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DCDS

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.

DCDS

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.

DCDS

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

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.

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.

DCDS

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.

DCDS

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.

DCDS-B

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.

DCDS

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.

DCDS-B

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].

CPAA

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.

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