460 research outputs found
On the flag curvature of Finsler metrics of scalar curvature
The flag curvature of a Finsler metric is called a Riemannian quantity
because it is an extension of sectional curvature in Riemannian geometry. In
Finsler geometry, there are several non-Riemannian quantities such as the
(mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature,
which all vanish for Riemannian metrics. It is important to understand the
geometric meanings of these quantities. In this paper, we study Finsler metrics
of scalar curvature (i.e., the flag curvature is a scalar function on the slit
tangent bundle) and partially determine the flag curvature when certain
non-Riemannian quantities are isotropic. Using the obtained formula for the
flag curvature, we classify locally projectively flat Randers metrics with
isotropic S-curvature.Comment: 23 page
A Study of the Navier-Stokes Equations with the Kinematic and Navier Boundary Conditions
We study the initial-boundary value problem of the Navier-Stokes equations
for incompressible fluids in a domain in with compact and smooth
boundary, subject to the kinematic and Navier boundary conditions. We first
reformulate the Navier boundary condition in terms of the vorticity, which is
motivated by the Hodge theory on manifolds with boundary from the viewpoint of
differential geometry, and establish basic elliptic estimates for vector fields
subject to the kinematic and Navier boundary conditions. Then we develop a
spectral theory of the Stokes operator acting on divergence-free vector fields
on a domain with the kinematic and Navier boundary conditions. Finally, we
employ the spectral theory and the necessary estimates to construct the
Galerkin approximate solutions and establish their convergence to global weak
solutions, as well as local strong solutions, of the initial-boundary problem.
Furthermore, we show as a corollary that, when the slip length tends to zero,
the weak solutions constructed converge to a solution to the incompressible
Navier-Stokes equations subject to the no-slip boundary condition for almost
all time. The inviscid limit of the strong solutions to the unique solutions of
the initial-boundary value problem with the slip boundary condition for the
Euler equations is also established.Comment: 30 page
Robust MIMO Channel Estimation from Incomplete and Corrupted Measurements
Location-aware communication is one of the enabling techniques for future 5G networks. It requires accurate temporal and spatial channel estimation from multidimensional data. Most of the existing channel estimation techniques assume that the measurements are complete and noise is Gaussian. While these approaches are brittle to corrupted or outlying measurements, which are ubiquitous in real applications. To address these issues, we develop a lp-norm minimization based iteratively reweighted higher-order singular value decomposition algorithm. It is robust to Gaussian as well as the impulsive noise even when the measurement data is incomplete. Compared with the state-of-the-art techniques, accurate estimation results are achieved for the proposed approach
New Approach for Vorticity Estimates of Solutions of the Navier-Stokes Equations
We develop a new approach for regularity estimates, especially vorticity
estimates, of solutions of the three-dimensional Navier-Stokes equations with
periodic initial data, by exploiting carefully formulated linearized vorticity
equations. An appealing feature of the linearized vorticity equations is the
inheritance of the divergence-free property of solutions, so that it can
intrinsically be employed to construct and estimate solutions of the
Navier-Stokes equations. New regularity estimates of strong solutions of the
three-dimensional Navier-Stokes equations are obtained by deriving new explicit
a priori estimates for the heat kernel (i.e., the fundamental solution) of the
corresponding heterogeneous drift-diffusion operator. These new a priori
estimates are derived by using various functional integral representations of
the heat kernel in terms of the associated diffusion processes and their
conditional laws, including a Bismut-type formula for the gradient of the heat
kernel. Then the a priori estimates of solutions of the linearized vorticity
equations are established by employing a Feynman-Kac-type formula. The
existence of strong solutions and their regularity estimates up to a time
proportional to the reciprocal of the square of the maximum initial vorticity
are established. All the estimates established in this paper contain known
constants that can be explicitly computed.Comment: 27 page
The Navier-Stokes Equations with the Kinematic and Vorticity Boundary Conditions on Non-Flat Boundaries
We study the initial-boundary value problem of the Navier-Stokes equations
for incompressible fluids in a general domain in with compact and smooth
boundary, subject to the kinematic and vorticity boundary conditions on the
non-flat boundary. We observe that, under the nonhomogeneous boundary
conditions, the pressure can be still recovered by solving the Neumann
problem for the Poisson equation. Then we establish the well-posedness of the
unsteady Stokes equations and employ the solution to reduce our
initial-boundary value problem into an initial-boundary value problem with
absolute boundary conditions. Based on this, we first establish the
well-posedness for an appropriate local linearized problem with the absolute
boundary conditions and the initial condition (without the incompressibility
condition), which establishes a velocity mapping. Then we develop
\emph{apriori} estimates for the velocity mapping, especially involving the
Sobolev norm for the time-derivative of the mapping to deal with the
complicated boundary conditions, which leads to the existence of the fixed
point of the mapping and the existence of solutions to our initial-boundary
value problem. Finally, we establish that, when the viscosity coefficient tends
zero, the strong solutions of the initial-boundary value problem in with nonhomogeneous vorticity boundary condition converges in to the
corresponding Euler equations satisfying the kinematic condition.Comment: 31 page
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