular, the constructed solutions start from smooth initial data and exhibit an instantaneous blowup saturating Type-I blowup rate at a finite time. Moreover, there are infinitely many such blowup solutions; and the non-uniqueness occurs in borderline spaces of known criteria which ensure uniqueness. This is joint work with Alexey Cheskidov and Stan Palasek.

Global well-posedness of a reduced model for micropolar fluids

Francesco Fanelli Basque Center for Applied Mathematics Spain Co-Author(s):

Abstract: In this talk, we introduce a reduced model for micropolar fluids. By resorting to a suitable ``good unknown`` for the system, we show global persistence of regularity for smooth initial data, as well as global well-posedness for Yudovich-type initial data. This talk is based on joint works with Pedro Fern\`andez-Dalgo (Basque Center for Applied Mathematics) and Mar\`ia Eugenia Mart\`inez Martini (Universidad de Chile).

Turbulent Dynamos in Bounded Domains

DAniel Faraco Universidad Autonoma de Madrid Spain Co-Author(s):    Giacomo del Nin, Francisco Mengual and Sauli Lindberg

Abstract: I will explain how an H principle obtained by simple convex integration scheme yields turbulent dynamos in arbitrary smooth bounded domains. The dynamo action happens simultaenously at various scales and can be made to be consistents with Taylor conjecture.

Desingularization of vortex sheets for the Euler equations

Antonio J. Fern\`andez Universidad Aut\`onoma de Madrid Spain Co-Author(s):

Abstract: In this talk we show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff-Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $\omega^0_{\mathrm{sing}}$, which is a signed Radon measure supported co-dimension one manifold. We construct a family of initial vorticities $\omega^0_\varepsilon \in C^\infty_c(\mathbb{R}^n)$, $n = 2,3$, converging to $\omega^0_{\mathrm{sing}}$ distributionally as $\varepsilon \to 0^+$, and show that the corresponding solutions $\omega_\varepsilon$ to the incompressible Euler equations (n = 2,3) converge to the measure defined by the Birkhoff-Rott system with initial datum $\omega^0_{\mathrm{sing}}$. The talk is based on joint works with A. Enciso and D. Meyer.

Stable and unstable interface dynamics for gravity Stokes flow

Francisco Gancedo Universidad de Sevilla Spain Co-Author(s):

Abstract: We study the evolution of the interface between two incompressible fluids with different densities in the Stokes regime under the effect of gravity. We analyze global-in-time regularity, stability, and instability results.

Leapfrogging motion for the Euler equations. Part 1.

Claudia Garcia Universidad de Granada Spain Co-Author(s):    Z. Hassainia, T. Hmidi

Abstract: Leapfrogging motion describes a phenomenon in which two or more vortex rings, or other entities such as point vortices, repeatedly pass through each other. This occurs because the induced flow generated by one vortex ring causes it to accelerate and move through the other, after which the roles reverse. The resulting motion is a time-periodic solution in certain fluid models. Although this behavior has long been observed in experiments and numerical simulations, a rigorous mathematical construction has remained elusive. In this talk, we will first review some known results on leapfrogging motion for point vortices and vortex rings, and then present new analytical approaches to describe such phenomena.

Nonlinear stability of the three-dimensional Peskin problem

Eduardo Garcia-Juarez Universidad de Sevilla Spain Co-Author(s):    S.V. Haziot, P.-C. Kuo, Y. Mori, H. Zhou

Abstract: The Peskin problem describes the motion of an elastic membrane immersed in an incompressible, viscous fluid, typically governed by the Stokes equations. The problem admits a contour dynamics formulation that leads to a nonlinear, nonlocal parabolic PDE. We study the well-posedness in critical spaces and the long-time behaviour of two-dimensional membranes in a three-dimensional fluid for initial interfaces with Lipschitz regularity, which is critical with respect to the natural scaling of the equation.

