1,675 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 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 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 the well-posedness of a mathematical model describing water-mud interaction
In this paper we consider a mathematical model describing the two-phase
interaction between water and mud in a water canal when the width of the canal
is small compared to its depth. The mud is treated as a non-Netwonian fluid and
the interface between the mud and fluid is allowed to move under the influence
of gravity and surface tension. We reduce the mathematical formulation, for
small boundary and initial data, to a fully nonlocal and nonlinear problem and
prove its local well-posedness by using abstract parabolic theory.Comment: 16 page
Global estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients
We consider a class of degenerate Ornstein-Uhlenbeck operators in
, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x)
\partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where
is symmetric uniformly positive definite on
(), with uniformly continuous and bounded entries, and
is a constant matrix such that the frozen operator
corresponding to is hypoelliptic. For this class of operators
we prove global estimates () of the kind:%
[|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(\mathbb{R}% ^{N})}\leq
c{|\mathcal{A}u|_{L^{p}(\mathbb{R}^{N})}+|u|_{L^{p}(\mathbb{R}% ^{N})}} for
i,j=1,2,...,p_{0}.] We obtain the previous estimates as a byproduct of the
following one, which is of interest in its own:%
[|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(S_{T})}\leq
c{|Lu|_{L^{p}(S_{T})}+|u|_{L^{p}(S_{T})}}] for any where is the strip
, small, and is the
Kolmogorov-Fokker-Planck operator% [L\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x,t)
\partial_{x_{i}x_{j}}%
^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}-\partial_{t}%] with uniformly
continuous and bounded 's
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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