Stability of stationary equivariant wave maps from the hyperbolic plane

In this paper we initiate the study of equivariant wave maps from 2d hyperbolic space, H², into rotationally symmetric surfaces. This problem exhibits markedly different phenomena than its Euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain. In particular, when the target is S², we find a family of equivariant harmonic maps H²→ S², indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere. These maps have energies taking all values between zero and the energy of the unique corotational Euclidean harmonic map, Q[subscript euc], from R² to S², given by stereographic projection. We prove that the harmonic maps are asymptotically stable for values of the parameter smaller than a threshold that is large enough to allow for maps that wrap more than halfway around the sphere. Indeed, we prove Strichartz estimates for the operator obtained by linearizing around such a harmonic map. However, for harmonic maps with energies approaching the Euclidean energy of Q[subscript euc], asymptotic stability via a perturbative argument based on Strichartz estimates is precluded by the existence of gap eigenvalues in the spectrum of the linearized operator. When the target is H², we find a continuous family of asymptotically stable equivariant harmonic maps H² → H² with arbitrarily small and arbitrarily large energies. This stands in sharp contrast to the corresponding problem on Euclidean space, where all finite energy solutions scatter to zero as time tends to infinity.National Science Foundation (U.S.) (Grant DMS-1302782

Equivariant wave maps on the hyperbolic plane with large energy

In this paper we continue the analysis of equivariant wave maps from 2-dimensional hyperbolic space H² into surfaces of revolution N that was initiated in [12, 13]. When the target N = H² we proved in [12] the existence and asymptotic stability of a 1-parameter family of finite energy harmonic maps indexed by how far each map wraps around the target. Here we conjecture that each of these harmonic maps is globally asymptotically stable, meaning that the evolution of any arbitrarily large finite energy perturbation of a harmonic map asymptotically resolves into the harmonic map itself plus free radiation. Since such initial data exhaust the energy space, this is the soliton resolution conjecture for this equation. The main result is a verification of this conjecture for a nonperturbative subset of the harmonic maps.National Science Foundation (U.S.) (Grant DMS-1302782)National Science Foundation (U.S.) (Grant 1045119

Optimization of an Air Film Cooled CFRP Panel with an Embedded Vascular Network

This paper summarizes research performed on thermodynamic simulation and design optimization of a composite panel cooled by an external cool film and an internal vascular network

The magic carpet: an arbitrary spectrum wave maker for internal waves

Abstract: We present a novel apparatus for generating internal waves of arbitrary size and shape, including both phase-locked and propagating waves. It is an actively driven, flexible “magic carpet” in the base of a tank. Our wave maker is computer-controlled to enable easy configuration. The actuation of a smooth, flexible surface produces clean waveforms with a predictable spectrum, for which we derive a theoretical model. We demonstrate the versatility of our wave maker through an experimental study of linear and nonlinear, isolated, and combined internal waves, including some that are sufficiently nonlinear to break remote from their source. Graphic abstract