387 research outputs found
Local boundedness and Harnack inequality for solutions of linear non-uniformly elliptic equations
We study local regularity properties for solutions of linear, non-uniformly
elliptic equations. Assuming certain integrability conditions on the
coefficient field, we prove local boundedness and Harnack inequality. The
assumed integrability assumptions are essentially sharp and improve upon
classical results by Trudinger [ARMA 1971]. We then apply the deterministic
regularity results to the corrector equation in stochastic homogenization and
establish sublinearity of the corrector
Quenched invariance principle for random walks with time-dependent ergodic degenerate weights
We study a continuous-time random walk, , on in an
environment of dynamic random conductances taking values in . We
assume that the law of the conductances is ergodic with respect to space-time
shifts. We prove a quenched invariance principle for the Markov process
under some moment conditions on the environment. The key result on the
sublinearity of the corrector is obtained by Moser's iteration scheme.Comment: 34 pages; in this version a minor technical gap in the proof of the
results in Section 5 has been remove
Test particle propagation in magnetostatic turbulence. 1. Failure of the diffusion approximation
The equation which governs the quasi-linear approximation to the ensemble and gyro-phase averaged one-body probability distribution function is constructed from first principles. This derived equation is subjected to a thorough investigation in order to calculate the possible limitations of the quasi-linear approximation. It is shown that the reduction of this equation to a standard diffusion equation in the Markovian limit can be accomplished through the application of the adiabatic approximation. A numerical solution of the standard diffusion equation in the Markovian limit is obtained for the narrow parallel beam injection. Comparison of the diabatic and adiabatic results explicitly demonstrates the failure of the Markovian description of the probability distribution function. Through the use of a linear time-scale extension the failure of the adiabatic approximation, which leads to the Markovian limit, is shown to be due to mixing of the relaxation and interaction time scales in the presence of the strong mean field
Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization
We consider degenerate elliptic equations of second order in divergence form
with a symmetric random coefficient field . Extending the work of the first
author, Fehrman, and Otto [Ann. Appl. Probab. 28 (2018), no. 3, 1379-1422], who
established the large-scale regularity of -harmonic functions
in a degenerate situation, we provide stretched exponential moments for the
minimal radius describing the minimal scale for this
regularity. As an application to stochastic homogenization, we partially
generalize results by Gloria, Neukamm, and Otto [Anal. PDE 14 (2021), no. 8,
2497-2537] on the growth of the corrector, the decay of its gradient, and a
quantitative two-scale expansion to the degenerate setting. On a technical
level, we demand the ensemble of coefficient fields to be stationary and
subject to a spectral gap inequality, and we impose moment bounds on and
. We also introduce the ellipticity radius which encodes the
minimal scale where these moments are close to their positive expectation
value
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