Abstract&Title
About stability and instability results for 2D Euler equations
Claude Bardos (Universit e Denis Diderot Paris)
In the recent years many results have contributed to our understanding (or lack of understanding) of phenomena that may be described by the 2d incompressible Euler equation. This would include examples as diverse as the Jupiter red spot, the shape of the hurricanes or the wake behind wing. In this spirit I would like to compare the Arnold stability criteria, the stability or unstability of the wake behind a wing in relation either with the Kelvin Helmholtz instability or with the zero viscosity limit in presenceof boundary effectsof the solutionsof the Navierstokesequations.
Metastability results for the Navier-Stokes equations and related models
MicheleCoti-Zelati(Imperial College)
We study diffusion and mixing in the incompressible Navier-Stokes equations and related scalar models. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequency. In turn, mixing acts to enhance the dissipative forces, giving rise to what we refer to as enhanced dissipation: this can be understood by the identification of a time-scale faster than the purely diffusive one. Wewill present two results:
(1) a general quantitative criterion that links mixing rates (in terms of decay of negative Sobolev norms) to enhanced dissipation time-scales, with several applications including passive scalar evolution in both planar and radial settings, fractionaldiffusion, andAnosov flows.
(2) a precise identification of the enhanced dissipation time-scale for the Navier-Stokes equations linearized around the Poiseuille flow, along with metastabilityresults and nonlinear transitionstability thresholds.
Global well-posedness of viscous surface waves without surface tension revisited
GuilongGui (桂贵龙 Northwest University)
Consideration in this talk is a viscous fluid of finite depth below the air, occupying a three-dimensional domain bounded below by a fixed solid boundary and above by a free moving boundary. The fluid dynamics are governed by the gravity-driven incompressible Navier-Stokes equations ( without the effect of surface tension on the free surface), which results in the anisotropic decay or growth of the free surface. By using the balance between the anisotropic decay and growth estimates, we prove global well-posedness of the surface wave problem with small initialputerbation near equilibrium.
Decoupled finite element methods for the 3D primitive equations of ocean
Yinnian He (何银年 Xi’an JiaotongUniversity)
In this talk, two decoupled finite element methods are proposed for solving the 3D primitive equations of ocean. Based on the finite element approximation, optimal error estimates are given under the convergence condition. And the detailed algorithms are given in the section of numerical tests. Further, numerical calculations are implemented to validate theoretical analysis and more calculations are implemented for a more meaningful problem. For both theoretical and numerical points of view, the proposed decoupled finite element methods are the effective strategiesto solve the 3D primitive equationsof ocean.
On the mathematical analysis of the synchronization theory with time-delayed effect
Chun-HsiungHsia(夏俊雄 NationalTaiwanUniversity)
This is joint work with Bongsuk Kwon, Chang-Yeol Jung and Yoshihiro Ueda. We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and frustration effect. Both the phase synchronization and frequencysynchronization are inview.
Instability of the abstract Rayleigh--Taylor problem and applications
Fei Jiang (江飞 Fuzhou University)
We prove the existence of a unique unstable strong solution in the sense of $L^1$-norm for an abstract Rayleigh--Taylor (RT) problem of stratified viscous fluids in Lagrangian coordinates based on a bootstrap instability method. In the proof, we develop a method to modify the initial data of the linearized abstract RT problem based on an existence theory of unique solution of stratified (steady) Stokes problem and an iterative technique, so that the obtained modified initial data satisfy necessary compatibility conditions of the (original) abstract RT problem. Applying an inverse transformation of Lagrangian coordinates to the obtained unstable solution, and then taking proper value of parameters, we can further get unstable solutions for the RT problems in viscoelastic fluids, magnetohydrodynamics (MHD) fluids with zero resistivity and pure viscous fluids (with or without interface intension) in Eulerian coordinates. Our results can be also extended to the corresponding inhomogeneous case(without interface).
Semigroup and maximal regularity approach to the primitive equations
TakahitoKashiwabara(The University ofTokyo)
We review the results so far obtained for the primitive equations (PEs), which describe large-scale motion of ocean or atmosphere. Most of them are concerned with the global-in-time existence and uniqueness of a strong solution provided by an analytic semigroup approach or by a maximal-regularity theory. In particular, investigation of the linear part of the PEs, i.e., the \emph{hydrostatic Stokes operator}, has a central importance. We will first present the $L^p$-theory where the strong solution is constructed for initial data belonging to $H^{2/p,p}$. We show that the solution becomes $C^\infty$ (even real analytic) in $x$ and $t$ after initial time. Then the endpoint case $p = \infty$ (more precisely, an anisotropic space $L^\infty_{xy}L^p_z$ will be considered) is discussed, which requires more delicate arguments due to the lack of boundedness in $L^\infty$ of the hydrostatic Helmholtz projector. If time permits, justification of hydrostatic approximation in the $L^p$-setting, that is, convergence from the Navier--Stokes equations to the PEs in the zero aspect-ratio limit, will alsobe mentioned.
