73 research outputs found
On the impossibility of finite-time splash singularities for vortex sheets
In fluid dynamics, an interface splash singularity occurs when a locally
smooth interface self-intersects in finite time. By means of elementary
arguments, we prove that such a singularity cannot occur in finite time for
vortex sheet evolution, i.e. for the two-phase incompressible Euler equations.
We prove this by contradiction; we assume that a splash singularity does indeed
occur in finite time. Based on this assumption, we find precise blow-up rates
for the components of the velocity gradient which, in turn, allow us to
characterize the geometry of the evolving interface just prior to
self-intersection. The constraints on the geometry then lead to an impossible
outcome, showing that our assumption of a finite-time splash singularity was
false.Comment: 39 pages, 8 figures, details added to proofs in Sections 5 and
Regularity of the velocity field for Euler vortex patch evolution
We consider the vortex patch problem for both the 2-D and 3-D incompressible
Euler equations. In 2-D, we prove that for vortex patches with
Sobolev-class contour regularity, , the velocity field on both sides
of the vortex patch boundary has regularity for all time. In 3-D, we
establish existence of solutions to the vortex patch problem on a finite-time
interval , and we simultaneously establish the regularity of
the two-dimensional vortex patch boundary, as well as the regularity of
the velocity fields on both sides of vortex patch boundary, for .Comment: 30 pages, added references and some details to Section
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
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
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
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
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
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