arXiv subjects

Jeffrey Rauch

Publications and source records attributed to Jeffrey Rauch.

16 recordsLinked to original sources

arXiv2022

Perfectly Matched Layers on Cubic Domains forPauli's Equations

This article proves the well posedness of the boundary value problemthat arises when PML algorithms are applied to Pauli's equationswith a three dimensional rectangle as computational domain. The absorptionsare positive near the boundary and zero far from the boundary so are always x-dependent. At the flat parts of the boundary of the rectangle, the natural absorbing boundary conditions are imposed.The difficulty addressed is the analysis of the resulting variable coeffi-cient problem on the rectanglar solid with its edges and corners. TheLaplace transform is analysed. It turns on the analysis of a boundaryvalue problem formally obtained by complex stretching. Existence isproved by deriving a boundary value problems for a complex stretchedHelmholtz equation on smoothed domains. This is the first stabilityproof with x-dependent absorptions on a bounded domain whoseboundary is not smooth.

math.AP↗

arXiv2019

A Discrete Algorithm for General Weakly Hyperbolic Systems

This paper studies the Cauchy problem for variable coefficient weakly hyperbolic first order systems of partial differential operators. The hyperbolicity assumption is that for each $t, x$ the principal symbol is hyperbolic. No hypothesis is imposed on lower order terms. For coefficients and Cauchy data sufficiently Gevrey regular the Cauchy problem has a unique sufficiently Gevrey regular solution. We prove stability and error estimates for the spectral Crank-Nicholson scheme. Approximate solutions can be computed with accuracy $epsilon$ in the supremum norm with cost growing at most polynomially in $epsilon^{-1}$. The proofs use the symmetrizers from [2].

math.AP↗

arXiv2019

Long Time Boundedness of Planar Jump Discontinuities for Homogeneous Hyperbolic Systems

Suppose that $L(\partial_t,\partial_x)$ is a homogeneous constant coefficient strongly hyperbolic partial differential operator on ${\mathbb R}^{1+d}$ and $H$ is a characteristic hyperplane. Suppose that in a conic neighborhood of the conormal variety of $H$, the characteristic variety of $L$ is the graph of a real analytic function $τ(ξ)$ with ${\rm rank}\,τ_{ξξ}$ identically equal to zero or the maximal possible value $d-1$. Suppose that the source function $f$ is compactly supported in $t\ge 0$ and piecewise smooth with singularities only on $H$. Then the solution of $Lu=f$ with $u=0$ for $t<0$ is uniformly bounded on ${\mathbb R}^{1+d}$. Typically when ${\rm rank}\,τ_{ξξ}\ne 0$ on the conormal variety, the sup norm of the the jump in the gradient of $u$ across $H$ grows linearly with $t$.

math.AP↗

arXiv2018

Crime Pays; Homogenized Wave Equations for Long Times

This article examines the accuracy for large times of asymptotic expansions from periodic homogenization of wave equations. As usual, $ε$ denotes the small period of the coefficients in the wave equation. We first prove that the standard two scale asymptotic expansion provides an accurate approximation of the exact solution for times $t$ of order $ε^{-2+δ}$ for any $δ>0$. Second, for longer times, we show that a different algorithm, that is called criminal because it mixes different powers of $ε$, yields an approximation of the exact solution with error $O(ε^N)$ for times $ε^{-N}$ with $N$ as large as one likes. The criminal algorithm involves high order homogenized equations that, in the context of the wave equation, were first proposed by Santosa and Symes and analyzed by Lamacz. The high order homogenized equations yield dispersive corrections for moderate wave numbers. We give a systematic analysis for all time scales and all high order corrective terms.

