Generic Regularity of Conservative Solutions to a Nonlinear Wave Equation

The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt} - c(u)\big(c(u) u_x\big)_x = 0$. For an open dense set of $C^3$ initial data, we prove that the solution is piecewise smooth in the $t$-$x$ plane, while the gradient $u_x$ can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem.Comment: 25 page

A nonlinear scalar model of extreme mass ratio inspirals in effective field theory I. Self force through third order

The motion of a small compact object in a background spacetime is investigated in the context of a model nonlinear scalar field theory. This model is constructed to have a perturbative structure analogous to the General Relativistic description of extreme mass ratio inspirals (EMRIs). We apply the effective field theory approach to this model and calculate the finite part of the self force on the small compact object through third order in the ratio of the size of the compact object to the curvature scale of the background (e.g., black hole) spacetime. We use well-known renormalization methods and demonstrate the consistency of the formalism in rendering the self force finite at higher orders within a point particle prescription for the small compact object. This nonlinear scalar model should be useful for studying various aspects of higher-order self force effects in EMRIs but within a comparatively simpler context than the full gravitational case. These aspects include developing practical schemes for higher order self force numerical computations, quantifying the effects of transient resonances on EMRI waveforms and accurately modeling the small compact object's motion for precise determinations of the parameters of detected EMRI sources.Comment: 30 pages, 8 figure

A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE

Time scale separation is a natural property of many control systems that can be ex- ploited, theoretically and numerically. We present a numerical scheme to solve optimal control problems with considerable time scale separation that is based on a model reduction approach that does not need the system to be explicitly stated in singularly perturbed form. We present examples that highlight the advantages and disadvantages of the method

Pipe Poiseuille flow of viscously anisotropic, partially molten rock

Laboratory experiments in which synthetic, partially molten rock is subjected to forced deformation provide a context for testing hypotheses about the dynamics and rheology of the mantle. Here our hypothesis is that the aggregate viscosity of partially molten mantle is anisotropic, and that this anisotropy arises from deviatoric stresses in the rock matrix. We formulate a model of pipe Poiseuille flow based on theory by Takei and Holtzman [2009a] and Takei and Katz [2013]. Pipe Poiseuille is a configuration that is accessible to laboratory experimentation but for which there are no published results. We analyse the model system through linearised analysis and numerical simulations. This analysis predicts two modes of melt segregation: migration of melt from the centre of the pipe toward the wall and localisation of melt into high-porosity bands that emerge near the wall, at a low angle to the shear plane. We compare our results to those of Takei and Katz [2013] for plane Poiseuille flow; we also describe a new approximation of radially varying anisotropy that improves the self-consistency of models over those of Takei and Katz [2013]. This study provides a set of baseline, quantitative predictions to compare with future laboratory experiments on forced pipe Poiseuille flow of partially molten mantle.Comment: 23 pages, 7 figures. Submitted to Geophysical Journal International on 25 April 2014. Revised after reviewer comments and resubmitted on 20 August 201

A variational framework for flow optimization using semi-norm constraints

When considering a general system of equations describing the space-time evolution (flow) of one or several variables, the problem of the optimization over a finite period of time of a measure of the state variable at the final time is a problem of great interest in many fields. Methods already exist in order to solve this kind of optimization problem, but sometimes fail when the constraint bounding the state vector at the initial time is not a norm, meaning that some part of the state vector remains unbounded and might cause the optimization procedure to diverge. In order to regularize this problem, we propose a general method which extends the existing optimization framework in a self-consistent manner. We first derive this framework extension, and then apply it to a problem of interest. Our demonstration problem considers the transient stability properties of a one-dimensional (in space) averaged turbulent model with a space- and time-dependent model "turbulent viscosity". We believe this work has a lot of potential applications in the fluid dynamics domain for problems in which we want to control the influence of separate components of the state vector in the optimization process.Comment: 30 page

Optimal control of multiscale systems using reduced-order models

We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy "first reduce, then optimize"--rather than "first optimize, then reduce"--is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessary conditions, under which the "first reduce, then control" strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the "first reduce, then optmize" approach and discuss possible pitfalls

A minimization principle for the description of time-dependent modes associated with transient instabilities

We introduce a minimization formulation for the determination of a finite-dimensional, time-dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime they can play a crucial role by either altering the system dynamics through the activation of other instabilities, or by creating sudden nonlinear energy transfers that lead to extreme responses. However, their essentially transient character makes their description a particularly challenging task. We develop a minimization framework that focuses on the optimal approximation of the system dynamics in the neighborhood of the system state. This minimization formulation results in differential equations that evolve a time-dependent basis so that it optimally approximates the most unstable directions. We demonstrate the capability of the method for two families of problems: i) linear systems including the advection-diffusion operator in a strongly non-normal regime as well as the Orr-Sommerfeld/Squire operator, and ii) nonlinear problems including a low-dimensional system with transient instabilities and the vertical jet in crossflow. We demonstrate that the time-dependent subspace captures the strongly transient non-normal energy growth (in the short time regime), while for longer times the modes capture the expected asymptotic behavior