1,818 research outputs found
Lyapunov functionals for boundary-driven nonlinear drift-diffusions
We exhibit a large class of Lyapunov functionals for nonlinear
drift-diffusion equations with non-homogeneous Dirichlet boundary conditions.
These are generalizations of large deviation functionals for underlying
stochastic many-particle systems, the zero range process and the
Ginzburg-Landau dynamics, which we describe briefly. As an application, we
prove linear inequalities between such an entropy-like functional and its
entropy production functional for the boundary-driven porous medium equation in
a bounded domain with positive Dirichlet conditions: this implies exponential
rates of relaxation related to the first Dirichlet eigenvalue of the domain. We
also derive Lyapunov functions for systems of nonlinear diffusion equations,
and for nonlinear Markov processes with non-reversible stationary measures
Supersymmetry in Boundary Integrable Models
Quantum integrable models that possess supersymmetry are investigated
on the half-space. Conformal perturbation theory is used to identify some
supersymmetric boundary integrable models, and the effective boundary
Landau-Ginzburg formulations are constructed. It is found that
supersymmetry largely determines the boundary action in terms of the bulk, and
in particular, the boundary bosonic potential is , where is the bulk
superpotential. Supersymmetry is also investigated using the affine quantum
group symmetry of exact scattering matrices, and the affine quantum group
symmetry of boundary reflection matrices is analyzed both for supersymmetric
and more general models. Some supersymmetry preserving boundary
reflection matrices are given, and their connection with the boundary
Landau-Ginzburg actions is discussed.Comment: 37 pages and two figures; lanlmac or harvmac, eps
Sharper global existence for the generalized 1D nonhomogeneous Ginzburg–Landau equation
AbstractWe study the following generalized 1D Ginzburg–Landau equation on Ω=(0,∞)×(0,∞): ut=(1+iμ)uxx+(a1+ib1)|u|2ux+(a2+ib2)u2ūx−(1+iν)|u|4u with initial and Dirichlet boundary conditions u(x,0)=h(x), u(0,t)=Q(t). Based on detail analysis, the sharper existence and uniqueness of global solutions are obtained under sufficient conditions
Front propagation in geometric and phase field models of stratified media
We study front propagation problems for forced mean curvature flows and their
phase field variants that take place in stratified media, i.e., heterogeneous
media whose characteristics do not vary in one direction. We consider phase
change fronts in infinite cylinders whose axis coincides with the symmetry axis
of the medium. Using the recently developed variational approaches, we provide
a convergence result relating asymptotic in time front propagation in the
diffuse interface case to that in the sharp interface case, for suitably
balanced nonlinearities of Allen-Cahn type. The result is established by using
arguments in the spirit of -convergence, to obtain a correspondence
between the minimizers of an exponentially weighted Ginzburg-Landau type
functional and the minimizers of an exponentially weighted area type
functional. These minimizers yield the fastest traveling waves invading a given
stable equilibrium in the respective models and determine the asymptotic
propagation speeds for front-like initial data. We further show that
generically these fronts are the exponentially stable global attractors for
this kind of initial data and give sufficient conditions under which complete
phase change occurs via the formation of the considered fronts
Critical Fields of mesoscopic superconductors
Recent measurements have shown oscillations in the upper critical field of
simply connected mesoscopic superconductors. A quantitative theory of these
effects is given here on the basis of a Ginzburg-Landau description. For small
fields, the phase boundary exhibits a cusp where the screening currents
change sign for the first time thus defining a lower critical field .
In the limit where many flux quanta are threading the sample, nucleation occurs
at the boundary and the upper critical field becomes identical with the surface
critical field .Comment: 5 pages (Revtex and 2 PostScript figures), to apppear in Z. Phys.
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