math.AP↗

arXiv2016

Convergence along mean flows

We develop a technique of multiple scale asymptotic expansions along mean flows and a corresponding notion of weak multiple scale convergence. These are applied to homogenize convection dominated parabolic equations with rapidly oscillating, locally periodic coefficients and $\mathcal{O}(\eps^{-1})$ mean convection term. Crucial to our analysis is the introduction of a fast time variable, $τ=\frac{t}{\eps}$, not apparent in the heterogeneous problem. The effective diffusion coefficient is expressed in terms of the average of Eulerian cell solutions along the orbits of the mean flow in the fast time variable. To make this notion rigorous, we use the theory of ergodic algebras with mean value.

math.AP↗

arXiv2016

Eigenvalues for Maxwell's equations with dissipative boundary conditions

Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup generated by Maxwell's equations in an exterior domain $Ω\subset {\mathbb R}^3$ with dissipative boundary condition $E_{tan}- γ(x) (ν\wedge B_{tan}) = 0, γ(x) > 0, \forall x \in Γ= \partial Ω.$ We prove that if $γ(x)$ is nowhere equal to 1, then for every $0 < ε\ll 1$ and every $N \in {\mathbb N}$ the eigenvalues of $G_b$ lie in the region $Λ_ε \cup {\mathcal R}_N,$ where $Λ_ε = \{ z \in {\mathbb C}:\: |\Re z | \leq C_ε (|\Im z|^{\frac{1}{2} + ε} + 1), \: \Re z < 0\},$ ${\mathcal R}_N = \{z \in {\mathbb C}:\: |\Im z| \leq C_N (|\Re z| + 1)^{-N},\: \Re z < 0\}.$

math.AP↗

arXiv2014

Spectral problems for non elliptic symmetric systems with dissipative boundary conditions

This paper considers and extends spectral and scattering theory to dissipative symmetric systems that may have zero speeds and in particular to strictly dissipative boundary conditions for Maxwell's equations. Consider symmetric systems $\partial_t - \sum_{j=1}^n A_j \partial_{x_j}$ in ${\mathbb R}^n,\: n \geq 3$, $n$ odd, in a smooth connected exterior domain $Ω:= {\mathbb R}^n \setminus \bar{K}$. Assume that the rank of $A(ξ) = \sum_{j= 1}^n A_j ξ_j$ is constant for $ξ\not= 0.$ For maximally dissipative boundary conditions on $Ω:={\mathbb R}^n \setminus \bar{K}$ with bounded open domain $K$ the solution of the boundary problem in ${\mathbb R}^{+} \times Ω$ is described by a contraction semigroup $V(t) = e^{t G_b},\:t \geq 0.$ Assuming coercive conditions for $G_b$ and its adjoint $G_b^*$ on the complement of their kernels, we prove that the spectrum of $G_b$ in the open half plane $\Re z < 0$ is formed only by isolated eigenvalues with finite multiplicities.

math.FA↗

arXiv2013

Diffraction of Bloch Wave Packets for Maxwell's Equations

We study, for times of order 1/h, solutions of Maxwell's equations in an O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer elliptic. Coercivity holds only on the complement of an infinite dimensional kernel. The system structure requires many innovations.

math.AP↗

arXiv2012

A bound on group velocity for Bloch wave packets

This short note is a sequel to our previous papers on the asymptotic behavior of Bloch wave packet solutions of the wave equation in periodic media. The purpose is to prove that the group velocity for these Bloch wave packets is bounded by the maximal speed of propagation for the original wave equation.

math.AP↗

arXiv2011

Global Stability of Steady Transonic Euler Shocks in Quasi-One-Dimensional Nozzles

We prove global in time dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. We assume neither the smallness of the relative slope of the nozzle nor the weakness of the shock. Key ingredients of the proof are an exponentially decaying energy estimate for a linearized problem together with methods from \cite{LRXX}.

