arXiv subjects

Walter Craig

Publications and source records attributed to Walter Craig.

12 recordsLinked to original sources

arXiv2019

On the initial value problem for the electromagnetic wave equation in Friedmann-Robertson-Walker space-times

We solve the source free electromagnetic wave equation in Friedmann-Robertson-Walker space-times for curvature $K=0$ and $K=-1$. Deriving a solution expression in the form of spherical means we deduce and compare two properties of the Maxwell propagator namely, decay rates, as well as continuity through the space-time singularity to that of the scalar wave equation presented by Abbasi and Craig [1].

math.AP↗

arXiv2019

Steady waves in flows over periodic bottoms

We study the formation of steady waves in two-dimensional fluids under a current with mean velocity $c$ flowing over a periodic bottom. Using a formulation based on the Dirichlet-Neumann operator, we establish the unique continuation of a steady solution from the trivial solution when a flat bottom is perturbed, except for a sequence of velocities $c_{k}$. The main contribution is the proof that at least two steady solutions exist close to a non-degenerate $S^{1}$-orbit of non-constant steady waves when a flat bottom is perturbed. Consequently, we obtain persistence of at least two steady waves close to a non-degenerate $S^{1}$-orbit of Stokes waves bifurcating from the velocities $c_{k}$.

math.AP↗

arXiv2019

Standing waves of fixed period for $n+1$ vortex filaments

The $n+1$ vortex filament problem has explicit solutions consisting of $n$ parallel filaments of equal circulation in the form of nested polygons uniformly rotating around a central filament which has circulation of opposite sign. We show that when the relation between temporal and spatial periods is fixed at certain rational numbers, these configurations have an infinite number of homographic time dependent standing wave patterns that bifurcate from these uniformly rotating central configurations.

math.AP↗

arXiv2017

Bloch theory and spectral gaps for linearized water waves

The system of equations for water waves, when linearized about equilibrium of a fluid body with a varying bottom boundary, is described by a spectral problem for the Dirichlet -- Neumann operator of the unperturbed free surface. This spectral problem is fundamental in questions of stability, as well as to the perturbation theory of evolution of the free surface in such settings. In addition, the Dirichlet -- Neumann operator is self-adjoint when given an appropriate definition and domain, and it is a novel but very natural spectral problem for a nonlocal operator. In the case in which the bottom boundary varies periodically, $\{y = -h + b(x)\}$ where $b(x+\gamma) = b(x)$, $\gamma \in \Gamma$ a lattice, this spectral problem admits a Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. In this article we describe this analytic construction in the case of a spatially periodic bottom variation from constant depth in two space dimensional water waves problem, giving a construction of the Bloch eigenfunctions and eigenvalues as a function of the band parameters and a description of the Dirichlet -- Neumann operator in terms of the bathymetry $b(x)$. One of the consequences of this description is that the spectrum consists of a series of bands separated by spectral gaps which are zones of forbidden energies. For a given generic periodic bottom profile $b(x)=\varepsilon \beta(x)$, every gap opens for a sufficiently small value of the perturbation parameter $\varepsilon$.

math.AP↗

arXiv2016

On the Hamiltonian for water waves

Many equations that arise in a physical context can be posed in the form of a Hamiltonian system, meaning that there is a symplectic structure on an appropriate phase space, and a Hamiltonian functional with respect to which time evolution of their solutions can be expressed in terms of a Hamiltonian vector field. It is known from the work of VE Zakharov that the equations for water waves can be posed as a Hamiltonian dynamical system, and that the equilibrium solution is an elliptic stationary point. In this article we generalize the Hamiltonian formulation of water waves by Zakharov to a general coordinatization of the dynamical free surface, which allows it to apply to situations that include overturning wave profiles. This answers a question posed to the author by T.~Nishida during the RIMS Symposium on Mathematical Analysis in Fluid and Gas Dynamics that took place during July 6 - 8 2016.

math.AP↗

arXiv2014

On the initial value problem for the wave equation in Friedmann -- Robertson -- Walker space-times

We study the wave propagator for a Friedmann - Robertson - Walker background space-time, which is singular at time t=0. Using a spherical means formulation for the solution of the wave equation that is due to Klainerman and Sarnak, we derive three properties of solutions. the first is sharp time decay properties, using ideas of Fritz John. the second is that the wave equation in Friedmann - Robertson - Walker space-time does not satisfy the sharp Huygens property. The third is that the initial value problem for a class of data posed at the tingular time is well defined. Since the wave equation is reversible, this represents a class of data which propagates information smoothly from the past to the future passing through the space-time singularity.

