3,191 research outputs found
Three-Scale Singular Limits of Evolutionary PDEs
Singular limits of a class of evolutionary systems of partial differential
equations having two small parameters and hence three time scales are
considered. Under appropriate conditions solutions are shown to exist and
remain uniformly bounded for a fixed time as the two parameters tend to zero at
different rates. A simple example shows the necessity of those conditions in
order for uniform bounds to hold. Under further conditions the solutions of the
original system tend to solutions of a limit equation as the parameters tend to
zero
Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains
We provide a self-contained proof of the solvability and regularity of a
Hodge-type elliptic system, wherein the divergence and curl of a vector field
are prescribed in an open, bounded, Sobolev-class domain, and either the normal
component or the tangential components of the vector field are prescribed on
the boundary. The proof is based on a regularity theory for vector elliptic
equations set on Sobolev-class domains and with Sobolev-class coefficients.Comment: 49 Pages, improved exposition and corrected typo
Convergence Rate Estimates for the Low Mach and Alfv\'en Number Three-Scale Singular Limit of Compressible Ideal Magnetohydrodynamics
Convergence rate estimates are obtained for singular limits of the
compressible ideal magnetohydrodynamics equations, in which the Mach and
Alfv\'en numbers tend to zero at different rates. The proofs use a detailed
analysis of exact and approximate fast, intermediate, and slow modes together
with improved estimates for the solutions and their time derivatives, and the
time-integration method. When the small parameters are related by a power law
the convergence rates are positive powers of the Mach number, with the power
varying depending on the component and the norm. Exceptionally, the convergence
rate for two components involve the ratio of the two parameters, and that rate
is proven to be sharp via corrector terms. Moreover, the convergence rates for
the case of a power-law relation between the small parameters tend to the
two-scale convergence rate as the power tends to one. These results demonstrate
that the issue of convergence rates for three-scale singular limits, which was
not addressed in the authors' previous paper, is much more complicated than for
the classical two-scale singular limits
On the Motion of Vortex Sheets with Surface Tension in the 3D Euler Equations with Vorticity
We prove well-posedness of vortex sheets with surface tension in the 3D
incompressible Euler equations with vorticity.Comment: 28 page
Navier-Stokes equations interacting with a nonlinear elastic fluid shell
We study a moving boundary value problem consisting of a viscous
incompressible fluid moving and interacting with a nonlinear elastic fluid
shell. The fluid motion is governed by the Navier-Stokes equations, while the
fluid shell is modeled by a bending energy which extremizes the Willmore
functional and a membrane energy that extremizes the surface area of the shell.
The fluid flow and shell deformation are coupled together by continuity of
displacements and tractions (stresses) along the moving material interface. We
prove existence and uniqueness of solutions in Sobolev spaces.Comment: 56 pages, 1 figur
Global existence and decay for solutions of the Hele-Shaw flow with injection
We study the global existence and decay to spherical equilibrium of Hele-Shaw
flows with surface tension. We prove that without injection of fluid,
perturbations of the sphere decay to zero exponentially fast. On the other
hand, with a time-dependent rate of fluid injection into the Hele-Shaw cell,
the distance from the moving boundary to an expanding sphere (with
time-dependent radius) also decays to zero but with an algebraic rate, which
depends on the injection rate of the fluid.Comment: 25 Page
A tool to aid redesign of flexible transport services to increase efficiency in rural transport service provision
This research was supported by the Research Councils UK Digital Economy programme award (reference: EP/G066051/1) to the dot.rural Digital Economy Hub, at the University of Aberdeen.Peer reviewedPublisher PD
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