Leapfrogging motion for the Euler equations. Part 2

Zineb Hassainia University of Granada Spain Co-Author(s):    Claudia Garc\`ia and Taoufik Hmidi

Abstract: In this talk, we present a rigorous construction of time-periodic leapfrogging vortex rings for the three-dimensional incompressible Euler equations. More precisely, we prove the existence of solutions in which two coaxial vortex rings periodically exchange positions, as observed in experiments and numerical simulations. The construction relies on a desingularization of two interacting vortex filaments within the contour dynamics formulation, leading to a Hamiltonian description of nearly concentric vortex rings. A central difficulty arises from a singular small-divisor problem in the linearized dynamics, where the effective time scale degenerates with the ring thickness parameter. We overcome this issue by combining a degenerate KAM-type analysis with pseudo-differential techniques and a Nash-Moser iteration scheme. This approach yields families of nontrivial time-periodic solutions in an almost uniformly translating frame and thereby provides a rigorous mathematical construction of classical leapfrogging motion for axisymmetric Euler flows without swirl, with no restriction on the time interval of existence.

Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces

Antonio Hidalgo Torne Max Planck Institute for Mathematics in the Sciences Germany Co-Author(s):    Jan Burczak

Abstract: For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the 2D Navier-Stokes equations, with their gradients in the Hardy space $\mathcal{H}^p$ with any $p\in (0,1)$. Thus, in terms of the path space $C(\mathcal{H}^p)$ for vorticity, $p=1$ is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains.

Existence of the Sadovskii vortex patch

De Huang Peking University Peoples Rep of China Co-Author(s):    Jiajun Tong

Abstract: The Sadovskii vortex patch--a steady contiguous anti-symmetric vortex-patch dipole solution of the 2D incompressible Euler qquation--was numerically discovered over 50 years ago, whose was also observed as an accurate approximation of the large-time asymptotic profile in the head-on collision of two anti-symmetric vortex rings. In this talk, we present the first proof of existence of the Sadovskii vortex patch with 90-degree touching angles via a fixed-point approach. In particular, we show that the upper boundary of the Sadovskii vortex patch is given by a smooth even function that is monotonic on one side.

Global weak solutions and Hamiltonian conservation for the SQG equation

Jaemin Park Yonsei University Korea Co-Author(s):    Luigi De Rosa, Mickael Latocca

Abstract: In this talk, I will discuss existence of weak solutions to the SQG equation with a rough initial data. We prove that when the initial data is has weak integrability, there exists a global weak solutions to the viscous SQG equation and the vanishing viscosity limit solves the inviscid SQG equation. We will also discuss Hamiltonian conservation of such weak solutions. The proof is based on an application of Lions` concentration compactness principle. This is a joint work with Luigi De Rosa (GSSI) and Mickael Latocca (Univ. Evry)

Self-Similar Solutions to the Hele-Shaw Problem with Surface Tension

Neel Patel University of Maine USA Co-Author(s):    Siddhant Agrawal

Abstract: The Hele-Shaw problem models the dynamics of the interface of a single viscous fluid domain in porous media. While the dynamics around a corner on the fluid interface are known in the absence of surface tension, less is known with the presence of surface tension. We demonstrate the existence of self-similar solutions that initially have a corner, but instantaneously smoothen out. Due to surface tension, the differential equation describing the self-similar solution is a third order nonlocal equation of elliptic type with coefficients that grow at infinity, and thus, requires an interesting linear analysis.

Small amplitude steady periodic water waves

Silvia Sastre Gómez Universidad de Sevilla Spain Co-Author(s):    David Henry

Abstract: In this talk, we apply bifurcation theory to prove the existence of small amplitude steady periodic water waves, which propagate over a flat bed with a specified fixed mean-depth, and where the underlying flow has a discontinuous vorticity distribution.

Unstable vortices and sharp nonuniqueness for the forced SQG equation

Marcos Solera Diana University of Valencia Spain Co-Author(s):    A. Castro, D. Faraco and F. Mengual

Abstract: We present a non-uniqueness result for the forced SQG equation in supercritical Sobolev spaces. A key step is the construction of smooth, compactly supported vortices that exhibit nonlinear instability. Moreover, we extend this approach to the forced 2D Navier-Stokes and dissipative SQG equations. This is joint work with A. Castro, D. Faraco and F. Mengual.

The Immersed Boundary Problem in 2-D: the Navier-Stokes Case

Jiajun Tong Beijing International Center for Mathematical Research, Peking University Peoples Rep of China Co-Author(s):    Jiajun Tong, Dongyi Wei

Abstract: We will report recent progress on the 2-D immersed boundary problem with the Navier-Stokes equation, which models coupled motion of a 1-D closed elastic string and ambient fluid in the entire plane. This is based on joint works with Dongyi Wei.