7,123 research outputs found
Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems
We show global well-posedness and exponential stability of equilibria for a
general class of nonlinear dissipative bulk-interface systems. They correspond
to thermodynamically consistent gradient structure models of bulk-interface
interaction. The setting includes nonlinear slow and fast diffusion in the bulk
and nonlinear coupled diffusion on the interface. Additional driving mechanisms
can be included and non-smooth geometries and coefficients are admissible, to
some extent. An important application are volume-surface reaction-diffusion
systems with nonlinear coupled diffusion.Comment: 21 page
On the flow of non-axisymmetric perturbations of cylinders via surface diffusion
We study the surface diffusion flow acting on a class of general
(non--axisymmetric) perturbations of cylinders in . Using tools from parabolic theory on uniformly regular manifolds, and
maximal regularity, we establish existence and uniqueness of solutions to
surface diffusion flow starting from (spatially--unbounded) surfaces defined
over via scalar height functions which are uniformly bounded
away from the central cylindrical axis. Additionally, we show that
is normally stable with respect to --axially--periodic
perturbations if the radius ,and unstable if . Stability is
also shown to hold in settings with axial Neumann boundary conditions.Comment: 20 pages, 1 figure, submitted for publicatio
Space-modulated Stability and Averaged Dynamics
In this brief note we give a brief overview of the comprehensive theory,
recently obtained by the author jointly with Johnson, Noble and Zumbrun, that
describes the nonlinear dynamics about spectrally stable periodic waves of
parabolic systems and announce parallel results for the linearized dynamics
near cnoidal waves of the Korteweg-de Vries equation. The latter are expected
to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees
partielles", Roscoff 201
Convective instability and transient growth in flow over a backward-facing step
Transient energy growths of two- and three-dimensional optimal linear perturbations to two-dimensional flow in a rectangular backward-facing-step geometry with expansion ratio two are presented. Reynolds numbers based on the step height and peak inflow speed are considered in the range 0–500, which is below the value for the onset of three-dimensional asymptotic instability. As is well known, the flow has a strong local convective instability, and the maximum linear transient energy growth values computed here are of order 80×103 at Re = 500. The critical Reynolds number below which there is no growth over any time interval is determined to be Re = 57.7 in the two-dimensional case. The centroidal location of the energy distribution for maximum transient growth is typically downstream of all the stagnation/reattachment points of the steady base flow. Sub-optimal transient modes are also computed and discussed. A direct study of weakly nonlinear effects demonstrates that nonlinearity is stablizing at Re = 500. The optimal three-dimensional disturbances have spanwise wavelength of order ten step heights. Though they have slightly larger growths than two-dimensional cases, they are broadly similar in character. When the inflow of the full nonlinear system is perturbed with white noise, narrowband random velocity perturbations are observed in the downstream channel at locations corresponding to maximum linear transient growth. The centre frequency of this response matches that computed from the streamwise wavelength and mean advection speed of the predicted optimal disturbance. Linkage between the response of the driven flow and the optimal disturbance is further demonstrated by a partition of response energy into velocity components
Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems
We study linear-quadratic stochastic optimal control problems with bilinear
state dependence for which the underlying stochastic differential equation
(SDE) consists of slow and fast degrees of freedom. We show that, in the same
way in which the underlying dynamics can be well approximated by a reduced
order effective dynamics in the time scale limit (using classical
homogenziation results), the associated optimal expected cost converges in the
time scale limit to an effective optimal cost. This entails that we can well
approximate the stochastic optimal control for the whole system by the reduced
order stochastic optimal control, which is clearly easier to solve because of
lower dimensionality. The approach uses an equivalent formulation of the
Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs
(FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares
Monte Carlo algorithm and show its applicability by a suitable numerical
example
Hyperbolic-parabolic singular perturbation for Kirchhoff equations with weak dissipation
We consider Kirchhoff equations with a small parameter epsilon in front of
the second-order time-derivative, and a dissipative term whose coefficient may
tend to 0 as t -> + infinity (weak dissipation).
In this note we present some recent results concerning existence of global
solutions, and their asymptotic behavior both as t -> + infinity and as epsilon
-> 0. Since the limit equation is of parabolic type, this is usually referred
to as a hyperbolic-parabolic singular perturbation problem.
We show in particular that the equation exhibits hyperbolic or parabolic
behavior depending on the values of the parameters.Comment: 20 pages, 2 tables, 1 figure, conference paper (7th ISAAC congress,
London 2009
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
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