CPAA

In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed Kähler manifolds.

CPAA

This paper aims to classify all the *traveling
fronts* of a curvature flow with external force fields in the
two-dimensional Euclidean space, i.e., the curve is evolved by the
sum of the curvature
and an external force field. We show that any *traveling front*
is either a line or Grim Reaper if the external force field is constant. However, we find
that the
*traveling fronts* are of completely different geometry for
non-constant external force fields.

DCDS-B

In this article, a locally stabilized finite element formulation
of the two-dimensional Navier-Stokes problem is used.
A macroelement condition which provides the stability
of the $Q_1-P_0$ quadrilateral element and
the $P_1-P_0$ triangular element is introduced.
Moreover, the $H^1$ and $L^2$-error estimates of optimal order
for finite element
solution $(u_h,p_h)$ are analyzed.
Finally, a uniform $H^1$ and $L^2$-error estimates of optimal order for finite
element solution $(u_h,p_h)$ is obtained
if the uniqueness condition is satisfied.

DCDS-B

We carry out error estimation of a class of immersed finite element
(IFE) methods for elliptic interface problems with both perfect and
imperfect interface jump conditions. A key feature of these methods
is that their partitions can be independent of the location of the
interface. These quadratic IFE spaces reduce to the standard
quadratic finite element space when the interface is not in the
interior of any element. More importantly, we demonstrate that these
IFE spaces have the optimal (slightly lower order in one case)
approximation capability expected from a finite element space using
quadratic polynomials.

KRM

We prove an $L^p$ compactness result for the gain parts of the
linearized Boltzmann collision operator associated with weakly cutoff
collision kernels that derive from a power-law intermolecular
potential. We replace the Grad cutoff assumption previously made by
Caflisch [1], Golse and Poupaud [7], and Guo [11] with a weaker local integrability assumption. This class
includes all classical kernels to which the DiPerna-Lions theory
applies that derive from a repulsive inverse-power intermolecular
potential. In particular, our approach allows the treatment of both
hard and soft potential cases.

DCDS-B

Recent models motivated by biological phenomena lead to non-local
PDEs or systems with singularities. It has been recently understood that these systems may have
traveling wave solutions that are not physically relevant
[19].
We present an original method that relies on the physical evolution
to capture the ``stable" traveling waves. This method allows us to
obtain the traveling wave profiles and their traveling speed
simultaneously. It is easy to implement, and it applies to classical
differential equations as well as nonlocal equations and systems
with singularities. We also show the convergence of the scheme
analytically for bistable reaction diffusion equations over the
whole space $\mathbb{R}$.

KRM

We establish asymptotic diffusion limits of the non-classical transport equation derived in [12]. By introducing appropriate scaling parameters, the limits will be either regular or fractional diffusion equations depending on the tail behaviour of the path-length distribution. Our analysis is based on a combination of the Fourier transform and a moment method. We put special focus on dealing with anisotropic scattering, which compared to the isotropic case makes the analysis significantly more involved.