PROC

The primitive equations (PEs) of large-scale oceanic flow formulated in
mean vorticity is proposed. In the reformulation of the PEs, the prognostic equation
for the horizontal velocity is replaced by evolutionary equations for the mean vorticity
field and the vertical derivative of the horizontal velocity. The total velocity field
(both horizontal and vertical) is statically determined by differential equations at each
fixed horizontal point. Its equivalence to the original formulation is also presented.

DCDS-B

A second order numerical method for the primitive equations
(PEs) of large-scale oceanic flow formulated in mean vorticity is proposed and
analyzed, and the full convergence in $L^2$ is established. In the reformulation of
the PEs, the prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative of
the horizontal velocity. The total velocity field (both horizontal and vertical)
is statically determined by differential equations at each fixed horizontal point.
The standard centered difference approximation is applied to the prognostic
equations and the determination of numerical values for the total velocity field
is implemented by FFT-based solvers. Stability of such solvers are established
and the convergence analysis for the whole scheme is provided in detail.

DCDS-B

A class of upwind flux splitting methods in the
Euler equations of compressible flow
is considered in this paper. Using the property that Euler flux $F(U)$
is a homogeneous function of degree one in $U$,
we reformulate the splitting fluxes with $F^{+}=A^{+} U$,
$F^{-}=A^{-} U$, and the corresponding matrices
are either symmetric or symmetrizable and keep only
non-negative and non-positive eigenvalues.
That leads to the conclusion that the first order schemes
are positive in the sense of Lax-Liu [18],
which implies that it is $L^2$-stable
in some suitable sense. Moreover, the second order scheme
is a stable perturbation of the first order scheme,
so that the positivity of the second order schemes
is also established, under
a CFL-like condition. In addition, these splitting methods preserve
the positivity of density and energy.

DCDS

We present unconditionally stable and convergent numerical sche- mes for gradient flows with energy of the form $ \int_\Omega( F(\nabla\phi(\x)) + \frac{\epsilon^2}{2}|\Delta\phi(\x)|^2 )$dx. The construction of the schemes involves an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. As an application, we derive unconditionally stable and convergent schemes for epitaxial film growth models with slope selection (F(y)= 1/4(|y|^{2}-1)^{2}) and without slope selection (F(y)= -1/2ln(1+|y|^{2})). We conclude the paper with some preliminary computations that employ the proposed schemes.

DCDS-B

In this paper, we prove the existence and uniqueness of a Gevrey regularity solution for a class of nonlinear bistable gradient flows, where with the energy may be decomposed into purely convex and concave parts. Example equations include certain epitaxial thin film growth models and phase field crystal models. The energy dissipation law implies a bound in the leading Sobolev norm. The polynomial structure of the nonlinear terms in the chemical potential enables us to derive a local-in-time solution with Gevrey regularity, with the existence time interval length dependent on a certain $H^m$ norm of the initial data. A detailed Sobolev estimate for the gradient equations results in a uniform-in-time-bound of that $H^m$ norm, which in turn establishes the existence of a global-in-time solution with Gevrey regularity.

IPI

Due to the restriction of the scanning environment and the energy of X-ray, few projections of an object can be obtained in some practical applications of computed tomography (CT).
In these situations, the projection data are incomplete and inconsistent, and the conventional analytic algorithm such as filtered backprojection (FBP) algorithm will not work.
The streak artifacts can be significantly reduced in few-view reconstruction if the total variation minimization (TVM) based CT reconstruction algorithm is used.
However, in the premise of preserving the resolution of image, it will not effectively suppress slope artifacts and metal artifacts when dealing with some few-view of the limited-angle reconstruction problems.
To solve this problem, we focus on the image reconstruction algorithm base on $\ell_{0}$ regularized of wavelet coefficients.
In this paper, the error bound between the reference or desire image and the reconstructed result, and the stability of solution were shown in theoretical and experimental, a reconstruction experiment on metal laths from few-view of the limited-angle projections was given. The experimental results indicate that this algorithm outperforms classical CT reconstruction algorithms in preserving the resolution of reconstructed image and suppressing the metal artifacts.

DCDS-B

A new generalization of the Poincaré-Birkhoff fixed point
theorem applying to small perturbations of finite-dimensional,
completely integrable Hamiltonian systems is formulated and
proved. The motivation for this theorem is an extension of some
recent results of Blackmore and Knio on the dynamics of three
coaxial vortex rings in an ideal fluid. In particular, it is
proved using KAM theory and this new fixed point theorem that if
$n>3$ coaxial rings all having vortex strengths of the same sign
are initially in certain positions sufficiently close to one
another in a three-dimensional ideal fluid environment, their
motion with respect to the center of vorticity exhibits invariant
$(n-1)$-dimensional tori comprised of quasiperiodic orbits
together with interspersed periodic trajectories.

CPAA

In this paper, we explore the relations between different kinds of Strichartz estimates
and give new estimates in Euclidean space $\mathbb{R}^n$. In particular, we prove the generalized and weighted Strichartz estimates for a large class of dispersive operators including the Schrödinger and wave equation. As a sample application of these new estimates,
we are able to prove the Strauss conjecture with low regularity for dimension $2$ and $3$.

IPI

The limited-angle projection data of an object, in some practical applications of computed tomography (CT), are obtained due to the restriction of scanning condition. In these situations, since the projection data are incomplete, some limited-angle artifacts will be presented near the edges of reconstructed image using some classical reconstruction algorithms, such as filtered backprojection (FBP). The reconstructed image can be fine approximated by sparse coefficients under a proper wavelet tight frame, and the quality of reconstructed image can be improved by an available prior image. To deal with limited-angle CT reconstruction problem, we propose a minimization model that is based on wavelet tight frame and a prior image, and perform this minimization problem efficiently by iteratively minimizing separately. Moreover, we show that each bounded sequence, which is generated by our method, converges to a critical or a stationary point. The experimental results indicate that our algorithm can efficiently suppress artifacts and noise and preserve the edges of reconstructed image, what's more, the introduced prior image will not miss the important information that is not included in the prior image.