Propagation of chaos for the Keller-Segel equation over bounded domains

In this paper we rigorously justify the propagation of chaos for the parabolic-elliptic Keller-Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller-Segel equation for bounded weak solutions

A unified structure preserving scheme for a multi-species model with a gradient flow structure and nonlocal interactions via singular kernels

In this paper, we consider a nonlinear and nonlocal parabolic model for multi-species ionic fluids and introduce a semi-implicit finite volume scheme, which is second order accurate in space, first order in time and satisfies the following properties: positivity preserving, mass conservation and energy dissipation. Besides, our scheme involves a fast algorithm on the convolution terms with singular but integrable kernels, which otherwise impedes the accuracy and efficiency of the whole scheme. Error estimates on the fast convolution algorithm are shown next. Numerous numerical tests are provided to demonstrate the properties, such as unconditional stability, order of convergence, energy dissipation and the complexity of the fast convolution algorithm. Furthermore, extensive numerical experiments are carried out to explore the modeling effects in specific examples, such as, the steric repulsion, the concentration of ions at the boundary and the blowup phenomenon of the Keller-Segel equations.Comment: 27 pages, 17 figure

Global existence and spatial analyticity for a nonlocal flux with fractional diffusion

In this paper, we study a one dimensional nonlinear equation with diffusion $-\nu(-\partial_{xx})^{\frac{\alpha}{2}}$ for $0\leq \alpha\leq 2$ and $\nu>0$. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space $L^1(\mathbb{R})\cap H^{1/2}(\mathbb{R})$ when $0\leq\alpha\leq 2$. For subcritical $1<\alpha\leq 2$ and critical case $\alpha=1$, we obtain global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrap method to improve the regularity of mild solutions in Bessel potential spaces for subcritical case $1<\alpha\leq 2$. Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case $\alpha=1$, if the initial data $\rho_0$ satisfies $-\nu<\inf\rho_0<0$, we use the characteristics methods for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If $\rho_0\geq0$, the solution exists globally and converges to steady state.Comment: Replace of previous version arXiv:2008.08860v