召集人：谢春景（上海交通大学）、章志飞（北京大学数学科学学院）

时间：2024.11.24—2024.11.30

Workshop on “Mixing, enhanced dissipation and stability effects in fluid dynamics” (Nov 24- Nov 30)

Conveners: Chunjing Xie, Zhifei Zhang

Talk title and abstract:

Speaker: Ken Abe, Osaka Metropolitan University

Title: Stationary self-similar profiles for the two-dimensional inviscid Boussinesq equations

Abstract: We consider homogeneous (stationary self-similar) solutions (of degree ) for the two-dimensional inviscid Boussinesq equations in a half-plane. We show their non-existence and existence with both regular and singular profile functions. More specifically, we demonstrate:

(i)  Non-existence of rotational homogeneous solutions with regular profiles for ,

(ii)  Existence of rotational homogeneous solutions with regular profiles for and ,

(iii)  Existence of rotational homogeneous solutions with -symmetric singular profiles for .

Speaker: Jeaheang Bang, Westlake University

Title: Self-Similar Solutions to the Stationary Navier-Stokes Equations in a Two-Dimensional Sector.

Abstract: Self-similar solutions of the stationary Navier-Stokes equations are useful to study the asymptotic behavior of a general solution at infinity. We investigated self-similar solutions in a two-dimensional sector with the no-slip boundary condition. We found necessary and sufficient conditions for the existence of a self-similar solution in terms of the angle of the sector and the flux. In addition, we established the uniqueness and non-uniqueness of flows with a given type. As an application, we identified the leading order term of a solution to the stationary Navier-Stokes equations in an aperture domain when the flux is small. The main idea is to study the reduced ODE system and use properties of both complete and incomplete elliptic functions. This is a joint work with Changfeng Gui, Chunjing Xie, Yun Wang and Hao Liu.

Speaker: Alexey Cheskidov, Westlake University

Title: Energy cascade in fluids: from convex integration to mixing

Abstract: In the past couple of decades, mathematical fluid dynamics has made significant strides with numerous constructions of solutions to fluid equations that exhibit pathological or wild behaviors. These include the loss of the energy balance, non-uniqueness, singularity formation, and dissipation anomaly. Interesting from the mathematical point of view, providing counterexamples to various well-posedness results in supercritical spaces, such constructions are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, it has been observed both experimentally and numerically, but had been difficult to produce analytically. In this talk I will overview new developments in discovering not only pathological mathematically, but also physically realistic solutions of fluid equations. I will focus on dissipation anomaly for viscous fluid flows as well as anomalous dissipation for the limiting inviscid flows. These two intrinsically linked laws of turbulence are postulated by Kolmogorov and Onsager’s empirical theories built on the persistence of the energy flux through the inertial range. I will first analyze these phenomena on a finite time interval and prove the existence of various scenarios in the limit of vanishing viscosity, ranging from the total dissipation anomaly to a pathological one where inviscid anomalous dissipation occurs without viscous dissipation anomaly, as well as the existence of infinitely many limiting solutions of the Euler equations in the limit of vanishing viscosity. Finally, I will show the existence of dissipation anomaly for long time averages, relevant for turbulent flows, proving that the Doering-Foias upper bound is sharp.

Speaker: Mimi Dai, University of Illinois at Chicago

Title: Onsager conjecture for SQG

Abstract: The Hamiltonian of the surface quasi-geostrophic (SQG) equation is an invariant quantity for regular enough solutions. It is postulated that the critical Holder regularity required to have the Hamiltonian conserved is , known as the Onsager type of conjecture for SQG. We give a proof of this conjecture using a two-step scheme of convex integration. This is joint work with Vikram Giri and Razvan-Octavian Radu.

Speaker: De Huang, Peking University

Title：On self-similar finite-time blowups of the incompressible Euler equations and related models.

Abstract：It remains an open problem whether the 3D incompressible Euler equations can develop finite-time singularity from smooth initial data in the whole space. In this talk, I will review some most recent results on finite-time blowups with self-similar features, divided into three parts. In the first part, I will talk about the dynamic rescaling method for establishing asymptotically self-similar finite-time blowup with rigorous computer-assisted proof. In the second part, I will introduce the constructions of exact self-similar blowup solutions for some simple models of the 3D Euler equations based on the fixed-point method. In the last part, I will talk about recent findings on potential self-similar finite-time blowups of the 3D Euler equations with multi-scale features, which are closely related to traveling wave solutions and may provide a new approach towards Euler singularity.

