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 $s>2$.

We study the coupled Navier-Stokes Ginzburg-Landau model of nematic liquid crystals introduced by F.H. Lin, which is a simplified version of the Ericksen-Leslie system. We generalize the model to compact n-dimensional Riemannian manifolds, and show that the system comes from a variational principle. We present a new simple proof for the local well-posedness of this coupled system without using the higher-order energy law. We then prove that this system is globally well-posed and has compact global attractors when the dimension of the manifold M is two.Finally, we introduce the Lagrangian averaged liquid crystal equations, which arise from averaging the Navier-Stokes fluid motion over small spatial scales in the variational principle. We show that this averaged system is globally well-posed and has compact global attractors even when M is three-dimensional.

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.

The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations along a moving interface. Unlike our approach for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in srong norms of our sequence of regularized problems.

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 $H^{k-0.5}$ Sobolev-class contour regularity, $k \ge 4$, the velocity field on both sides of the vortex patch boundary has $H^k$ regularity for all time. In 3-D, we establish existence of solutions to the vortex patch problem on a finite-time interval $[0,T]$, and we simultaneously establish the $H^{k-0.5}$ regularity of the two-dimensional vortex patch boundary, as well as the $H^k$ regularity of the velocity fields on both sides of vortex patch boundary, for $k \ge 3$.

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.

The classical one-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition, such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free-boundary. We establish a global-in-time stability result for nearly spherical geometries and small temperatures, using a novel hybrid methodology, which combines energy estimates, decay estimates, and Hopf-type inequalities.

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.

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 (Hadžić & Shkoller 2014 Commun. Pure Appl. Math. 68, 689-757 (doi:10.1002/cpa.21522)) in which we studied nearly spherical shapes.

The free-boundary compressible 1-D Euler equations with moving physical vacuum boundary are a system of hyperbolic conservation laws which are both characteristic and degenerate. The physical vacuum singularity (or rate-of-degeneracy) requires the sound speed $c= \gamma \rho^{\gamma -1}$ to scale as the square-root of the distance to the vacuum boundary, and has attracted a great deal of attention in recent years. We establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary. The proof is founded on a new higher-order Hardy-type inequality in conjunction with an approximation of the Euler equations consisting of a particular degenerate parabolic regularization. Our regular solutions can be viewed as degenerate viscosity solutions.