Showing papers in "Quarterly of Applied Mathematics in 2017"

On singular limit equations for incompressible fluids in moving thin domains

Abstract: We consider the incompressible Euler and Navier-Stokes equations in a three-dimensional moving thin domain. Under the assumption that the moving thin domain degenerates into a two-dimensional moving closed surface as the width of the thin domain goes to zero, we give a heuristic derivation of singular limit equations on the degenerate moving surface of the Euler and Navier-Stokes equations in the moving thin domain and investigate relations between their energy structures. We also compare the limit equations with the Euler and Navier-Stokes equations on a stationary manifold, which are described in terms of the Levi-Civita connection.

The Bramson logarithmic delay in the cane toads equations

Abstract: We study a nonlocal reaction-diffusion-mutation equation modeling the spreading of a cane toads population structured by a phenotypical trait responsible for the spatial diffusion rate. When the trait space is bounded, the cane toads equation admits traveling wave solutions [7]. Here, we prove a Bramson type spreading result: the lag between the position of solutions with localized initial data and that of the traveling waves grows as (3/(2$\lambda$ *)) log t. This result relies on a present-time Harnack inequality which allows to compare solutions of the cane toads equation to those of a Fisher-KPP type equation that is local in the trait variable.

On derivation of compressible fluid systems on an evolving surface

Abstract: We consider the governing equations for the motion of compressible fluid on an evolving surface from both energetic and thermodynamic points of view. We employ our energetic variational approaches to derive the momentum equation of our compressible fluid systems on the evolving surface. Applying the first law of thermodynamics and the Gibbs equation, we investigate the internal energy, enthalpy, entropy, and free energy of the fluid on the evolving surface. We also study conservative forms and conservation laws of our compressible fluid systems on the evolving surface. Moreover, we derive the generalized heat and diffusion systems on an evolving surface from an energetic point of view. This paper gives a mathematical validity of the surface stress tensor determined by the Boussinesq-Scriven law. Using a flow map on an evolving surface and applying the Riemannian metric induced by the flow map are key ideas to analyze fluid flow on the evolving surface.

Fokas's Uniform Transform Method for linear systems

Abstract: We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constant-coefficient linear partial differential equations. We discuss how the apparent branch singularities typically appearing in the global relation are removable, allowing the method to proceed, in essence, as for scalar problems. We illustrate the use of the method with boundary value problems for the Klein-Gordon equation and the linearized Fitzhugh-Nagumo system. The case of wave equations is treated separately in an appendix.

A note on deconvolution with completely monotone sequences and discrete fractional calculus

Abstract: We study in this work convolution groups generated by completely monotone sequences related to the ubiquitous time-delay memory effect in physics and engineering. In the first part, we give an accurate description of the convolution inverse of a completely monotone sequence and show that the deconvolution with a completely monotone kernel is stable. In the second part, we study a discrete fractional calculus defined by the convolution group generated by the completely monotone sequence c = (1, 1, 1, . . .), and show the consistency with time-continuous Riemann-Liouville calculus, which may be suitable for modeling memory kernels in discrete time series.

Universal halting times in optimization and machine learning

Abstract: The authors present empirical distributions for the halting time (measured by the number of iterations to reach a given accuracy) of optimization algorithms applied to two random systems: spin glasses and deep learning. Given an algorithm, which we take to be both the optimization routine and the form of the random landscape, the fluctuations of the halting time follow a distribution that, after centering and scaling, remains unchanged even when the distribution on the landscape is changed. We observe two qualitative classes: A Gumbel-like distribution that appears in Google searches, human decision times, the QR eigenvalue algorithm and spin glasses, and a Gaussian-like distribution that appears in conjugate gradient method, deep network with MNIST input data and deep network with random input data. This empirical evidence suggests presence of a class of distributions for which the halting time is independent of the underlying distribution under some conditions.

Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation

Abstract: The dynamics of the magnetic distribution in a ferromagnetic material is governed by the Landau-Lifshitz equation, which is a nonlinear geometric dispersive equation with a nonconvex constraint that requires the magnetization to remain of unit length throughout the domain. In this article, we present a mass-lumped finite element method for the Landau-Lifshitz equation. This method preserves the nonconvex constraint at each node of the finite element mesh, and is energy nonincreasing. We show that the numerical solution of our method for the Landau-Lifshitz equation converges to a weak solution of the Landau-Lifshitz-Gilbert equation using a simple proof technique that cancels out the product of weakly convergent sequences. Numerical tests for both explicit and implicit versions of the method on a unit square with periodic boundary conditions are provided for structured and unstructured meshes.

