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

Existence of global strong solutions to a beam-fluid interaction system

We study an unsteady non linear fluid-structure interaction problem which is a simplified model to describe blood flow through viscoleastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid-structure interface and the action-reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain in particular that contact between the viscoleastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, but also of existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure

On the existence of solutions to the relativistic Euler equations in 2 spacetime dimensions with a vacuum boundary

We prove the existence of a wide class of solutions to the isentropic relativistic Euler equations in 2 spacetime dimensions with an equation of state of the form $p=K\rho^2$ that have a fluid vacuum boundary. Near the fluid vacuum boundary, the sound speed for these solutions are monotonically decreasing, approaching zero where the density vanishes. Moreover, the fluid acceleration is finite and bounded away from zero as the fluid vacuum boundary is approached. The existence results of this article also generalize in a straightforward manner to equations of state of the form $p=K\rho^\frac{\gamma+1}{\gamma}$ with $\gamma > 0$.Comment: A major revision of the second half of the pape

Dynamical elastic bodies in Newtonian gravity

Well-posedness for the initial value problem for a self-gravitating elastic body with free boundary in Newtonian gravity is proved. In the material frame, the Euler-Lagrange equation becomes, assuming suitable constitutive properties for the elastic material, a fully non-linear elliptic-hyperbolic system with boundary conditions of Neumann type. For systems of this type, the initial data must satisfy compatibility conditions in order to achieve regular solutions. Given a relaxed reference configuration and a sufficiently small Newton's constant, a neigborhood of initial data satisfying the compatibility conditions is constructed

Particle trajectories in linearized irrotational shallow water flows

We investigate the particle trajectories in an irrotational shallow water flow over a flat bed as periodic waves propagate on the water's free surface. Within the linear water wave theory, we show that there are no closed orbits for the water particles beneath the irrotational shallow water waves. Depending on the strength of underlying uniform current, we obtain that some particle trajectories are undulating path to the right or to the left, some are looping curves with a drift to the right and others are parabolic curves or curves which have only one loop

Global well-posedness of the 3-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 3-D infinite depth full water wave problem. We show that the nature of the nonlinearity of the water wave equation is essentially of cubic and higher orders. For any initial interface that is sufficiently small in its steepness and velocity, we show that there exists a unique smooth solution of the full water wave problem for all time, and the solution decays at the rate $1/t$.Comment: 60 page

Large time existence for 3D water-waves and asymptotics

We rigorously justify in 3D the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili (KP) approximation, Green-Naghdi equations, Serre approximation and full-dispersion model. We first introduce a ``variable'' nondimensionalized version of the water-waves equations which vary from shallow to deep water, and which involves four dimensionless parameters. Using a nonlocal energy adapted to the equations, we can prove a well-posedness theorem, uniformly with respect to all the parameters. Its validity ranges therefore from shallow to deep-water, from small to large surface and bottom variations, and from fully to weakly transverse waves. The physical regimes corresponding to the aforementioned models can therefore be studied as particular cases; it turns out that the existence time and the energy bounds given by the theorem are always those needed to justify the asymptotic models. We can therefore derive and justify them in a systematic way.Comment: Revised version of arXiv:math.AP/0702015 (notations simplified and remarks added) To appear in Inventione

Phylogenetic Study of Plant Q-type C2H2 Zinc Finger Proteins and Expression Analysis of Poplar Genes in Response to Osmotic, Cold and Mechanical Stresses

Plant Q-type C2H2 zinc finger transcription factors play an important role in plant tolerance to various environmental stresses such as drought, cold, osmotic stress, wounding and mechanical loading. To carry out an improved analysis of the specific role of each member of this subfamily in response to mechanical loading in poplar, we identified 16 two-fingered Q-type C2H2-predicted proteins from the poplar Phytozome database and compared their phylogenetic relationships with 152 two-fingered Q-type C2H2 protein sequences belonging to more than 50 species isolated from the NR protein database of NCBI. Phylogenetic analyses of these Q-type C2H2 proteins sequences classified them into two groups G1 and G2, and conserved motif distributions of interest were established. These two groups differed essentially in their signatures at the C-terminus of their two QALGGH DNA-binding domains. Two additional conserved motifs, MALEAL and LVDCHY, were found only in sequences from Group G1 or from Group G2, respectively. Functional significance of these phylogenetic divergences was assessed by studying transcript accumulation of six poplar C2H2 Q-type genes in responses to abiotic stresses; but no group specificity was found in any organ. Further expression analyses focused on PtaZFP1 and PtaZFP2, the two genes strongly induced by mechanical loading in poplars. The results revealed that these two genes were regulated by several signalling molecules including hydrogen peroxide and the phytohormone jasmonate