3,611 research outputs found
Parabolic equations with singular divergence-free drift vector fields
In this paper, we study an elliptic operator in divergence-form but not
necessary symmetric. In particular, our results can be applied to elliptic
operator , where is a
time-dependent vector field in , which is divergence-free in
distribution sense, i.e. . Suppose . We show the existence of the
fundamental solution of the parabolic operator
, and show that satisfies the Aronson estimate with
a constant depending only on the dimension , the elliptic constant and
the norm . Therefore
the existence and uniqueness of the parabolic equation
are established for initial data in
-space, and their regularity is obtained too. In fact, we establish
these results for a general non-symmetric elliptic operator in divergence form.Comment: 28 page
Markov semi-groups generated by elliptic operators with divergence-free drift
In this paper we construct a conservative Markov semi-group with generator
on , where is a divergence-free
vector field which belongs to with . The
research is motivated by the question of understanding the blow-up solutions of
the fluid dynamic equations, which attracts a lot of attention in recent years.Comment: 12 page
Parabolic equations with divergence-free drift in space
In this paper we study the fundamental solution of
the parabolic operator , where for
every , is a divergence-free vector field, and we consider the
case that belongs to the Lebesgue space
. The regularity of
weak solutions to the parabolic equation depends critically on the
value of the parabolic exponent . Without the
divergence-free condition on , the regularity of weak solutions has been
established when , and the heat kernel estimate has been obtained
as well, except for the case that . The regularity of weak
solutions was deemed not true for the critical case
for a general
, while it is true for the divergence-free case, and a written proof can be
deduced from the results in [Semenov, 2006]. One of the results obtained in the
present paper establishes the Aronson type estimate for critical and
supercritical cases and for vector fields which are divergence-free. We
will prove the best possible lower and upper bounds for the fundamental
solution one can derive under the current approach. The significance of the
divergence-free condition enters the study of parabolic equations rather
recently, mainly due to the discovery of the compensated compactness. The
interest for the study of such parabolic equations comes from its connections
with Leray's weak solutions of the Navier-Stokes equations and the Taylor
diffusion associated with a vector field where the heat operator
appears naturally.Comment: 31 page
Bayesian modeling longitudinal dyadic data with nonignorable dropout, with application to a breast cancer study
Dyadic data are common in the social and behavioral sciences, in which
members of dyads are correlated due to the interdependence structure within
dyads. The analysis of longitudinal dyadic data becomes complex when
nonignorable dropouts occur. We propose a fully Bayesian selection-model-based
approach to analyze longitudinal dyadic data with nonignorable dropouts. We
model repeated measures on subjects by a transition model and account for
within-dyad correlations by random effects. In the model, we allow subject's
outcome to depend on his/her own characteristics and measure history, as well
as those of the other member in the dyad. We further account for the
nonignorable missing data mechanism using a selection model in which the
probability of dropout depends on the missing outcome. We propose a Gibbs
sampler algorithm to fit the model. Simulation studies show that the proposed
method effectively addresses the problem of nonignorable dropouts. We
illustrate our methodology using a longitudinal breast cancer study.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS515 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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