DCDS
Global structure of 2-D incompressible flows
Tian Ma Shouhong Wang
The main objective of this article is to classify the structure of divergence-free vector fields on general two-dimensional compact manifold with or without boundaries. First we prove a Limit Set Theorem, Theorem 2.1, a generalized version of the Poincaré-Bendixson to divergence-free vector fields on 2-manifolds of nonzero genus. Namely, the $\omega$ (or $\alpha$) limit set of a regular point of a regular divergence-free vector field is either a saddle point, or a closed orbit, or a closed domain with boundaries consisting of saddle connections. We call the closed domain ergodic set. Then the ergodic set is fully characterized in Theorem 4.1 and Theorem 5.1. Finally, we obtain a global structural classification theorem (Theorem 3.1), which amounts to saying that the phase structure of a regular divergence-free vector field consists of finite union of circle cells, circle bands, ergodic sets and saddle connections.
keywords: Compact manifold global structural classification theorem. Limit Set Theorem
DCDS
Tropical atmospheric circulations: Dynamic stability and transitions
Tian Ma Shouhong Wang
In this article, we present a mathematical theory of the Walker circulation of the large-scale atmosphere over the tropics. This study leads to a new metastable state oscillation theory for the El Niño Southern Oscillation (ENSO), a typical inter-annual climate low frequency oscillation. The mathematical analysis is based on 1) the dynamic transition theory, 2) the geometric theory of incompressible flows, and 3) the scaling law for proper effect of the turbulent friction terms, developed recently by the authors.
keywords: tropical atmospheric circulation dynamic transition Walker circulation metastable states. new mechanism of ENSO El Niño Southern Oscillation (ENSO)
DCDS
Unified field equations coupling four forces and principle of interaction dynamics
Tian Ma Shouhong Wang
The main objective of this article is to postulate a principle of interaction dynamics (PID) and to derive field equations coupling the four fundamental interactions based on first principles. PID is a least action principle subject to div$_A$-free constraints for the variational element with $A$ being gauge potentials. The Lagrangian action is uniquely determined by 1) the principle of general relativity, 2) the $U(1)$, $SU(2)$ and $SU(3)$ gauge invariances, 3) the Lorentz invariance, and 4) the principle of representation invariance (PRI), introduced in [11]. The unified field equations are then derived using PID. The field model spontaneously breaks the gauge symmetries, and gives rise to a new mass generation mechanism. The unified field model introduces a natural duality between the mediators and their dual mediators, and can be easily decoupled to study each individual interaction when other interactions are negligible. The unified field model, together with PRI and PID applied to individual interactions, provides clear explanations and solutions to a number of outstanding challenges in physics and cosmology, including e.g. the dark energy and dark matter phenomena, the quark confinement, asymptotic freedom, short-range nature of both strong and weak interactions, decay mechanism of sub-atomic particles, baryon asymmetry, and the solar neutrino problem.
keywords: unified field equations dual particle fields solar neutrino problem Higgs mechanism dark energy short-range nature of strong and weak interactions baryon asymmetry. Principle of interaction dynamics (PID) fundamental interactions asymptotic freedom quark confinement principle of representation invariance gauge symmetry breaking dark matter
DCDS-B
Structure of 2D incompressible flows with the Dirichlet boundary conditions
Tian Ma Shouhong Wang
We study in this article the structure and its stability of 2-D divergence-free vector fields with the Dirichlet boundary conditions. First we classify boundary points into two new categories: $\partial$−singular points and $\partial$−regular points, and establish an explicit formulation of divergence-free vector fields near the boundary. Second, local orbit structure near the boundary is classified. Then a structural stability theorem for divergence-free vector fields with the Dirichlet boundary conditions is obtained, providing necessary and sufficient conditions of a divergence-free vector fields. These structurally stability conditions are extremely easy to verify, and examples on stability of typical flow patterns are given.
The main motivation of this article is to provide an important step for a forthcoming paper, where, for the first time, we are able to establish precise rigorous criteria on boundary layer separations of incompressible fluid flows, a long standing problem in fluid mechanics.
keywords: structural stability Divergence-free vector fields Dirichlet boundary conditions.
DCDS
Asymptotic structure for solutions of the Navier--Stokes equations
Tian Ma Shouhong Wang
We study in this article the large time asymptotic structural stability and structural evolution in the physical space for the solutions of the 2-D Navier-Stokes equations with the periodic boundary conditions. Both the Hamiltonian and block structural stabilities and structural evolutions are considered, and connections to the Lyapunov stability are also given.
keywords: divergence-free vector fields Navier-Stokes equations Lyapunov stability. periodic boundary conditions structural stability block stability Hamiltonian stability
DCDS
Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows
Tian Ma Shouhong Wang
