Abstract

We prove the existence of global strong solutions of the primitive equations of the ocean in the case of the Dirichlet boundary conditions and variable boundary. We also discuss the dissipativity of the system and uniform gradient bounds for solutions. This is a joint work with M. Ziane.

Space charge limiting (SCL) current is a fundamental constraint on the flow of an electron beam in a diode. The Child-Langmuir law describes the resulting maximal current that can be sustained in a steady flow across a diode. This talk presents a new formulation of the solution for fully nonlinear and unsteady planar flow of an electron beam in a diode. Using characteristic variables - i.e., variables that follow particle paths - the solution is expressed through an exact analytic, but implicit, formula for any choice of incoming boundary data. For steady solutions, this approach clarifies the origin of the maximal current. The implicit formulation is used to find (1) a simplified derivation of the maximal current, (2) unsteady solutions having constant incoming flux that exceeds the Child-Langmuir limit, which leads to formation of a “virtual cathode,” and (3) time-periodic solutions whose flux exceeds the Child-Langmuir limit on average.

We introduce a PDE model in frequency space for the inertial energy cascade that reproduces the classical scaling laws of Kolmogorov's theory of turbulence. The resulting model is a variant of Burgers equation on the half line with a boundary condition which represents a constant energy input at integral scales. We show the existence of a unique stationary solution, both in the viscous and inviscid cases, which replicates the classical dissipation anomaly in the limit of vanishing viscosity.

We address the problem of global regularity of solutions of the Navier-Stokes equations in a three-dimensional periodic domain. We show that, in the class of solutions oscillating in the vertical direction, the solutions are smooth under certain conditions on the derivatives in the horizontal direction and the vertical and horizontal averages of the initial data. The imposed conditions involve only the size of initial data and are easy to verify. They also admit a class of large data. The results are obtain in collaboration with Igor Kukavica and Mohammed Ziane.

Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in mathematics and fluid dynamics. In this talk, we will present a class of potentially singular solutions of the 3D Euler and Navier-Stokes eequations based on our recent numerical study. An interesting feature of these solutions is that their velocity fields produce a “tornado” like structure. Near the center of the “tornado,” the angular velocity develops a very sharp gradient and becomes almost discontinuous. As a result, the solution approaches to a vortex sheet like structure as time evolves. Near the center of the tornado, there is a strong nonlinear alignment in the vortex stretching term, and the solution becomes increasingly singular with a scaling consistent with a finite time blow-up. However, as the thickness of the vortex sheet becomes smaller and smaller, the Kelvin-Helmholtz instability of the fluid flow eventually kicks in and destroys such nonlinear alignment, leading to the subsequent development of turbulent flow. We will also discuss the possibility of adding a regular nonlinear forcing based on feedback control to maintain the dynamic stability of the vortex sheet structure. If this could be done, it may provide a way to produce a potentially highly unstable singular solution of the 3D Euler equation in a finite time. This is joint work with Tom Hou.

Helical symmetry is invariance under a one-dimensional group of rigid motions generated by a simultaneous rotation around a fixed axis and translation along the same axis. This symmetry is preserved by both the Navier-Stokes and Euler equations and the vortex stretching term in the vorticity equation is non-trivial. Still, global well-posedness of strong solutions was established for incompressible viscous flows with helical symmetry by Mahalov, Titi, and Leibowich in 1990. Under an additional geometric condition which we call “vanishing helical swirl,” global well-posedness was proved, in the smooth setting, by Dutrifoy in 1999, and, in the weak setting, by Ettinger and Titi in 2009, for finite-energy velocities whose vorticity is bounded. In this talk we will present a series of recent results for helical flows, including the vanishing viscosity limit, planar limits, and extensions of the inviscid existence results.