5,964 research outputs found
contraction for bounded (non-integrable) solutions of degenerate parabolic equations
We obtain new contraction results for bounded entropy solutions of
Cauchy problems for degenerate parabolic equations. The equations we consider
have possibly strongly degenerate local or non-local diffusion terms. As
opposed to previous results, our results apply without any integrability
assumption on the %(the positive part of the difference of) solutions. They
take the form of partial Duhamel formulas and can be seen as quantitative
extensions of finite speed of propagation local contraction results for
scalar conservation laws. A key ingredient in the proofs is a new and
non-trivial construction of a subsolution of a fully non-linear (dual)
equation. Consequences of our results are maximum and comparison principles,
new a priori estimates, and in the non-local case, new existence and uniqueness
results
Constructing a partially transparent computational boundary for UPPE using leaky modes
In this paper we introduce a method for creating a transparent computational
boundary for the simulation of unidirectional propagation of optical beams and
pulses using leaky modes. The key element of the method is the introduction of
an artificial-index material outside a chosen computational domain and
utilization of the quasi-normal modes associated with such artificial
structure. The method is tested on the free space propagation of TE
electromagnetic waves. By choosing the material to have appropriate optical
properties one can greatly reduce the reflection at the computational boundary.
In contrast to the well-known approach based on a perfectly matched layer, our
method is especially well suited for spectral propagators.Comment: 32 pages, 19 figure
Analysis and assessment of film materials and associated manufacturing processes for a solar sail
Candidate resin manufacturers and film producers were surveyed to determine the availability of key materials and to establish the capabilities of fabricators to prepare ultrathin films of these materials within the capacity/cost/time constraints of the Halley program. Infrared spectra of three candidate samples were obtained by pressing each sample against an internal reflection crystal with the polymer sandwiched between the crystal and the metal backing. The sample size was such that less than one-fourth of the surface of the crystal was covered with the sample. This resulted in weak spectra requiring a six-fold expansion. Internal reflection spectra of the three samples were obtained using both a KRS-5 and a Ge internal reflection crystal. Subtracted infrared spectra of the three samples are presented
Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory
Abstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations
âtu â LĎ,Îź[Ď(u)] = f(x,t) in RN Ă (0,T),
where LĎ,Îź is a general symmetric diffusion operator of L Ěevy type and Ď is
merely continuous and non-decreasing. We then use this theory to prove con-
vergence for many different numerical schemes. In the nonlocal case most of
the results are completely new. Our theory covers strongly degenerate Stefan
problems, the full range of porous medium equations, and for the first time
for nonlocal problems, also fast diffusion equations. Examples of diffusion op-
Ď,Îź Îą
are the (fractional) Laplacians â and â(ââ)2 for Îą â (0,2),
erators L
discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L Ěevy operators, allows us to give a unified and compact nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and conver- gence of the methods under minimal assumptions â including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in [28]. We also present some numerical tests, but extensive testing is deferred to the companion paper [31] along with a more detailed discussion of the numerical methods included in our theory
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