Speaker: In-Jee Jeong, Seoul National University

Title: On the optimal rate of vortex stretching for axisymmetric Euler flows without swirl

Abstract: For axisymmetric flows without swirl, we prove the upper bound of  for the growth of the vorticity maximum, which was conjectured by Childress [Phys. D, 2008] and supported by numerical computations by Childress--Gilbert--Valiant [J. Fluid Mech. 2016]. This is joint work with Deokwoo Lim (SNU).

Speaker: Ning Jiang, Wuhan University

Title: Recent progress on kinetic boundary layer equations and applications to fluid limits

Abstract: We review recent progress on well-posedness of the Knudsen boundary layer equations, including Maxwell reflection and absorbing boundary conditions. Our results are valid for all range of accommodation coefficients and moving boundaries. These results could be applied to the fluid limits from kinetic equations, such as compressible Euler and compressible Navier-Stokes. We also will list many open problems involving both kinetic and fluid boundary layers arising fluid limits of Boltzmann equations.  

Speaker: Baishun Lai, Hunan Normal University

Title: Localization analysis of incompressible 3D Navier-Stokes equation and its application

Abstract: In this talk, I will introduce the local-in-space smoothing technique near the initial time established by Jia-Sverak， and the localized argument in frequency space given by Terence Tao. At the same time,  I will present some applications of these techniques , such as  quantitative regularity for the Navier-Stokes equations via Carleman inequalities, construction of large self-similar solution of Navier-Stokes by Leary Shaulder fixed point Theorem.

Speaker: Te Li, Chinese Academy of Sciences

Title: Linear enhanced dissipation for the 2D Taylor-Couette flow in the exterior region: A supplementary example for Gearhart-Prüss type lemma

Abstract: In this paper, we show that for the 2D TC flow in the exterior region, it is not possible to expect enhanced dissipation estimates similar to the time pointwise estimate and the space-time estimate. The reason is that derivatives of any order of the TC flow degenerate in the exterior region.

Speaker: Zhiwu Lin, Fudan University

Title: Sharp instability criteria and nonlinear dynamics of zonal flows on a sphere
Abstract: The classical Rayleigh criterion only gives a necessary condition for the linear instability of a zonal flow on a sphere. Noting that the 1, 2-jet zonal flows are stable, the 3-jet flow is the first jet to experience instability. In this talk, we present the sharp linear stability/instability criteria for the 3-jet flow, from which it turns out that Rayleigh's criterion is far from being sufficient. This confirms the previous numerical computations and gives an accurate description for the critical rotation rates. We also discuss the mechanisms that induce instability. Then we show that linear instability implies nonlinear instability for general steady flows. For any non-zonal Rossby-Haurwitz wave, we further show that linear instability implies nonlinear orbital instability. Finally, we construct non-zonal travelling wave solutions near the 3-jet flow, which are not discrete but form curves.

Speaker: Hao Liu, Shanghai Jiaotong University

Title: Rigidity of Steady Solutions to the Navier-Stokes Equations in High Dimensions and its applications

Abstract: Solutions with scaling-invariant bounds such as self-similar solutions, play important roles in the understanding of the regularity and asymptotic structures of solutions to the Navier-Stokes equations.

In this talk, we will discuss some recent rigidity results about the solutions satisfying some scaling-invariant bounds to the steady Navier-Stokes equations.Then we talk about some applications of these rigidity results such as the regularity and the leading order behaviors of the solutions in the exterior domain. This talk is based on the joint work with Jeaheang Bang, Changfeng Gui, Yun Wang and Chunjing Xie.

Speaker: Shuang Miao, Wuhan University

Title: Well-posedness for fluid free boundary problems in general relativity

Abstract: In the framework of general relativity, the dynamics of a barotropic fluid are coupled to the Einstein equations, which govern the structure of the underlying spacetime. The free boundary problem of such models can describe the motion of isolated bodies in universe and appears in many physical contexts. In this talk I shall present our recent progress on well-posedness of such free boundary problems in Sobolev spaces. This talk is based on joint works with Sohrab Shahshahani, Zeming Hao and Wei Huo.

Speaker: Feng Shao, Peking University

Title: On global well-posedness of inhomogeneous Navier-Stokes equations with bounded density allowing vacuum

Abstract: In this talk, I will discuss the joint work with Tiantian Hao, Dongyi Wei and Zhifei Zhang on the global well-posedness of inhomogeneous Navier-Stokes equations (INS) with bounded density. We solve Lions' density patch problem (without compatibility conditions) about the preseving of boundary regularity of a density patch and Lions' open problem on the 2-D uniqueness of weak solutions. Moreover, we extend Leray's 2-D global well-posedness result of weak solutions in on the classical Navier-Stokes equations (NS) to (INS) (in which case our uniqueness requires the positive lower bound of the density), and we also extend Fujita-Kato's celebrated result on the global well-posednesss of (NS) in  to (INS). Particularly, our proof of uniqueness is based on a duality argument and a surprising finding that the estimate instead of  is enough to ensure the uniqueness of the solution.

