Weiran Sun

On a singular incompressible porous media equation

This paper considers a family of active scalar equations with transport velocities which are more singular by a derivative of order β than the active scalar. We prove that the equations with 0 < β ⩽ 2 are Lipschitz ill-posed for regular initial data. On the contrary, when 0 < β < 1 we show local well-posedness for patch-type weak solutions.

A Low Mach Number Limit of a Dispersive Navier–Stokes System

We establish a low Mach number limit for classical solutions over the whole space of a compressible fluid dynamic system that includes dispersive corrections to the Navier–Stokes equations. The limiting system is similar to a ghost effect system [Y. Sone, Kinetic Theory and Fluid Dynamics, Model. Simul. Sci. Eng. Technol., Birkhauser, Boston, 2002]. Our analysis builds upon the framework developed by Metivier and Schochet [Arch. Ration. Mech. Anal., 158 (2001), pp. 61–90] and Alazard [Arch. Ration. Mech. Anal., 180 (2006), pp. 1–73] for nondispersive systems. The strategy involves establishing a priori estimates for the slow motion as well as a priori estimates for the fast motion. The desired convergence is obtained by establishing the local decay of the energy of the fast motion.

Half-space Kinetic Equations with General Boundary Conditions

We study half-space linear kinetic equations with general boundary conditions that consist of both given incoming data and various type of reflections, extending our previous work [LLS14] on half-space equations with incoming boundary conditions. As in [LLS14], the main technique is a damping adding-removing procedure. We establish the well-posedness of linear (or linearized) half-space equations with general boundary conditions and quasi-optimality of the numerical scheme. The numerical method is validated by examples including a two-species transport equation, a multi-frequency transport equation, and the linearized BGK equation in 2D velocity space.

First-order aggregation models and zero inertia limits

Abstract We consider a first-order aggregation model in both discrete and continuum formulations and show rigorously how it can be obtained as zero inertia limits of second-order models. In the continuum case the procedure consists in a macroscopic limit, enabling the passage from a kinetic model for aggregation to an evolution equation for the macroscopic density. We work within the general space of measure solutions and use mass transportation ideas and the characteristic method as essential tools in the analysis.

Diffusion approximations and domain decomposition method of linear transport equations

In this paper we construct numerical schemes to approximate linear transport equations with slab geometry by diffusion equations. We treat both the case of pure diffusive scaling and the case where kinetic and diffusive scalings coexist. The diffusion equations and their data are derived from asymptotic and layer analysis which allows general scattering kernels and general data. We apply the half-space solver in 20] to resolve the boundary layer equation and obtain the boundary data for the diffusion equation. The algorithms are validated by numerical experiments and also by error analysis for the pure diffusive scaling case.

Validity and Regularization of Classical Half-Space Equations

Recent result (Wu and Guo in Commun Math Phys 336(3):1473–1553, 2015) has shown that over the 2D unit disk, the classical half-space equation (CHS) for the neutron transport does not capture the correct boundary layer behaviour as long believed. In this paper we develop a regularization technique for CHS to any arbitrary order and use its first-order regularization to show that in the case of the 2D unit disk, although CHS misrepresents the boundary layer behaviour, it does give the correct boundary condition for the interior macroscopic (Laplace) equation. Therefore CHS is still a valid equation to recover the correct boundary condition for the interior Laplace equation over the 2D unit disk.

First-order aggregation models with alignment

Abstract We include alignment interactions in a well-studied first-order attractive–repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the population density, becomes implicit, and can have non-unique solutions. We investigate the well-posedness of the model and show rigorously how it can be obtained as a macroscopic limit of a second-order kinetic equation. We work within the space of probability measures with compact support and use mass transportation ideas and the characteristic method as essential tools in the analysis. A discretization procedure that parallels the analysis is formulated and implemented numerically in one and two dimensions.

Macroscopic Limits of Pathway-Based Kinetic Models for E. coli Chemotaxis in Large Gradient Environments

It is of great biological interest to understand the molecular origins of chemotactic behavior of E. coli by developing population-level models based on the underlying signaling pathway dynamics. We derive macroscopic models for E.coli chemotaxis that match quantitatively with the agent-based model (SPECS) for all ranges of the spacial gradient, in particular when the chemical gradient is large such that the standard Keller-Segel model is no longer valid. These equations are derived both formally and rigorously as asymptotic limits for pathway-based kinetic equations. We also present numerical results that show good agreement between the macroscopic models and SPECS. Our work provides an answer to the question of how to determine the population-level diffusion coefficient and drift velocity from the molecular mechanisms of chemotaxis, for both shallow gradients and large gradients environments.

SMALL INERTIA REGULARIZATION OF AN ANISOTROPIC AGGREGATION MODEL

We consider an anisotropic first-order ODE aggregation model and its approximation by a second-order relaxation system. The relaxation model contains a small parameter

The fractional diffusion limit of a kinetic model with biochemical pathway

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