Some results on incompressible fluids with helical symmetry
Dongjuan Niu (Capital Normal University)
In this talk, I will present the lower bounds of the lifespan of the solutions to the incompressible Euler equations with helical symmetry. It is the first results to the three-dimensional incompressible helical flows with the nonzero helical swirl. It is a joint workwithA. Swierczewska.Gwiazda.
Analysis of oceanic and tropical atmospheric models with moisture: global regularity, finite-time blowup and singular limit behavior
Edriss S.Titi(TexasA&M University&TheWeizmann Institute of Science)
In this talk I will present some recent results concerning global regularity of certain geophysical models. This will include the three-dimensional primitive equations with various anisotropic viscosity and turbulence mixing diffusion, and certain tropical atmospheric models with moisture. Moreover, I will show that in the non-viscous (inviscid) case there is a one-parameter family of initial data for which the corresponding smooth solutions of the primitive equations develop finite-time singularities (blowup).
Capitalizing on the above results, I will also provide rigorous justification of the derivation ofthe Primitive Equations of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations as the vanishing limit of the small aspect ratio of the depth to horizontal width. Specifically, I will show that the Navier-Stokes equations, after being scaled appropriately by the small aspect ratio parameter of the physical domain, converge strongly to the primitive equations, globally and uniformly in time, and the convergence rate is of the same order as the aspect ratio parameter. Furthermore, I will also consider the singular limit behavior of a tropical atmospheric model with moisture, as ε→0, where ε> 0 is a moisture phase transition small convectiveadjustmentrelaxation time parameter.
On the boundary layer equations from the Navier-Stokes equations and MHD equations
WendongWang(王文栋 DalianUniversity ofTechnology)
In this talk, we will recall some known results on boundary layer equations from the Navier-Stokes equations and MHD equations including the local wellposedness theory and the vanishing viscosity limit analysis. We also introduce some recent developmentson thistopic.Thisis jointworkswithY.-J. Li,T.Tao and Z.-F. Zhang.
A deformation-curl decomposition to the steady Euler equations
ShangkunWeng (翁上昆 Wuhan University)
This talk will discuss the hyperbolic-elliptic coupled structure in steady compressible Euler equations. We will give a deformation-curl decomposition of 3D steady Euler system. This is based on the reformulation of the density equation by usingthe Bernoulli's law.Thisis jointwith Prof. Zhouping Xin.
The well-posedness of 2D Boussinesq system
XiaojingXu (许孝精 Beijing Normal University)
In this talk, I shall give some results on the global well-posedness of 2D Boussinesq equations with some kinds of dissipation terms, including the subcritical or critical fractional Laplacian, damping term, anisotropic dissipation and
viscosity dependingon temperature，and with othercaseson bounded domain.
On the free boundary problem of the non-isentropic compressible Naiver-Stokes equations in 3D
YuanYuan(袁源 South China Normal University)
In this talk we will introduce the strong solutions to the free boundary problem of the full compressible Navier-Stokes equations in three-dimensional space. The vanishing density and temperature condition is imposed on the free boundary, which captures the motions of the non-isentropic viscous gas surrounded by vacuum with bounded entropy. We also assume some proper decay rates of the density towards the boundary and singularities of derivatives of the temperature across the boundary on the initial data, which coincides with the physical vacuum condition for the isentropic flows.
Euler system is approximated by turbulence model with Dirichlet boundary conditions
AibinZang (臧爱彬 Yichun University)
In this talk, I start to present previous some results for the convergence of α-model, which is regularized Euler equations, to Euler equations. I will show that the existence of global weak solutions for inviscid Leray-α equations and obtain the solutions of Leray-α equations converges to the solutions for Euler equations by Kato corrector as α goes to zero.This work joint with Claude Bardos and Edirss S. Titi.
Convergence to equilibrium for the solution of the full compressible Navier-Stokes equations
Ruzhao Zi (訾瑞昭 Central China Normal University)
We study the convergence to equilibrium for the full compressible Navier-Stokes equations on the torus T3. Under the conditions that both the density ρ and the temperature θ possess uniform in time positive lower and upper bounds, it is shown that global regular solutions converge to equilibrium with exponential rate. We improve the previous result obtained by Villani in [Mem. Amer. Math. Soc., 202(2009), no. 950] on two levels: weaker conditions on solutions and faster decay rates.This is a jointwork withProf. Zhifei Zhang.