Speaker: Jiajun Tong, Peking University

Title: Steady Contiguous Vortex-Patch Dipole Solutions of the 2D Incompressible Euler Equation

Abstract: It is of great mathematical and physical interest to study traveling wave solutions to the 2D incompressible Euler equation in the form of a touching pair of symmetric vortex patches with opposite signs. Such a solution was numerically illustrated by Sadovskii in 1971, but its rigorous existence was left as an open problem. In this talk, we will rigorously construct such a solution by a novel fixed-point approach that determines the patch boundary as a fixed point of a nonlinear map. Smoothness and other properties of the patch boundary will also be characterized. This is based on a joint work with De Huang.

Speaker: Yun Wang, Soochow University

Title: Structural stability of Poiseuille flows

Abstract: In this talk, I will list some recent results on the structural stability of Poiseuille flows. We study the  flows in a channel and prove the existence and local uniqueness of periodic solutions of arbitrary period and arbitrary flux to the Navier-Stokes equations. This is a joint work with Chunjing Xie and Kaijian Sha.

Speaker: Di Wu, South China University of Technology

Title: Linear instability of compressible boundary layer flow

Abstract: Understanding the transition mechanism of boundary layer flows is of great significance in physics and engineering, especially due to the current development of supersonic and hypersonic aircraft. In this talk, we show some results about the linear instability of boundary layer flow: 1. We construct the unstable Tollmien-Schlichting waves of both temporal and spatial mode to the linearized compressible Navier-Stokes system around the boundary layer flow in the whole subsonic regime. 2. We construct multiple unstable acoustic modes so called Mack modes, which plays a crucial role during the early stage of transition in the supersonic boundary layer

Speaker: Xiaojing Xu, Beijing Normal University

Title: On the Sobolev stability threshold for 3D Navier-Stokes equations with rotation near the Couette flow

Abstract: In this talk, we investigate the dynamic stability of periodic, plane Couette flow in the three-dimensional Navier-Stokes equations with rotation at high Reynolds number . Our aim is to determine the stability threshold index on : the maximum range of perturbations within which the solution

remains stable. Initially, we examine the linear stability effects of a linearized perturbed system. Comparing our results with those obtained by Bedrossian, Germain, and Masmoudi [Ann.  Math. 185(2): 541–608 (2017)], we observe that mixing effects (which correspond to enhanced dissipation and inviscid damping) arise from Couette flow while Coriolis force acts as a restoring force inducing a dispersion mechanism for inertial waves that cancels out lift-up effects occurred at zero frequency velocity. This dispersion mechanism exhibits favorable algebraic decay properties distinct from those observed in classical 3D Navier-Stokes equations. Consequently, we demonstrate that if initial data satisfies for any  and some  depending only on  , then the solution to the 3D Navier-Stokes equations with rotation is  global in time without transitioning away from Couette flow. In this sense, Coriolis force contributes as a factor enhancing fluid stability by improving its threshold from  to 1. This is a joint work with Sun Ying and Huang Wenting.

Speaker: Yao Yao, National University of Singapore

Title: Stability of vortex quadrupoles with odd-odd symmetry

Abstract: In this talk, I will discuss a recent result on the 2D incompressible Euler equation, giving global-in-time stability of vortex quadrupoles satisfying odd symmetry with respect to both axes. Specifically, if the vorticity restricted to a quadrant is signed, sufficiently concentrated and close to its radial rearrangement up to a translation in , we prove that it remains so for all times. Moreover, with a similar strategy we obtain stability of a pair of opposite-signed Lamb dipoles moving away from each other. (joint work with Kyudong Choi and In-Jee Jeong).

Speaker: Ruizhao Zi, Central China Normal University

Title: Stability of Couette flow in Stokes-transport equations

Abstract: In this talk, we will present some resent progress on the asymptotic stability of Couette flow for 2D and 3D Stokes-transport equations in stratified fluids. In the two dimensional case, the boundary is involved, and in the three dimensional case, the lift-up effect is treated carefully. This talk is based on joint works with Daniel Sinabela and Weiren Zhao.