Songmu Zheng

Global solutions to the equations of a Ginzburg-Landau theory for structural phase transitions in shape memory alloys

Abstract Global existence of smooth solutions is proved for the system of partial differential equations governing the dynamics of martensic phase transitions in shape memory alloys. The free energy is assumed in Ginzburg-Landau form and nonconvex in the order parameter.

Asymptotic behavior of the solutions to a Landau-Ginzburg system with viscosity for martensitic phase transitions in shape memory alloys

In this paper, we investigate the system of partial differential equations governing the dynamics of martensitic phase transitions in shape memory alloys under the presence of a (possibly small) viscous stress. The corresponding free energy is assumed in Landau--Ginzburg form and nonconvex as a function of the order parameter. Results concerning the asymptotic behavior of the solution as time tends to infinity are proved, and the compactness of the orbit is shown.

CONVERGENCE TO EQUILIBRIUM FOR A PARABOLIC–HYPERBOLIC PHASE-FIELD SYSTEM WITH NEUMANN BOUNDARY CONDITIONS

This paper is concerned with the asymptotic behavior of global solutions to a parabolic–hyperbolic coupled system which describes the evolution of the relative temperature θ and the order parameter χ in a material subject to phase transitions. For the system with homogeneous Neumann boundary conditions for both ¸ and χ, under the assumption that the nonlinearities λ and ϕ are real analytic functions, we prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Łojasiewicz–Simon type inequality.

Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys

Abstract In this paper, a system of nonlinear partial differential equations modelling the dynamics of martensitic phase transitions in shape memory alloys is further investigated. In a previous paper published in SIAM J. Math. Anal., the global existence of a unique solution and the asymptotic behaviour of the solution as time goes to infinity have been established; in the present paper the existence of a compact maximal attractor is proved.

Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition

This paper is concerned with the asymptotic behavior of the solution to the following damped semilinear wave equation with critical exponent: u tt + u t - Δu + f(x, u) = 0, (x,t) ∈ Ω × R + (0.1) subject to the dissipative boundary condition ∂ ν u+u+u t = 0, t > 0, x ∈ Γ (0.2) and the initial conditions u| t=0 = u 0 (x), u t | t=0 = u 1 (x), x ∈ Ω, (0.3) where Ω is a bounded domain in R 3 with smooth boundary F, v is the outward normal direction to the boundary, and f is analytic in u. In this paper convergence of the solution to an equilibrium as time goes to infinity is proved. While these types of results are known for the damped semilinear wave equation with interior dissipation and Dirichlet boundary condition, this is, to our knowledge, the first result with dissipative boundary condition and critical growth exponent.

Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions

In this paper a nonlocal phase-field model for non-isothermal phase transitions with a non-conserved order parameter is studied. The paper extends recent investigations to the non-isothermal situation, complementing results obtained by H. Gajewski for the non-isothermal case for conserved order parameters in phase separation phenomena. The resulting field equations studied in this paper form a system of integro-partial differential equations which are highly nonlinearly coupled. For this system, results concerning global existence, uniqueness and large-time asymptotic behaviour are derived. The main results are proved using techniques that have been recently developed by P. Krejci and the authors for phase-field systems involving hysteresis operators.

Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains

In this paper, we investigate an initial-boundary value problem for a chemotaxis-fluid system in a general bounded regular domain Omega subset of R-N (N is an element of{2, 3}), not necessarily being convex. Thanks to the elementary lemma given by Mizoguchi and Souplet [Ann. Inst. H. Poincare - AN 31 (2014), 851-875], we can derive a new type of entropy-energy estimate, which enables us to prove the following: (1) for N = 2, there exists a unique global classical solution to the full chemotaxis-Navier-Stokes system, which converges to a constant steady state (n(infinity), 0, 0) as t -> +infinity, and (2) for N = 3, the existence of a global weak solution to the simplified chemotaxis-Stokes system. Our results generalize the recent work due to Winkler [Commun. Partial Diff. Equ. 37 (2012), 319-351; Arch. Rational Mech. Anal. 211 (2014), 455-487], in which the domain Omega is essentially assumed to be convex.

Uniform Exponential Stability and Approximation in Control of a Thermoelastic System

This paper has two objectives. First, necessary and sufficient conditions are given to characterize the uniform exponential stability of a sequence of

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR A SEMICONDUCTOR DRIFT-DIFFUSION-POISSON MODEL

c_0

Convergence to Equilibrium for Parabolic-Hyperbolic Time-Dependent Ginzburg–Landau–Maxwell Equations

-semigroups