Mihaela Ifrim, Ben Pineau, Daniel Tataru, Mitchell A. Taylor
In this article, we provide a definitive well-posedness theory for the free boundary problem in incompressible magnetohyrodynamics. Despite the clear physical interest in this system and the remarkable progress in the study of the free boundary Euler equations in recent decades, the low regularity well-posedness of the free boundary MHD equations has remained completely open. This is due, in large part, to the highly nonlinear wave-type coupling between the velocity, magnetic field and free boundary, which has forced previous works to impose restrictive geometric constraints on the data. To address this problem, we introduce a novel Eulerian approach and an entirely new functional setting, which better captures the wave equation structure of the MHD equations and permits a complete Hadamard well-posedness theory in low-regularity Sobolev spaces. In particular, we give the first proofs of existence, uniqueness and continuous dependence on the data at the sharp $s>\frac{d}{2}+1$ Sobolev regularity, in addition to a blowup criterion for smooth solutions at the same low regularity scale. Moreover, we provide a completely new method for constructing smooth solutions which, to our knowledge, gives the first proof of existence (at any regularity) in our new functional setting. All of our results hold in arbitrary dimensions and in general, not necessarily simply connected, domains. By taking the magnetic field to be zero, they also recover the corresponding sharp well-posedness theorems for the free boundary Euler equations. The methodology and tools that we employ here can likely be fruitfully implemented in other free boundary models.
An industrial scheme, to simulate the compressible two-phase flow in porous media, consists of a finite volume method together with a phase-by-phase upstream scheme. The implicit finite volume scheme satisfies industrial constraints of robustness since the proposed scheme discretizes the equations with gravity and capillary terms. We show that the proposed scheme satisfies the maximum principle for the saturation, a discrete-energy estimate on the pressures, and a function of the saturation that denotes capillary terms. These stability results allow us to derive the convergence of a subsequence to a weak solution of the continuous equations as the size of the discretization tends to zero. To our knowledge, this is the first convergence result of a finite volume scheme in the case of two-phase compressible flow in several space dimensions. The proof is given for the complete system when the density of each phase depends on its own pressure.
Advanced Numerical Methods in Computational Mathematics
Jayanth N. Kudva, Kari Appa, Christopher A. Martin, A. P. Jardine · 16 authors
The concept of an adaptive aircraft wing, i.e., whose shape parameters such as camber, span-wise twist, and thickness can be varied to optimize the wing shape for various flight conditions, has been extensively studied by numerous researchers [1-8]. While the aerodynamic benefits (in terms of increased lift/drag ratios, improved maneuverability, and delayed flow separation) have been analytically and experimentally established, the complexity and weight penalty of the designs and actuation mechanisms have limited their practical implementation. Recent developments in sensors and actuators using materials could potentially alleviate the shortcomings of prior designs, leading the way to a more practical smart adaptive wing which responds to changes in flight and environmental conditions by optimally modifying its shape. This paper presents the results of recent work conducted under a Defense Advanced Research Projects Agency (DARPA) contract entitled Structures and Materials Development - Smart Wing. In particular, development and testing of the wing wind tunnel model are presented. Limitations and potential benefits of adaptive wing designs are also discussed, along with recommendations for future work required to develop an operational adaptive wing.