1,544 research outputs found
On the global regularity for anisotropic dissipative surface quasi-geostrophic equation
In this paper, we consider the two-dimensional surface quasi-geostrophic
equation with fractional horizontal dissipation and fractional vertical thermal
diffusion. Global existence of classical solutions is established when the
dissipation powers are restricted to a suitable range. Due to the nonlocality
of these 1D fractional operators, some of the standard energy estimate
techniques no longer apply, to overcome this difficulty, we establish several
anisotropic embedding and interpolation inequalities involving fractional
derivatives. In addition, in order to bypass the unavailability of the
classical Gronwall inequality, we establish a new logarithmic type Gronwall
inequality, which may be of independent interest and potential applications.Comment: To appear in Nonlinearity. We have added some more details. The new
title is "On the global regularity for anisotropic dissipative surface
quasi-geostrophic equation
Global regularity of the two-dimensional regularized MHD equations
In this paper, we consider the Cauchy problem of the two-dimensional
regularized incompressible magnetohydrodynamics equations. The main objective
of this paper is to establish the global regularity of classical solutions of
the magnetohydrodynamics equations with the minimal dissipation. Consequently,
our results significantly improve the previous works.Comment: This is the final version published in Dynamics of Partial
Differential Equation
A note on global regularity results for 2D Boussinesq equations with fractional dissipation
In this paper we study the Cauchy problem for the two-dimensional (2D)
incompressible Boussinesq equations with fractional Laplacian dissipation and
thermal diffusion. Invoking the energy method and several commutator estimates,
we get the global regularity result of the 2D Boussinesq equations as long as
with . As a result, this result is a
further improvement of the previous two works \cite{MX,YXX}.Comment: 15 page
Global regularity of 2D tropical climate model with zero thermal diffusion
This article studies the global regularity problem of the two-dimensional
zero thermal diffusion tropical climate model with fractional dissipation,
given by in the barotropic mode equation and by
in the first baroclinic mode of the vector velocity
equation. More precisely, we show that the global regularity result holds true
as long as with . In addition, with no
dissipation from both the temperature and the first baroclinic mode of the
vector velocity, we also establish the global regularity result with the
dissipation strength at the logarithmically supercritical level. Finally, our
arguments can be extended to obtain the corresponding global regularity results
of the higher dimensional cases.Comment: 21 pages, submitted for publication in 201
Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation
As a continuation of the previous work [40], in this paper we focus on the
Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations
with fractional Laplacian dissipation. We give an elementary proof of the
global regularity of the smooth solutions of the 2D Boussinesq equations with a
new range of fractional powers of the Laplacian. The argument is based on the
nonlinear lower bounds for the fractional Laplacian established in [12].
Consequently, this result significantly improves the recent works [12, 38, 40].Comment: This version fix several typos of the previous one. 25 page
Shortcut Matrix Product States and its applications
Matrix Product States (MPS), also known as Tensor Train (TT) decomposition in
mathematics, has been proposed originally for describing an (especially
one-dimensional) quantum system, and recently has found applications in various
applications such as compressing high-dimensional data, supervised kernel
linear classifier, and unsupervised generative modeling. However, when applied
to systems which are not defined on one-dimensional lattices, a serious
drawback of the MPS is the exponential decay of the correlations, which limits
its power in capturing long-range dependences among variables in the system. To
alleviate this problem, we propose to introduce long-range interactions, which
act as shortcuts, to MPS, resulting in a new model \textit{ Shortcut Matrix
Product States} (SMPS). When chosen properly, the shortcuts can decrease
significantly the correlation length of the MPS, while preserving the
computational efficiency. We develop efficient training methods of SMPS for
various tasks, establish some of their mathematical properties, and show how to
find a good location to add shortcuts. Finally, using extensive numerical
experiments we evaluate its performance in a variety of applications, including
function fitting, partition function calculation of d Ising model, and
unsupervised generative modeling of handwritten digits, to illustrate its
advantages over vanilla matrix product states.Comment: 15pages, 11 figure
Global regularity for the 2D Oldroyd-B model in the corotational case
This paper is dedicated to the Oldroyd-B model with fractional dissipation
for any . We establish the global smooth
solutions to the Oldroyd-B model in the corotational case with arbitrarily
small fractional powers of the Laplacian in two spatial dimensions. The methods
described here are quite different from the tedious iterative approach used in
recent paper \cite{XY}. Moreover, in the Appendix we provide some a priori
estimates to the Oldroyd-B model in the critical case which may be useful and
of interest for future improvement. Finally, the global regularity to to the
Oldroyd-B model in the corotational case with replaced by
for are also collected in the Appendix.
Therefore our result is more closer to the resolution of the well-known global
regularity issue on the critical 2D Oldroyd-B model.Comment: 23 pages, Submitted August 201
Global existence and exponential decay of strong solutions for the inhomogeneous incompressible Navier-Stokes equations with vacuum
The inhomogeneous incompressible Navier-Stokes equations with fractional
Laplacian dissipations in the multi-dimensional whole space are considered. The
existence and uniqueness of global strong solution with vacuum are established
for large initial data. The exponential decay-in-time of the strong solution is
also obtained, which is different from the homogeneous case. The initial
density may have vacuum and even compact support.Comment: 31 page
On the Differentiability issue of the drift-diffusion equation with nonlocal L\'evy-type diffusion
We investigate the differentiability issue of the drift-diffusion equation
with nonlocal L\'evy-type diffusion at either supercritical or critical type
cases. Under the suitable conditions on the drift velocity and the forcing term
in terms of the spatial H\"older regularity, we prove that the vanishing
viscosity solution is differentiable with some H\"older continuous derivatives
for any positive time.Comment: 24 pages. Submitte
Stabilizing Weighted Graphs
An edge-weighted graph is called stable if the value of a
maximum-weight matching equals the value of a maximum-weight fractional
matching. Stable graphs play an important role in some interesting game theory
problems, such as network bargaining games and cooperative matching games,
because they characterize instances which admit stable outcomes. Motivated by
this, in the last few years many researchers have investigated the algorithmic
problem of turning a given graph into a stable one, via edge- and
vertex-removal operations. However, all the algorithmic results developed in
the literature so far only hold for unweighted instances, i.e., assuming unit
weights on the edges of .
We give the first polynomial-time algorithm to find a minimum cardinality
subset of vertices whose removal from yields a stable graph, for any
weighted graph . The algorithm is combinatorial and exploits new structural
properties of basic fractional matchings, which are of independent interest. In
particular, one of the main ingredients of our result is the development of a
polynomial-time algorithm to compute a basic maximum-weight fractional matching
with minimum number of odd cycles in its support. This generalizes a
fundamental and classical result on unweighted matchings given by Balas more
than 30 years ago, which we expect to prove useful beyond this particular
application.
In contrast, we show that the problem of finding a minimum cardinality subset
of edges whose removal from a weighted graph yields a stable graph, does
not admit any constant-factor approximation algorithm, unless . In this
setting, we develop an -approximation algorithm for the problem,
where is the maximum degree of a node in
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