126 research outputs found
Dynamics of the Ericksen-Leslie Equations with General Leslie Stress I: The Incompressible Isotropic Case
The Ericksen-Leslie model for nematic liquid crystals in a bounded domain
with general Leslie and isotropic Ericksen stress is studied in the case of a
non-isothermal and incompressible fluid. This system is shown to be locally
well-posed in the -setting, and a dynamic theory is developed. The
equilibria are identified and shown to be normally stable. In particular, a
local solution extends to a unique, global strong solution provided the initial
data are close to an equilibrium or the solution is eventually bounded in the
topology of the natural state manifold. In this case, the solution converges
exponentially to an equilibrium, in the topology of the state manifold. The
above results are proven {\em without} any structural assumptions on the Leslie
coefficients and in particular {\em without} assuming Parodi's relation
On a Class of Energy Preserving Boundary Conditions for Incompressible Newtonian Flows
We derive a class of energy preserving boundary conditions for incompressible
Newtonian flows and prove local-in-time well-posedness of the resulting initial
boundary value problems, i.e. the Navier-Stokes equations complemented by one
of the derived boundary conditions, in an Lp-setting in domains, which are
either bounded or unbounded with almost flat, sufficiently smooth boundary. The
results are based on maximal regularity properties of the underlying
linearisations, which are also established in the above setting.Comment: 53 page
On quasilinear parabolic evolution equations in weighted Lp-spaces II
Our study of abstract quasi-linear parabolic problems in time-weighted
L_p-spaces, begun in [17], is extended in this paper to include singular lower
order terms, while keeping low initial regularity. The results are applied to
reaction-diffusion problems, including Maxwell-Stefan diffusion, and to
geometric evolution equations like the surface-diffusion flow or the Willmore
flow. The method presented here will be applicable to other parabolic systems,
including free boundary problems.Comment: 21 page
On thermodynamically consistent Stefan problems with variable surface energy
A thermodynamically consistent two-phase Stefan problem with
temperature-dependent surface tension and with or without kinetic undercooling
is studied. It is shown that these problems generate local semiflows in
well-defined state manifolds. If a solution does not exhibit singularities, it
is proved that it exists globally in time and converges towards an equilibrium
of the problem. In addition, stability and instability of equilibria is
studied. In particular, it is shown that multiple spheres of the same radius
are unstable if surface heat capacity is small; however, if kinetic
undercooling is absent, they are stable if surface heat capacity is
sufficiently large.Comment: To appear in Arch. Ration. Mech. Anal. The final publication is
available at Springer via http://dx.doi.org/10.1007/s00205-015-0938-y. arXiv
admin note: substantial text overlap with arXiv:1101.376
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