On uniform estimate of complex elliptic equations on closed Hermitian manifolds
Wei Sun

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.

keywords: Uniform estimate complex elliptic equation Hermitian manifold
Traveling fronts of curve flow with external force field
Huaiyu Jian Hongjie Ju Wei Sun
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.
keywords: nonlinear ODE traveling front Curve flow phase analysis.
Stabilized finite element method for the non-stationary Navier-Stokes problem
Yinnian He Yanping Lin Weiwei Sun
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.
keywords: stabilized finite element uniform error estimate. Navier-Stokes problem
Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems
Tao Lin Yanping Lin Weiwei Sun
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.
keywords: interface error estimates. immersed finite element method Elliptic jump condition
Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels
C. David Levermore Weiran Sun
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.
keywords: Compactness Fredholm property linearized Boltzmann operator.
A relaxation method for one dimensional traveling waves of singular and nonlocal equations
Weiran Sun Min Tang
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 $\mathbb{R}$.
keywords: Reaction-diffusion equations traveling wave numerical simulation.
Fractional diffusion limits of non-classical transport equations
Martin Frank Weiran Sun

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.

keywords: Fractional diffusion limit non-classical transport

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