math.AP↗

arXiv2011

The Analysis of Matched Layers

A systematic analysis of matched layers is undertaken with special attention to better understand the remarkable method of B\'erenger. We prove that the B\'erenger and closely related layers define well posed transmission problems in great generality. When the B\'erenger method or one of its close relatives is well posed, perfect matching is proved. The proofs use the energy method, Fourier-Laplace transform, and real coordinate changes for Laplace transformed equations. It is proved that the loss of derivatives associated with the B\'erenger method does not occur for elliptic generators. More generally, an essentially necessary and sufficient condition for loss of derivatives in B\'erenger's method is proved. The sufficiency relies on the energy method with pseudodifferential multiplier. Amplifying and nonamplifying layers are identified by a geometric optics computation. Among the various flavors of B\'erenger's algorithm for Maxwell's equations our favorite choice leads to a strongly well posed augmented system and is both perfect and nonamplifying in great generality. We construct by an extrapolation argument an alternative matched layer method which preserves the strong hyperbolicity of the original problem and though not perfectly matched has leading reflection coefficient equal to zero at all angles of incidence.

math.NA↗

arXiv2010

Incoming and disappearing solutions for Maxwell's equations

We prove that in contrast to the free wave equation in $\R^3$ there are no incoming solutions of Maxwell's equations in the form of spherical or modulated spherical waves. We construct solutions which are corrected by lower order incoming waves. With their aid, we construct dissipative boundary conditions and solutions to Maxwell's equations in the exterior of a sphere which decay exponentially as $t \to +\infty$. They are asymptotically disappearing. Disappearing solutions which are identically zero for $t \geq T > 0$ are constructed which satisfy maximal dissipative boundary conditions which depend on time $t$. Both types are invisible in scattering theory.

math-ph↗

arXiv2010

Optimal Focusing for Monochromatic Scalar and Electromagnetic Waves

For monochromatic solutions of D'Alembert's wave equation and Maxwell's equations, we obtain sharp bounds on the sup norm as a function of the far field energy. The extremizer in the scalar case is radial. In the case of Maxwell's equation, the electric field maximizing the value at the origin follows longitude lines on the sphere at infinity. In dimension $d=3$ the highest electric field for Maxwell's equation is smaller by a factor 2/3 than the highest corresponding scalar waves. The highest electric field densities on the balls $B_R(0)$ occur as $R\to 0$. The density dips to half max at $R$ approximately equal to one third the wavelength. The extremizing fields are identical to those that attain the maximum field intensity at the origin.

math.AP↗

arXiv2010

Stability of Transonic Shock Solutions for One-Dimensional Euler-Poisson Equations

In this paper, both structural and dynamical stabilities of steady transonic shock solutions for one-dimensional Euler-Poission system are investigated. First, a steady transonic shock solution with supersonic backgroumd charge is shown to be structurally stable with respect to small perturbations of the background charge, provided that the electric field is positive at the shock location. Second, any steady transonic shock solution with the supersonic background charge is proved to be dynamically and exponentially stable with respect to small perturbation of the initial data, provided the electric field is not too negative at the shock location. The proof of the first stability result relies on a monotonicity argument for the shock position and the downstream density, and a stability analysis for subsonic and supersonic solutions. The dynamical stability of the steady transonic shock for the Euler-Poisson equations can be transformed to the global well-posedness of a free boundary problem for a quasilinear second order equation with nonlinear boundary conditions. The analysis for the associated linearized problem plays an essential role.

math.AP↗

arXiv2009

Dispersive Stabilization

Ill posed linear and nonlinear initial value problems may be stabilized, that it converted to to well posed initial value problems, by the addition of purely nonscalar linear dispersive terms. This is a stability analog of the Turing instability. This idea applies to systems of quasilinear Schrödinger equations from nonlinear optics.

math.AP↗

arXiv2002

Focusing of Spherical Nonlinear Pulses in ${\mathbb R}^{1+3}$, III. Sub and Supercritical cases

We study the validity of geometric optics in $L^\infty$ for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is subcritical, then no nonlinear effect occurs at leading order. If the amplitude of the initial data is sufficiently big, strong nonlinear effects occur; we study the cases where the equation is either dissipative or accretive. When the equation is dissipative, pulses are absorbed before reaching the focal point. When the equation is accretive, the family of pulses becomes unbounded.

math.AP↗