140 research outputs found
Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations
We present a very simple proof of the global existence of a
Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a
compact Riemannian manifold with boundary) which has dependence on
initial data in the class of divergence-free vector fields for
On Incompressible Averaged Lagrangian Hydrodynamics
This paper is devoted to the geometric analysis of the incompressible
averaged Euler equations on compact Riemannian manifolds with boundary. The
equation also coincides with the model for a second-grade non-Newtonian fluid.
We study the analytical and geometrical properties of the Lagrangian flow map.
We prove existence and uniqueness of smooth-in-time solutions for initial data
in , by establishing the existence of smooth geodesics of a
new weak right invariant metric on new subgroups of the volume-preserving
diffeomorphism group. We establish smooth limits of zero viscosity for the
second-grade fluids equations even on manifolds with boundary. We prove that
the weak curvature operator of the weak invariant metric is continuous in the
topology for , thus proving existence and uniqueness for the
Jacobi equation. We show that this new metric stabilizes the Lagrangian flow of
the original Euler equations by changing the sign of the sectional curvature.Comment: 35 page
Geometry and curvature of diffeomorphism groups with metric and mean hydrodynamics
Recently, Holm, Marsden, and Ratiu [1998] have derived a new model for the
mean motion of an ideal fluid in Euclidean space given by the equation
where , and . In this model, the momentum is transported by the velocity
, with the effect that nonlinear interaction between modes corresponding to
length scales smaller than is negligible. We generalize this equation
to the setting of an dimensional compact Riemannian manifold. The resulting
equation is the Euler-Poincar\'{e} equation associated with the geodesic flow
of the right invariant metric on , the group of
volume preserving Hilbert diffeomorphisms of class . We prove that the
geodesic spray is continuously differentiable from
into so that a standard Picard iteration argument
proves existence and uniqueness on a finite time interval. Our goal in this
paper is to establish the foundations for Lagrangian stability analysis
following Arnold [1966]. To do so, we use submanifold geometry, and prove that
the weak curvature tensor of the right invariant metric on is a bounded trilinear map in the topology, from which it
follows that solutions to Jacobi's equation exist. Using such solutions, we are
able to study the infinitesimal stability behavior of geodesics.Comment: AMS-LaTeX, 22 pages, To appear in J. Func. Ana
Well-posedness of the free-surface incompressible Euler equations with or without surface tension
We provide a new method for treating free boundary problems in perfect
fluids, and prove local-in-time well-posedness in Sobolev spaces for the
free-surface incompressible 3D Euler equations with or without surface tension
for arbitrary initial data, and without any irrotationality assumption on the
fluid. This is a free boundary problem for the motion of an incompressible
perfect liquid in vacuum, wherein the motion of the fluid interacts with the
motion of the free-surface at highest-order.Comment: To appear in J. Amer. Math. Soc., 96 page
On the splash singularity for the free-surface of a Navier-Stokes fluid
In fluid dynamics, an interface splash singularity occurs when a locally
smooth interface self-intersects in finite time. We prove that for
-dimensional flows, or , the free-surface of a viscous water wave,
modeled by the incompressible Navier-Stokes equations with moving
free-boundary, has a finite-time splash singularity. In particular, we prove
that given a sufficiently smooth initial boundary and divergence-free velocity
field, the interface will self-intersect in finite time.Comment: 21 pages, 5 figure
The vortex blob method as a second-grade non-Newtonian fluid
We show that a certain class of vortex blob approximations for ideal
hydrodynamics in two dimensions can be rigorously understood as solutions to
the equations of second-grade non-Newtonian fluids with zero viscosity, and
initial data in the space of Radon measures . The
solutions of this regularized PDE, also known as the averaged Euler or
Euler- equations, are geodesics on the volume preserving diffeomorphism
group with respect to a new weak right invariant metric. We prove global
existence of unique weak solutions (geodesics) for initial vorticity in
such as point-vortex data, and show that the
associated coadjoint orbit is preserved by the flow. Moreover, solutions of
this particular vortex blob method converge to solutions of the Euler equations
with bounded initial vorticity, provided that the initial data is approximated
weakly in measure, and the total variation of the approximation also converges.
In particular, this includes grid-based approximation schemes of the type that
are usually used for vortex methods
Well-posedness for the classical Stefan problem and the zero surface tension limit
We develop a framework for a unified treatment of well-posedness for the
Stefan problem with or without surface tension. In the absence of surface
tension, we establish well-posedness in Sobolev spaces for the classical Stefan
problem. We introduce a new velocity variable which extends the velocity of the
moving free-boundary into the interior domain. The equation satisfied by this
velocity is used for the analysis in place of the heat equation satisfied by
the temperature. Solutions to the classical Stefan problem are then constructed
as the limit of solutions to a carefully chosen sequence of approximations to
the velocity equation, in which the moving free-boundary is regularized and the
boundary condition is modified in a such a way as to preserve the basic
nonlinear structure of the original problem. With our methodology, we
simultaneously find the required stability condition for well-posedness and
obtain new estimates for the regularity of the moving free-boundary. Finally,
we prove that solutions of the Stefan problem with positive surface tension
converge to solutions of the classical Stefan problem as .Comment: Various typos corrected and references adde
Unique solvability of the free-boundary Navier-Stokes equations with surface tension
We prove the existence and uniqueness of solutions to the time-dependent
incompressible Navier-Stokes equations with a free-boundary governed by surface
tension. The solution is found using a topological fixed-point theorem for a
nonlinear iteration scheme, requiring at each step, the solution of a model
linear problem consisting of the time-dependent Stokes equation with linearized
mean-curvature forcing on the boundary. We use energy methods to establish new
types of spacetime inequalities that allow us to find a unique weak solution to
this problem. We then prove regularity of the weak solution, and establish the
a priori estimates required by the nonlinear iteration process.Comment: 73 pages; typos corrected; minor details adde
Global stability of steady states in the classical Stefan problem
The classical one-phase Stefan problem (without surface tension) allows for a
continuum of steady state solutions, given by an arbitrary (but sufficiently
smooth) domain together with zero temperature. We prove global-in-time
stability of such steady states, assuming a sufficient degree of smoothness on
the initial domain, but without any a priori restriction on the convexity
properties of the initial shape. This is an extension of our previous result
[28] in which we studied nearly spherical shapes.Comment: 14 pages. arXiv admin note: substantial text overlap with
arXiv:1212.142
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