The main objective of this article and the previous articles [2, 3, 7] is to provide a rigorous characterization of the boundary layer separation of 2-D incompressible viscous fluids. First we establish a simple equation linking the separation location and time with the Reynolds number, the external forcing the boundary curvature, and the initial velocity field. Second, we show that external forcing with reverse orientation to the initial velocity field leads to structural bifurcation at a degenerate singular point with integer index of the velocity field at the critical bifurcation time. Necessary and sufficient kinematic conditions are given to identify the case for boundary layer separation.
keywords: location and time of separation kinematic conditions. Navier-Stokes equations structural bifurcation boundary layer separation
DCDS-B
Cahn-Hilliard equations and phase transition dynamics for binary systems
Tian Ma Shouhong Wang
The process of phase separation of binary systems is described by the Cahn-Hilliard equation. The main objective of this article is to give a classification on the dynamic phase transitions for binary systems using either the classical Cahn-Hilliard equation or the Cahn-Hilliard equation coupled with entropy, leading to some interesting physical predictions. The analysis is based on dynamic transition theory for nonlinear systems and new classification scheme for dynamic transitions, developed recently by the authors.
keywords: dynamic transition theory unified time-dependent Ginzburg-Landau equation. binary system Cahn-Hilliard equation phase transition dynamics dynamic classification scheme of phase transitions
DCDS
Gravitational Field Equations and Theory of Dark Matter and Dark Energy
Tian Ma Shouhong Wang
The main objective of this article is to derive new gravitational field equations and to establish a unified theory for dark energy and dark matter. The gravitational field equations with a scalar potential $\varphi$ function are derived using the Einstein-Hilbert functional, and the scalar potential $\varphi$ is a natural outcome of the divergence-free constraint of the variational elements. Gravitation is now described by the Riemannian metric $g_{\mu\nu}$, the scalar potential $\varphi$ and their interactions, unified by the new field equations. From quantum field theoretic point of view, the vector field $\Phi_\mu=D_\mu \varphi$, the gradient of the scalar function $\varphi$, is a spin-1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin-2 massless bosonic particle) and this spin-1 massless bosonic particle. Both particles can be considered as gravitational force carriers, and as they are massless, the induced forces are long-range forces. The (nonlinear) interaction between these bosonic particle fields leads to a unified theory for dark energy and dark matter. Also, associated with the scalar potential $\varphi$ is the scalar potential energy density $\frac{c^4}{8\pi G} \Phi=\frac{c^4}{8\pi G} g^{\mu\nu}D_\mu D_\nu \varphi$, which represents a new type of energy caused by the non-uniform distribution of matter in the universe. The negative part of this potential energy density produces attraction, and the positive part produces repelling force. This potential energy density is conserved with mean zero: $\int_M \Phi dM=0$. The sum of this potential energy density $\frac{c^4}{8\pi G} \Phi$ and the coupling energy between the energy-momentum tensor $T_{\mu\nu}$ and the scalar potential field $\varphi$ gives rise to a unified theory for dark matter and dark energy: The negative part of this sum represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. In addition, the scalar curvature of space-time obeys $R=\frac{8\pi G}{c^4} T + \Phi$. Furthermore, the proposed field equations resolve a few difficulties encountered by the classical Einstein field equations.
keywords: gravitational eld equations spin-1 massless particle spin-2 massless graviton gravitational force formula. scalar potential energy dark matter Dark energy
DCDS-B
Block structure and block stability of two-dimensional incompressible flows
Tian Ma Shouhong Wang
We study in this article topological structure of divergence-free vector fields on general two-dimensional manifolds. We introduce a new concept called block structural stability (or block stability for simplicity) and prove that the block stable divergence-free vector fields form a dense and open set. Furthermore, we show that a block stable divergence-free vector field, which we call a basic vector field, is fully characterized by a nice and simple structure, which we call block structure. The results and ideas presented in this article have been applied to studies on structure and its evolutions of the solutions of the Navier-Stokes equations; see [4, 9, 10].
keywords: Divergence-free vector fields block structural stability ergodic sets structural classification. block structure
CPAA
Attractor bifurcation theory and its applications to Rayleigh-Bénard convection
Tian Ma Shouhong Wang
In this note, we present a fast communication of a new bifurcation theory for nonlinear evolution equations, and its application to Rayleigh-Bénard Convection. The proofs of the main theorems presented will appear elsewhere. The bifurcation theory is based on a new notion of bifurcation, called attractor bifurcation. We show that as the parameter crosses certain critical value, the system bifurcates from a trivial steady state solution to an attractor with dimension between $m$ and $m+1$, where $m+1$ is the number of eigenvalues crosses the imaginary axis. Based on this new bifurcation theory, we obtain a nonlinear theory for bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Bénard convection. In particular, we show that the problem bifurcates from the trivial solution an attractor $\mathcal A_R$ when the Rayleigh number $R$ crosses the first critical Rayleigh number $R_c$ for all physically sound boundary conditions.
keywords: asymptotic stability central manifold theory Attractor bifurcation semigroup roll structure. structural stability Rayleigh-Bénard convection

Year of publication

Related Authors

Related Keywords

[Back to Top]