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 $C^\infty$ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has $C^\infty$ dependence on initial data $u_0$ in the class of $H^s$ divergence-free vector fields for $s>2$

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 $H^s$, $s > n/2 +1$ 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 $H^s$ topology for $s> n/2+2$, 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 H1H^1 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 $\dot{V}(t) + \nabla_{U(t)} V(t) - \alpha^2 [\nabla U(t)]^t \cdot \triangle U(t) = -\text{grad} p(t)$ where $\text{div} U=0$, and $V = (1- \alpha^2 \triangle)U$. In this model, the momentum $V$ is transported by the velocity $U$, with the effect that nonlinear interaction between modes corresponding to length scales smaller than $\alpha$ is negligible. We generalize this equation to the setting of an $n$ dimensional compact Riemannian manifold. The resulting equation is the Euler-Poincar\'{e} equation associated with the geodesic flow of the $H^1$ right invariant metric on ${\mathcal D}^s_\mu$, the group of volume preserving Hilbert diffeomorphisms of class $H^s$. We prove that the geodesic spray is continuously differentiable from $T{\mathcal D}_\mu^s(M)$ into $TT{\mathcal D}_\mu^s(M)$ 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 $H^1$ metric on ${\mathcal D}^s_\mu$ is a bounded trilinear map in the $H^s$ 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 $d$-dimensional flows, $d=2$ or $3$, 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 ${\mathcal M}({\mathbb R}^2)$. The solutions of this regularized PDE, also known as the averaged Euler or Euler-$\alpha$ 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 ${\mathcal M}({\mathbb R}^2)$ 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 $\sigma$ converge to solutions of the classical Stefan problem as $\sigma \to 0$.Comment: Various typos corrected and references adde

On the stability of periodic 2D Euler-alpha flows

An explicit expression is obtained for the sectional curvature in the plane spanned by two stationary flows, cos(k, x) and cos(l, x). It is shown that for certain values of the wave vectors k and l the curvature becomes positive for alpha > alpha_0, where 0 < alpha_0 < 1 is of the order 1/k. This suggests that the flow corresponding to such geodesics becomes more stable as one goes from usual Eulerian description to the Euler-alpha model

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