Spatial decay in transient heat conduction for general elongated regions

Abstract: Zanaboni's procedure for establishing Saint-Venant's principle is ex- tended to anisotropic homogeneous transient heat conduction on regions that are successively embedded in each other to become indefinitely elon- gated. No further geometrical restrictions are imposed. The boundary of each region is maintained at zero temperature apart from the common surface of intersection which is heated to the same temperature assumed to be of bounded time variation. Heat sources are absent. Subject to these conditions, the thermal energy, supposed bounded in each region, becomes vanishingly small in those parts of the regions suficiently remote from the heated common surface. As with the original treatment, the proof involves certain monotone bounded sequences, and does not depend upon differential inequalities or the maximum principle. A definition is presented of an elongated region.

Balanced flux formulations for multidimensional Evans-function computations for viscous shocks

Abstract: The Evans function is a powerful tool for the stability analysis of viscous shock profiles; zeros of this function carry stability information. In the one-dimensional case, it is typical to compute the Evans function using Goodman's integrated coordinates [G1]; this device facilitates the search for zeros of the Evans function by winding number arguments. Although integrated coordinates are not available in the multidimensional case, we show here that there is a choice of coordinates which gives similar advantages.

A novel stochastic method for the solution of direct and inverse exterior elliptic problems

Abstract: A new method, in the interface of stochastic differential equations with boundary value problems, is developed in this work, aiming at representing solutions of exterior boundary value problems in terms of stochastic processes. The main effort concerns exterior harmonic problems but furthermore special attention has been paid on the investigation of time-reduced scattering processes (involving the Helmholtz operator) in the realm of low frequencies. The method, in principle, faces the construction of the solution of the direct versions of the aforementioned boundary value problems but the special features of the method assure definitely the usefulness of the approach to the solution of the corresponding inverse problems as clearly indicated herein.

Concentration inequalities for a removal-driven thinning process

Abstract: We prove exponential concentration estimates and a strong law of large numbers for a particle system that is the simplest representative of a general class of models for 2D grain boundary coarsening. The system consists of $n$ particles in $(0,\infty)$ that move at unit speed to the left. Each time a particle hits the boundary point $0$, it is removed from the system along with a second particle chosen uniformly from the particles in $(0,\infty)$. Under the assumption that the initial empirical measure of the particle system converges weakly to a measure with density $f_0(x) \in L^1_+(0,\infty)$, the empirical measure of the particle system at time $t$ is shown to converge to the measure with density $f(x,t)$, where $f$ is the unique solution to the kinetic equation with nonlinear boundary coupling $$\partial_t f (x,t) - \partial_x f(x,t) = -\frac{f(0,t)}{\int_0^\infty f(y,t)\, dy} f(x,t), \quad 0

Revisiting Diffusion: Self-similar Solutions and the $t^{-1/2}$ Decay in Initial and Initial-Boundary Value Problems

Abstract: The diffusion equation is a universal and standard textbook model for partial differential equations (PDEs). In this work, we revisit its solutions, seeking, in particular, self-similar profiles. This problem connects to the classical theory of special functions and, more specifically, to the Hermite as well as the Kummer hypergeometric functions. Reconstructing the solution of the original diffusion model from novel self-similar solutions of the associated self-similar PDE, we infer that the $t^{-1/2}$ decay law of the diffusion amplitude is {\it not necessary}. In particular, it is possible to engineer setups of {\it both} the Cauchy problem and the initial-boundary value problem in which the solution decays at a {\it different rate}. Nevertheless, we observe that the $t^{-1/2}$ rate corresponds to the dominant decay mode among integrable initial data, i.e., ones corresponding to finite mass. Hence, unless the projection to such a mode is eliminated, generically this decay will be the slowest one observed. In initial-boundary value problems, an additional issue that arises is whether the boundary data are \textit{consonant} with the initial data; namely, whether the boundary data agree at all times with the solution of the Cauchy problem associated with the same initial data, when this solution is evaluated at the boundary of the domain. In that case, the power law dictated by the solution of the Cauchy problem will be selected. On the other hand, in the non-consonant cases a decomposition of the problem into a self-similar and a non-self-similar one is seen to be beneficial in obtaining a systematic understanding of the resulting solution.

Oscillatory traveling wave solutions for coagulation equations

Abstract: We consider Smoluchowski's coagulation equation with kernels of homogeneity one of the form $K_{\\varepsilon }(\\xi,\\eta) =\\big( \\xi^{1-\\varepsilon }+\\eta^{1-\\varepsilon }\\big)\\big ( \\xi\\eta\\big) ^{\\frac{\\varepsilon }{2}}$. Heuristically, in suitable exponential variables, one can argue that in this case the long-time behaviour of solutions is similar to the inviscid Burgers equation and that for Riemann data solutions converge to a traveling wave for large times. Numerical simulations in \\cite{HNV16} indeed support this conjecture, but also reveal that the traveling waves are oscillatory and the oscillations become stronger with smaller $\\varepsilon$. The goal of this paper is to construct such oscillatory traveling wave solutions and provide details of their shape via formal matched asymptotic expansions.