1,873 research outputs found
On the two-phase Navier-Stokes equations with surface tension
The two-phase free boundary problem for the Navier-Stokes system is
considered in a situation where the initial interface is close to a halfplane.
By means of -maximal regularity of the underlying linear problem we show
local well-posedness of the problem, and prove that the solution, in particular
the interface, becomes instantaneously real analytic.Comment: 34 page
On the microscopic bidomain problem with FitzHugh-Nagumo ionic transport
The microscopic bidomain problem with FitzHhugh-Nagumo ionic transport is
studied in the -framework. Reformulating the problem as a
semilinear evolution equation on the interface, local well-posedness is proved
in strong as well as in weak settings. We obtain solvability for initial data
in the critical spaces of the problem. For dimension , by means of
energy estimates and a recent result of Serrin type, global existence is shown.
Finally, stability of spatially constant equilibria is investigated, to the
result that the stability properties of such equilibria parallel those of the
classical FitzHugh-Nagumo system in ODE's. These properties of the bidomain
equations are obtained combining recent results on Dirichlet-to-Neumann
operators, on critical spaces for parabolic evolution equations, and
qualitative theory of evolution equations.Comment: 15 page
On conserved Penrose-Fife type models
In this paper we investigate quasilinear parabolic systems of conserved
Penrose-Fife type. We show maximal - regularity for this problem with
inhomogeneous boundary data. Furthermore we prove global existence of a
solution, provided that the absolute temperature is bounded from below and
above. Moreover, we apply the Lojasiewicz-Simon inequality to establish the
convergence of solutions to a steady state as time tends to infinity.Comment: 32 page
On the Rayleigh-Taylor Instability for the two-Phase Navier-Stokes Equations
The two-phase free boundary problem with surface tension and downforce
gravity for the Navier-Stokes system is considered in a situation where the
initial interface is close to equilibrium. The boundary symbol of this problem
admits zeros in the unstable halfplane in case the heavy fluid is on top of the
light one, which leads to the well-known Rayleigh-Taylor instability.
Instability is proved rigorously in an -setting by means of an abstract
instability result due to Henry.Comment: 16 page
On the manifold of closed hypersurfaces in R^n
Several results from differential geometry of hypersurfaces in R^n are
derived to form a tool box for the direct mapping method. The latter technique
has been widely employed to solve problems with moving interfaces, and to study
the asymptotics of the induced semiflows.Comment: 21 pages. To appea
The Verigin problem with and without phase transition
Isothermal compressible two-phase flows with and without phase transition are
modeled, employing Darcy's and/or Forchheimer's law for the velocity field. It
is shown that the resulting systems are thermodynamically consistent in the
sense that the available energy is a strict Lyapunov functional. In both cases,
the equilibria are identified and their thermodynamical stability is
investigated by means of a variational approach. It is shown that the problems
are well-posed in an -setting and generate local semiflows in the proper
state manifolds. It is further shown that a non-degenerate equilibrium is
dynamically stable in the natural state manifold if and only if it is
thermodynamically stable. Finally, it is shown that a solution which does not
develop singularities exists globally and converges to an equilibrium in the
state manifold
Analytic solutions for the two-phase Navier-Stokes equations with surface tension and gravity
We consider the motion of two superposed immiscible, viscous, incompressible,
capillary fluids that are separated by a sharp interface which needs to be
determined as part of the problem. Allowing for gravity to act on the fluids,
we prove local well-posedness of the problem. In particular, we obtain
well-posedness for the case where the heavy fluid lies on top of the light one,
that is, for the case where the Rayleigh-Taylor instability is present.
Additionally we show that solutions become real analytic instantaneously.Comment: 31 page
On the Muskat flow
Of concern is the motion of two fluids separated by a free interface in a
porous medium, where the velocities are given by Darcy's law. We consider the
case with and without phase transition. It is shown that the resulting models
can be understood as purely geometric evolution laws, where the motion of the
separating interface depends in a non-local way on the mean curvature. It turns
out that the models are volume preserving and surface area reducing, the latter
property giving rise to a Lyapunov function. We show well-posedness of the
models, characterize all equilibria, and study the dynamic stability of the
equilibria. Lastly, we show that solutions which do not develop singularities
exist globally and converge exponentially fast to an equilibrium.Comment: 15 page
On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension
We study the regularity of the free boundary arising in a thermodynamically
consistent two-phase Stefan problem with surface tension by means of a family
of parameter-dependent diffeomorphisms, -maximal regularity theory, and
the implicit function theorem.Comment: 42 page
On the motion of a fluid-filled rigid body with Navier Boundary conditions
We consider the inertial motion of a system constituted by a rigid body with
an interior cavity entirely filled with a viscous incompressible fluid. Navier
boundary conditions are imposed on the cavity surface. We prove the existence
of weak solutions and determine the critical spaces for the governing evolution
equation. Using parabolic regularization in time-weighted spaces, we establish
regularity of solutions and their long-time behavior. We show that every weak
solution \`a la Leray-Hopf to the equations of motion converges to an
equilibrium at an exponential rate in the -topology for every fluid-solid
configuration. A nonlinear stability analysis shows that equilibria associated
with the largest moment of inertia are asymptotically (exponentially) stable,
whereas all other equilibria are normally hyperbolic and unstable in an
appropriate topology.Comment: arXiv admin note: text overlap with arXiv:1804.0540
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