math-ph↗

arXiv2013

Global Wellposeness for the 3D inhomogeneous incompressible Navier-Stokes equations

This paper addresses the three-dimensional Navier-Stokes equations for an incompressible fluid whose density is permitted to be inhomogeneous. We establish a theorem of global existence and uniqueness of strong solutions for initial data with small $\dot{H}^{\frac12}$-norm, which also satisfies a natural compatibility condition. A key point of the theorem is that the initial density need not be strictly positive.

math.AP↗

arXiv2011

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.

math.AP↗

arXiv2009

Bounds on Kolmogorov spectra for the Navier - Stokes equations

Let $u(x,t)$ be a (possibly weak) solution of the Navier - Stokes equations on all of ${\mathbb R}^3$, or on the torus ${\mathbb R}^3/ {\mathbb Z}^3$. The {\it energy spectrum} of $u(\cdot,t)$ is the spherical integral \[ E(κ,t) = \int_{|k| = κ} |\hat{u}(k,t)|^2 dS(k), \qquad 0 \leq κ< \infty, \] or alternatively, a suitable approximate sum. An argument involking scale invariance and dimensional analysis given by Kolmogorov (1941) and Obukhov (1941) predicts that large Reynolds number solutions of the Navier - Stokes equations in three dimensions should obey \[ E(κ, t) \sim C_0\varepsilon^{2/3}κ^{-5/3} \] over an inertial range $κ_1 \leq κ\leq κ_2$, at least in an average sense. We give a global estimate on weak solutions in the norm $\|{\mathcal F}\partial_x u(\cdot, t)\|_\infty$ which gives bounds on a solution's ability to satisfy the Kolmogorov law. A subsequent result is for rigorous upper and lower bounds on the inertial range, and an upper bound on the time of validity of the Kolmogorov spectral regime.

math-ph↗

arXiv2008

On determinism and well-posedness in multiple time dimensions

We study the initial value problem for the wave equation and the ultrahyperbolic equation for data posed on initial surface of mixed signature (both spacelike and timelike). Under a nonlocal constraint, we show that the Cauchy problem on codimension-one hypersurfaces has global unique solutions in the Sobolev spaces $H^{m}$, thus it is well-posed. In contrast, we show that the initial value problem on higher codimension hypersurfaces is ill-posed, at least when specifying a finite number of derivatives of the data, due to the failure of uniqueness. This is in contrast to a uniqueness result which Courant and Hilbert deduce from Asgeirsson's mean value theorem, for which we give an independent derivation. The proofs use Fourier synthesis and the Holmgren-John uniqueness theorem.

math-ph↗

arXiv2008

Towards a New Proof of Anderson Localization

The wave function of a non-relativistic particle in a periodic potential admits oscillatory solutions, the Bloch waves. In the presence of a random noise contribution to the potential the wave function is localized. We outline a new proof of this Anderson localization phenomenon in one spatial dimension, extending the classical result to the case of a periodic background potential. The proof makes use of techniques previously developed to study the effects of noise on reheating in inflationary cosmology, employing methods of random matrix theory.

hep-th↗

arXiv2007

Long wave expansions for water waves over random topography

In this paper, we study the motion of the free surface of a body of fluid over a variable bottom, in a long wave asymptotic regime. We assume that the bottom of the fluid region can be described by a stationary random process $β(x, ω)$ whose variations take place on short length scales and which are decorrelated on the length scale of the long waves. This is a question of homogenization theory in the scaling regime for the Boussinesq and KdV equations. The analysis is performed from the point of view of perturbation theory for Hamiltonian PDEs with a small parameter, in the context of which we perform a careful analysis of the distributional convergence of stationary mixing random processes. We show in particular that the problem does not fully homogenize, and that the random effects are as important as dispersive and nonlinear phenomena in the scaling regime that is studied. Our principal result is the derivation of effective equations for surface water waves in the long wave small amplitude regime, and a consistency analysis of these equations, which are not necessarily Hamiltonian PDEs. In this analysis we compute the effects of random modulation of solutions, and give an explicit expression for the scattered component of the solution due to waves interacting with the random bottom. We show that the resulting influence of the random topography is expressed in terms of a canonical process, which is equivalent to a white noise through Donsker's invariance principle, with one free parameter being the variance of the random process $β$. This work is a reappraisal of the paper by Rosales & Papanicolaou \cite{RP83} and its extension to general stationary mixing processes.

math.AP↗