338 research outputs found
A quasinonlocal coupling method for nonlocal and local diffusion models
In this paper, we extend the idea of "geometric reconstruction" to couple a
nonlocal diffusion model directly with the classical local diffusion in one
dimensional space. This new coupling framework removes interfacial
inconsistency, ensures the flux balance, and satisfies energy conservation as
well as the maximum principle, whereas none of existing coupling methods for
nonlocal-to-local coupling satisfies all of these properties. We establish the
well-posedness and provide the stability analysis of the coupling method. We
investigate the difference to the local limiting problem in terms of the
nonlocal interaction range. Furthermore, we propose a first order finite
difference numerical discretization and perform several numerical tests to
confirm the theoretical findings. In particular, we show that the resulting
numerical result is free of artifacts near the boundary of the domain where a
classical local boundary condition is used, together with a coupled fully
nonlocal model in the interior of the domain
Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms
In this work we study a system of parabolic reaction-diffusion equations which are coupled not only through the reaction terms but also by way of nonlocal diffusivity functions. For the associated initial problem, endowed with homogeneous Dirichlet or Neumann boundary conditions, we prove the existence of global solutions. We also prove the existence of local solutions but with less assumptions on the boundedness of the nonlocal terms. The uniqueness result is established next and then we find the conditions under which the existence of strong solutions is assured. We establish several bow-up results for the strong solutions to our problem and we give a criterium for the convergence of these solutions towards a homogeneous state.CAPES [BEX 2478-12-8]; MEC/MCTI/CAPES/CNPq/FAPs, Brazil [71/2013, 88881.030388/2013-01]; Fundacao para a Ciencia e a Tecnologia, Portugal [UID/MAT/04561/2013-2015]info:eu-repo/semantics/publishedVersio
Exponential quasi-ergodicity for processes with discontinuous trajectories
This paper establishes exponential convergence to a unique quasi-stationary
distribution in the total variation norm for a very general class of strong
Markov processes. Specifically, we can treat nonreversible processes with
discontinuous trajectories, which seems to be a substantial breakthrough.
Considering jumps driven by Poisson Point Processes in two different
applications, we intend to illustrate the potential of these results and
motivate our criteria. Our set of conditions is expected to be much easier to
verify than an implied property which is crucial in our proof, namely a
comparison of asymptotic extinction rates between different initial conditions.
Keywords : continuous-time and continuous-space Markov process , jumps ,
quasi-stationary distribution , survival capacity , Q-process , Harris
recurrenc
Large Time Behavior of Periodic Viscosity Solutions for Uniformly Parabolic Integro-Differential Equations
International audienceIn this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form . We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of ''mixed operators'' for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis
A domain decomposition scheme for couplings between local and nonlocal equations
We study a natural alternating method of Schwarz type (domain decomposition)
for certain class of couplings between local and nonlocal operators. We show
that our method fits into Lion's framework and prove, as a consequence,
convergence in both, the continuous and the discrete settings
Wasserstein decay of one dimensional jump-diffusions
This work is devoted to the Lipschitz contraction and the long time behavior
of certain Markov processes. These processes diffuse and jump. They can
represent some natural phenomena like size of cell or data transmission over
the Internet. Using a Feynman-Kac semigroup, we prove a bound in Wasserstein
metric. This bound is explicit and optimal in the sense of Wasserstein
curvature. This notion of curvature is relatively close to the notion of
(coarse) Ricci curvature or spectral gap. Several consequences and examples are
developed, including an spectral for general Markov processes, explicit
formulas for the integrals of compound Poisson processes with respect to a
Brownian motion, quantitative bounds for Kolmogorov-Langevin processes and some
total variation bounds for piecewise deterministic Markov processes
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