98 research outputs found
On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow
In this article, we study the axisymmetric surface diffusion flow (ASD), a
fourth-order geometric evolution law. In particular, we prove that ASD
generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older
regular surfaces of revolution embedded in R^3 and satisfying periodic boundary
conditions. We also give conditions for global existence of solutions and prove
that solutions are real analytic in time and space. Further, we investigate the
geometric properties of solutions to ASD. Utilizing a connection to
axisymmetric surfaces with constant mean curvature, we characterize the
equilibria of ASD. Then, focusing on the family of cylinders, we establish
results regarding stability, instability and bifurcation behavior, with the
radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana
The surface diffusion and the Willmore flow for uniformly regular hypersurfaces
We consider the surface diffusion and Willmore flows acting on a general
class of (possibly non-compact) hypersurfaces parameterized over a uniformly
regular reference manifold possessing a tubular neighborhood with uniform
radius. The surface diffusion and Willmore flows each give rise to a
fourth-order quasilinear parabolic equation with nonlinear terms satisfying a
specific singular structure. We establish well-posedness of both flows for
initial surfaces that are -regular and parameterized over a
uniformly regular hypersurface. For the Willmore flow, we also show long-term
existence for initial surfaces which are -close to a sphere, and
we prove that these solutions become spherical as time goes to infinity.Comment: 22 page
On quasilinear parabolic equations and continuous maximal regularity
We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings
Structure And Dynamics Of Modulated Traveling Waves In Cellular Flames
We describe spatial and temporal patterns in cylindrical premixed flames in
the cellular regime, , where the Lewis number is the ratio of
thermal to mass diffusivity of a deficient component of the combustible
mixture. A transition from stationary, axisymmetric flames to stationary
cellular flames is predicted analytically if is decreased below a critical
value. We present the results of numerical computations to show that as is
further decreased traveling waves (TWs) along the flame front arise via an
infinite-period bifurcation which breaks the reflection symmetry of the
cellular array. Upon further decreasing different kinds of periodically
modulated traveling waves (MTWs) as well as a branch of quasiperiodically
modulated traveling waves (QPMTWs) arise. These transitions are accompanied by
the development of different spatial and temporal symmetries including period
doublings and period halvings. We also observe the apparently chaotic temporal
behavior of a disordered cellular pattern involving creation and annihilation
of cells. We analytically describe the stability of the TW solution near its
onset+ using suitable phase-amplitude equations. Within this framework one of
the MTW's can be identified as a localized wave traveling through an underlying
stationary, spatially periodic structure. We study the Eckhaus instability of
the TW and find that in general they are unstable at onset in infinite systems.
They can, however, become stable for larger amplitudes.Comment: to appear in Physica D 28 pages (LaTeX), 11 figures (2MB postscript
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Dynamics of vorticity
Remarks are made about the status of research on the role of vorticity in fluid dynamics and some unsolved problems of current interest are described
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