Tian Ma

Bifurcation theory and applications

# Introduction to Steady State Bifurcation Theory # Introduction to Dynamic Bifurcation # Reduction Procedures and Stability # Steady State Bifurcation # Dynamic Bifurcation Theory: Finite Dimensional Case # Dynamic Bifurcation Theory: Infinite Dimensional Case # Bifurcations for Nonlinear Elliptic Equations # Reaction-Diffusion Equations # Pattern Formation and Wave Equations # Fluid Dynamics

Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics

Introduction Structure classification of divergence-free vector fields Structural stability of divergence-free vector fields Block stability of divergence-free vector fields on manifolds with nonzero genus Structural stability of solutions of Navier-Stokes equations Structural bifurcation for one-parameter families of divergence-free vector fields Two examples Bibliography Index.

CAHN-HILLIARD EQUATIONS AND PHASE TRANSITION DYNAMICS FOR BINARY SYSTEMS

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.

DYNAMIC BIFURCATION OF NONLINEAR EVOLUTION EQUATIONS

The authors introduce a notion of dynamic bifurcation for nonlinear evolution equations, which can be called attractor bifurcation. It is proved that as the control 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 crossing the imaginary axis. The attractor bifurcation theory presented in this article generalizes the existing steady state bifurcations and the Hopf bifurcations. It provides a unified point of view on dynamic bifurcation and can be applied to many problems in physics and mechanics.

Dynamic phase transition theory in PVT systems

Department of Mathematics, Indiana University, Bloomington, IN 47405(Dated: February 4, 2008)The main objective of this article are two-fold. First, we introduce some general principles on phasetransition dynamics, including a new dynamic transition classi cation scheme, and a Ginzburg-Landau theory for modeling equilibrium phase transitions. Second, apply the general principlesand the recently developed dynamic transition theory to study dynamic phase transitions of PVTsystems. In particular, we establish a new time-dependent Ginzburg-Landau model, whose dynamictransition analysis is carried out. It is worth pointing out that the new dynamic transition theory,along with the dynamic classi cation scheme and new time-dependent Ginzburg Landau models forequilibrium phase transitions can be used in other phase transition problems, including e.g. theferromagnetism and superuidity, which will be reported elsewhere. In addition, the analysis forthe PVT system in this article leads to a few physical predications, which are otherwise unclearfrom the physical point of view.

Dynamic transition theory for thermohaline circulation

Abstract The main objective of this and its accompanying articles is to derive a mathematical theory associated with the thermohaline circulations (THC). This article provides a general transition and stability theory for the Boussinesq system, governing the motion and states of the large scale ocean circulation. First, it is shown that the first transition is either to multiple steady states or to oscillations (periodic solutions), determined by the sign of a nondimensional parameter K , depending on the geometry of the physical domain and the thermal and saline Rayleigh numbers. Second, for both the multiple equilibria and periodic solutions transitions, both Type-I (continuous) and Type-II (jump) transitions can occur, and precise criteria are derived in terms of two computable nondimensional parameters b 1 and b 2 . Associated with Type-II transitions are the hysteresis phenomena, and the physical reality is represented by either metastable states or by a local attractor away from the basic solution, showing more complex dynamical behavior. Third, a convection scale law is introduced, leading to an introduction of proper friction terms in the model in order to derive the correct circulation length scale. In particular, the dynamic transitions of the model with the derived friction terms suggest that the THC favors the continuous transitions to stable multiple equilibria. Applications of the theoretical analysis and results to different flow regimes and to more realistic models will be explored in the accompanying articles.

Bifurcation and stability of superconductivitya)

In this article, we present a bifurcation and stability analysis on time-dependent Ginzburg–Landau model of superconductivity. It is proved in particular that there are two different phase transitions from the normal state to superconducting states or vice versa: one is continuous, and the other is jump. These two transitions are precisely determined by a simple nondimensional parameter, which links the superconducting behavior with the geometry of the material, the applied field and the physical parameters. The rigorous analysis is conducted using a bifurcation theory newly developed by the authors, and provides some interesting physical predictions.

Dynamic model and phase transitions for liquid helium

This article presents a phenomenological dynamic phase transition theory—modeling and analysis—for liquid helium-4. First we derive a time-dependent Ginzburg–Landau model for helium-4 by (1) separating the superfluid and the normal fluid densities with the superfluid density given in terms of a wave function and (2) using a unified dynamical Ginzburg–Landau model. One the one hand, the analysis of this model leads to phase diagrams consistent with the classical ones for liquid helium-4. On the other hand, it leads to predictions of (1) the existence of a metastable region H, where both solid and liquid He II states are metastable and appear randomly depending on fluctuations and (2) the existence of a switch point M on the λ curve, where the transitions changes types. It is hoped that these predictions will be useful for designing better physical experiments and lead to better understanding of the physical mechanism of superfluidity.

Stratified rotating Boussinesq equations in geophysical fluid dynamics: Dynamic bifurcation and periodic solutions

The main objective of this article is to study the dynamics of the stratified rotating Boussinesq equations, which are a basic model in geophysical fluid dynamics. First, for the case where the Prandtl number is greater than 1, a complete stability and bifurcation analysis near the first critical Rayleigh number is carried out. Second, for the case where the Prandtl number is smaller than 1, the onset of the Hopf bifurcation near the first critical Rayleigh number is established, leading to the existence of nontrivial periodic solutions. The analysis is based on a newly developed bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) by two of the authors [T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Sciences Vol. 53 (World Scientific, Singapore, 2005)].

Phase separation of binary systems

In this paper, three physical predictions on the phase separation of binary systems are derived based on a dynamic transition theory developed recently by the authors. First, the order of phase transitions is precisely determined by the sign of a nondimensional parameter K such that if K>0, the transition is first order with latent heat and if K<0, the transition is second order. Here the parameter K is defined in terms of the coefficients in the quadratic and cubic nonlinear terms of the Cahn–Hilliard equation and the typical length scale of the container. Second, a phase diagram is derived, characterizing the order of phase transitions, and leading in particular to a prediction that there is only a second-order transition for molar fraction near 1/2. This is different from the prediction made by the classical phase diagram. Third, a TL-phase diagram is derived, characterizing the regions of both homogeneous and separation phases and their transitions.